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4.4 Test file Number [63] 3-Logarithms/3.4-u-a+b-log-c-d+e-x^m-^n-^p

4.4.1 Mathematica

Integral number [98] \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx \]

[B]   time = 3.753 (sec), size = 909 ,normalized size = 2.4 \[ \frac{\left (-48 \left (4 \sqrt{b x^2} \tanh ^{-1}\left (\frac{\sqrt{b x^2}}{\sqrt{-a}}\right ) \left (\log \left (b x^2+a\right )-\log \left (\frac{b x^2}{a}+1\right )\right )-\sqrt{-a} \sqrt{-\frac{b x^2}{a}} \left (\log ^2\left (\frac{b x^2}{a}+1\right )-4 \log \left (\frac{1}{2} \left (\sqrt{-\frac{b x^2}{a}}+1\right )\right ) \log \left (\frac{b x^2}{a}+1\right )+2 \log ^2\left (\frac{1}{2} \left (\sqrt{-\frac{b x^2}{a}}+1\right )\right )-4 \text{PolyLog}\left (2,\frac{1}{2}-\frac{1}{2} \sqrt{-\frac{b x^2}{a}}\right )\right )\right ) a^2+416 \sqrt{-a} \sqrt{\frac{b x^2}{b x^2+a}} \sqrt{b x^2+a} \sin ^{-1}\left (\frac{\sqrt{a}}{\sqrt{b x^2+a}}\right ) a^{3/2}+36 \sqrt{-a} \sqrt{\frac{b x^2}{b x^2+a}} \left (8 \sqrt{a} \, _4F_3\left (\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{a}{b x^2+a}\right )+\log \left (b x^2+a\right ) \left (4 \sqrt{a} \, _3F_2\left (\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{a}{b x^2+a}\right )+\sqrt{b x^2+a} \sin ^{-1}\left (\frac{\sqrt{a}}{\sqrt{b x^2+a}}\right ) \log \left (b x^2+a\right )\right )\right ) a^{3/2}+\frac{2}{3} \sqrt{-a} b x^2 \left (9 b x^2 \log ^3\left (b x^2+a\right )+18 \left (3 a-b x^2\right ) \log ^2\left (b x^2+a\right )+\left (24 b x^2-288 a\right ) \log \left (b x^2+a\right )-16 b x^2+624 a\right )\right ) p^3}{18 \sqrt{-a} b^2 x}+3 \left (\log \left (c \left (b x^2+a\right )^p\right )-p \log \left (b x^2+a\right )\right ) \left (\frac{1}{3} x^3 \log ^2\left (b x^2+a\right )-\frac{4 \left (9 i a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2+3 a^{3/2} \left (6 \log \left (\frac{2 \sqrt{a}}{i \sqrt{b} x+\sqrt{a}}\right )+3 \log \left (b x^2+a\right )-8\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+\sqrt{b} x \left (-2 b x^2+24 a+\left (3 b x^2-9 a\right ) \log \left (b x^2+a\right )\right )+9 i a^{3/2} \text{PolyLog}\left (2,\frac{\sqrt{b} x+i \sqrt{a}}{\sqrt{b} x-i \sqrt{a}}\right )\right )}{27 b^{3/2}}\right ) p^2+\frac{2 a x \left (\log \left (c \left (b x^2+a\right )^p\right )-p \log \left (b x^2+a\right )\right )^2 p}{b}-\frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (\log \left (c \left (b x^2+a\right )^p\right )-p \log \left (b x^2+a\right )\right )^2 p}{b^{3/2}}+x^3 \log \left (b x^2+a\right ) \left (\log \left (c \left (b x^2+a\right )^p\right )-p \log \left (b x^2+a\right )\right )^2 p+\frac{1}{3} x^3 \left (\log \left (c \left (b x^2+a\right )^p\right )-p \log \left (b x^2+a\right )\right )^2 \left (-\log \left (b x^2+a\right ) p-2 p+\log \left (c \left (b x^2+a\right )^p\right )\right ) \]

[In]

Integrate[x^2*Log[c*(a + b*x^2)^p]^3,x]

[Out]

(2*a*p*x*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2)/b - (2*a^(3/2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-(p*Log
[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2)/b^(3/2) + p*x^3*Log[a + b*x^2]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2
)^p])^2 + (x^3*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2*(-2*p - p*Log[a + b*x^2] + Log[c*(a + b*x^2)^p])
)/3 + 3*p^2*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])*((x^3*Log[a + b*x^2]^2)/3 - (4*((9*I)*a^(3/2)*ArcTan[
(Sqrt[b]*x)/Sqrt[a]]^2 + 3*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-8 + 6*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)
] + 3*Log[a + b*x^2]) + Sqrt[b]*x*(24*a - 2*b*x^2 + (-9*a + 3*b*x^2)*Log[a + b*x^2]) + (9*I)*a^(3/2)*PolyLog[2
, (I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b]*x)]))/(27*b^(3/2))) + (p^3*(416*Sqrt[-a]*a^(3/2)*Sqrt[(b*x^2
)/(a + b*x^2)]*Sqrt[a + b*x^2]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]] + (2*Sqrt[-a]*b*x^2*(624*a - 16*b*x^2 + (-288*a
 + 24*b*x^2)*Log[a + b*x^2] + 18*(3*a - b*x^2)*Log[a + b*x^2]^2 + 9*b*x^2*Log[a + b*x^2]^3))/3 + 36*Sqrt[-a]*a
^(3/2)*Sqrt[(b*x^2)/(a + b*x^2)]*(8*Sqrt[a]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, a/(a + b*
x^2)] + Log[a + b*x^2]*(4*Sqrt[a]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, a/(a + b*x^2)] + Sqrt[a + b*x
^2]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]]*Log[a + b*x^2])) - 48*a^2*(4*Sqrt[b*x^2]*ArcTanh[Sqrt[b*x^2]/Sqrt[-a]]*(Lo
g[a + b*x^2] - Log[1 + (b*x^2)/a]) - Sqrt[-a]*Sqrt[-((b*x^2)/a)]*(Log[1 + (b*x^2)/a]^2 - 4*Log[1 + (b*x^2)/a]*
Log[(1 + Sqrt[-((b*x^2)/a)])/2] + 2*Log[(1 + Sqrt[-((b*x^2)/a)])/2]^2 - 4*PolyLog[2, 1/2 - Sqrt[-((b*x^2)/a)]/
2]))))/(18*Sqrt[-a]*b^2*x)

Integral number [99] \[ \int \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx \]

[B]   time = 3.35596 (sec), size = 789 ,normalized size = 2.73 \[ \frac{p^3 \left (-6 \sqrt{-a^2} \sqrt{\frac{b x^2}{a+b x^2}} \left (8 \sqrt{a} \, _4F_3\left (\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{a}{b x^2+a}\right )+\log \left (a+b x^2\right ) \left (4 \sqrt{a} \, _3F_2\left (\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{a}{b x^2+a}\right )+\sqrt{a+b x^2} \log \left (a+b x^2\right ) \sin ^{-1}\left (\frac{\sqrt{a}}{\sqrt{a+b x^2}}\right )\right )\right )+6 (-a)^{3/2} \sqrt{-\frac{b x^2}{a}} \left (-4 \text{PolyLog}\left (2,\frac{1}{2}-\frac{1}{2} \sqrt{-\frac{b x^2}{a}}\right )+\log ^2\left (\frac{b x^2}{a}+1\right )+2 \log ^2\left (\frac{1}{2} \left (\sqrt{-\frac{b x^2}{a}}+1\right )\right )-4 \log \left (\frac{1}{2} \left (\sqrt{-\frac{b x^2}{a}}+1\right )\right ) \log \left (\frac{b x^2}{a}+1\right )\right )-48 \sqrt{-a^2} \sqrt{\frac{b x^2}{a+b x^2}} \sqrt{a+b x^2} \sin ^{-1}\left (\frac{\sqrt{a}}{\sqrt{a+b x^2}}\right )+\sqrt{-a} b x^2 \left (\log ^3\left (a+b x^2\right )-6 \log ^2\left (a+b x^2\right )+24 \log \left (a+b x^2\right )-48\right )+24 a \sqrt{b x^2} \left (\log \left (a+b x^2\right )-\log \left (\frac{b x^2}{a}+1\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b x^2}}{\sqrt{-a}}\right )\right )}{\sqrt{-a} b x}-\frac{3 p^2 \left (p \log \left (a+b x^2\right )-\log \left (c \left (a+b x^2\right )^p\right )\right ) \left (4 i \sqrt{a} \text{PolyLog}\left (2,\frac{\sqrt{b} x+i \sqrt{a}}{\sqrt{b} x-i \sqrt{a}}\right )+\sqrt{b} x \left (\log ^2\left (a+b x^2\right )-4 \log \left (a+b x^2\right )+8\right )+4 \sqrt{a} \left (\log \left (a+b x^2\right )+2 \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )-2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+4 i \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2\right )}{\sqrt{b}}+3 p x \log \left (a+b x^2\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^2+x \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^2 \left (\log \left (c \left (a+b x^2\right )^p\right )+p \left (-\log \left (a+b x^2\right )\right )-6 p\right )+\frac{6 \sqrt{a} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^2}{\sqrt{b}} \]

[In]

Integrate[Log[c*(a + b*x^2)^p]^3,x]

[Out]

(6*Sqrt[a]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2)/Sqrt[b] + 3*p*x*Log[a
 + b*x^2]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2 + x*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2*(-
6*p - p*Log[a + b*x^2] + Log[c*(a + b*x^2)^p]) - (3*p^2*(p*Log[a + b*x^2] - Log[c*(a + b*x^2)^p])*((4*I)*Sqrt[
a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2 + 4*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-2 + 2*Log[(2*Sqrt[a])/(Sqrt[a] + I*
Sqrt[b]*x)] + Log[a + b*x^2]) + Sqrt[b]*x*(8 - 4*Log[a + b*x^2] + Log[a + b*x^2]^2) + (4*I)*Sqrt[a]*PolyLog[2,
 (I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b]*x)]))/Sqrt[b] + (p^3*(-48*Sqrt[-a^2]*Sqrt[(b*x^2)/(a + b*x^2)
]*Sqrt[a + b*x^2]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]] + Sqrt[-a]*b*x^2*(-48 + 24*Log[a + b*x^2] - 6*Log[a + b*x^2]
^2 + Log[a + b*x^2]^3) - 6*Sqrt[-a^2]*Sqrt[(b*x^2)/(a + b*x^2)]*(8*Sqrt[a]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1
/2}, {3/2, 3/2, 3/2}, a/(a + b*x^2)] + Log[a + b*x^2]*(4*Sqrt[a]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}
, a/(a + b*x^2)] + Sqrt[a + b*x^2]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]]*Log[a + b*x^2])) + 24*a*Sqrt[b*x^2]*ArcTanh
[Sqrt[b*x^2]/Sqrt[-a]]*(Log[a + b*x^2] - Log[1 + (b*x^2)/a]) + 6*(-a)^(3/2)*Sqrt[-((b*x^2)/a)]*(Log[1 + (b*x^2
)/a]^2 - 4*Log[1 + (b*x^2)/a]*Log[(1 + Sqrt[-((b*x^2)/a)])/2] + 2*Log[(1 + Sqrt[-((b*x^2)/a)])/2]^2 - 4*PolyLo
g[2, 1/2 - Sqrt[-((b*x^2)/a)]/2])))/(Sqrt[-a]*b*x)

Integral number [100] \[ \int \frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx \]

[C]   time = 0.781377 (sec), size = 505 ,normalized size = 10.1 \[ \frac{p^3 \left (-96 \sqrt{a} \sqrt{1-\frac{a}{a+b x^2}} \, _4F_3\left (\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{a}{b x^2+a}\right )-48 \sqrt{a} \sqrt{1-\frac{a}{a+b x^2}} \log \left (a+b x^2\right ) \, _3F_2\left (\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{a}{b x^2+a}\right )-2 \log ^2\left (a+b x^2\right ) \left (\sqrt{a} \log \left (a+b x^2\right )+6 \sqrt{a+b x^2} \sqrt{1-\frac{a}{a+b x^2}} \sin ^{-1}\left (\frac{\sqrt{a}}{\sqrt{a+b x^2}}\right )\right )\right )}{2 \sqrt{a} x}+3 p^2 \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right ) \left (-\frac{\log ^2\left (a+b x^2\right )}{x}+\frac{4 \sqrt{b} \left (i \text{PolyLog}\left (2,\frac{\sqrt{b} x+i \sqrt{a}}{\sqrt{b} x-i \sqrt{a}}\right )+\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (\log \left (a+b x^2\right )+2 \log \left (\frac{2 i}{-\frac{\sqrt{b} x}{\sqrt{a}}+i}\right )+i \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right )\right )}{\sqrt{a}}\right )-\frac{3 p \log \left (a+b x^2\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^2}{x}-\frac{\left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^3}{x}+\frac{6 \sqrt{b} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^2}{\sqrt{a}} \]

[In]

Integrate[Log[c*(a + b*x^2)^p]^3/x^2,x]

[Out]

(p^3*(-96*Sqrt[a]*Sqrt[1 - a/(a + b*x^2)]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, a/(a + b*x^
2)] - 48*Sqrt[a]*Sqrt[1 - a/(a + b*x^2)]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, a/(a + b*x^2)]*Log[a +
 b*x^2] - 2*Log[a + b*x^2]^2*(6*Sqrt[a + b*x^2]*Sqrt[1 - a/(a + b*x^2)]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]] + Sqrt
[a]*Log[a + b*x^2])))/(2*Sqrt[a]*x) + (6*Sqrt[b]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-(p*Log[a + b*x^2]) + Log[c*(a
 + b*x^2)^p])^2)/Sqrt[a] - (3*p*Log[a + b*x^2]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2)/x - (-(p*Log[a
+ b*x^2]) + Log[c*(a + b*x^2)^p])^3/x + 3*p^2*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])*(-(Log[a + b*x^2]^2
/x) + (4*Sqrt[b]*(ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(I*ArcTan[(Sqrt[b]*x)/Sqrt[a]] + 2*Log[(2*I)/(I - (Sqrt[b]*x)/Sq
rt[a])] + Log[a + b*x^2]) + I*PolyLog[2, (I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b]*x)]))/Sqrt[a])

Integral number [101] \[ \int \frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx \]

[B]   time = 2.62218 (sec), size = 851 ,normalized size = 3.36 \[ \frac{\left (-a^2 \log ^3\left (b x^2+a\right )-6 a b x^2 \log ^2\left (b x^2+a\right )+6 \sqrt{a} \left (\frac{b x^2}{b x^2+a}\right )^{3/2} \left (b x^2+a\right )^{3/2} \sin ^{-1}\left (\frac{\sqrt{a}}{\sqrt{b x^2+a}}\right ) \log ^2\left (b x^2+a\right )+24 \sqrt{-a} \left (b x^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b x^2}}{\sqrt{-a}}\right ) \log \left (b x^2+a\right )+24 a b x^2 \sqrt{\frac{b x^2}{b x^2+a}} \, _3F_2\left (\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{a}{b x^2+a}\right ) \log \left (b x^2+a\right )-6 a^2 \left (-\frac{b x^2}{a}\right )^{3/2} \log ^2\left (\frac{b x^2}{a}+1\right )-12 a^2 \left (-\frac{b x^2}{a}\right )^{3/2} \log ^2\left (\frac{1}{2} \left (\sqrt{-\frac{b x^2}{a}}+1\right )\right )+48 a b x^2 \sqrt{\frac{b x^2}{b x^2+a}} \, _4F_3\left (\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{a}{b x^2+a}\right )-24 \sqrt{-a} \left (b x^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b x^2}}{\sqrt{-a}}\right ) \log \left (\frac{b x^2}{a}+1\right )+24 a^2 \left (-\frac{b x^2}{a}\right )^{3/2} \log \left (\frac{b x^2}{a}+1\right ) \log \left (\frac{1}{2} \left (\sqrt{-\frac{b x^2}{a}}+1\right )\right )+24 a^2 \left (-\frac{b x^2}{a}\right )^{3/2} \text{PolyLog}\left (2,\frac{1}{2}-\frac{1}{2} \sqrt{-\frac{b x^2}{a}}\right )\right ) p^3+3 \sqrt{a} \left (p \log \left (b x^2+a\right )-\log \left (c \left (b x^2+a\right )^p\right )\right ) \left (4 b \left (i \sqrt{b} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2+\sqrt{b} x \left (2 \log \left (\frac{2 \sqrt{a}}{i \sqrt{b} x+\sqrt{a}}\right )+\log \left (b x^2+a\right )-2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+\sqrt{a} \log \left (b x^2+a\right )+i \sqrt{b} x \text{PolyLog}\left (2,\frac{\sqrt{b} x+i \sqrt{a}}{\sqrt{b} x-i \sqrt{a}}\right )\right ) x^2+a^{3/2} \log ^2\left (b x^2+a\right )\right ) p^2-6 a b x^2 \left (\log \left (c \left (b x^2+a\right )^p\right )-p \log \left (b x^2+a\right )\right )^2 p-6 \sqrt{a} b^{3/2} x^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (\log \left (c \left (b x^2+a\right )^p\right )-p \log \left (b x^2+a\right )\right )^2 p-3 a^2 \log \left (b x^2+a\right ) \left (\log \left (c \left (b x^2+a\right )^p\right )-p \log \left (b x^2+a\right )\right )^2 p+a^2 \left (p \log \left (b x^2+a\right )-\log \left (c \left (b x^2+a\right )^p\right )\right )^3}{3 a^2 x^3} \]

[In]

Integrate[Log[c*(a + b*x^2)^p]^3/x^4,x]

[Out]

(a^2*(p*Log[a + b*x^2] - Log[c*(a + b*x^2)^p])^3 - 6*a*b*p*x^2*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2
- 6*Sqrt[a]*b^(3/2)*p*x^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2 - 3*a^2*p
*Log[a + b*x^2]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2 + 3*Sqrt[a]*p^2*(p*Log[a + b*x^2] - Log[c*(a +
b*x^2)^p])*(a^(3/2)*Log[a + b*x^2]^2 + 4*b*x^2*(I*Sqrt[b]*x*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2 + Sqrt[a]*Log[a + b*
x^2] + Sqrt[b]*x*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-2 + 2*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)] + Log[a + b*x^2]
) + I*Sqrt[b]*x*PolyLog[2, (I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b]*x)])) + p^3*(48*a*b*x^2*Sqrt[(b*x^2
)/(a + b*x^2)]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, a/(a + b*x^2)] + 24*Sqrt[-a]*(b*x^2)^(
3/2)*ArcTanh[Sqrt[b*x^2]/Sqrt[-a]]*Log[a + b*x^2] + 24*a*b*x^2*Sqrt[(b*x^2)/(a + b*x^2)]*HypergeometricPFQ[{1/
2, 1/2, 1/2}, {3/2, 3/2}, a/(a + b*x^2)]*Log[a + b*x^2] - 6*a*b*x^2*Log[a + b*x^2]^2 + 6*Sqrt[a]*((b*x^2)/(a +
 b*x^2))^(3/2)*(a + b*x^2)^(3/2)*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]]*Log[a + b*x^2]^2 - a^2*Log[a + b*x^2]^3 - 24*
Sqrt[-a]*(b*x^2)^(3/2)*ArcTanh[Sqrt[b*x^2]/Sqrt[-a]]*Log[1 + (b*x^2)/a] - 6*a^2*(-((b*x^2)/a))^(3/2)*Log[1 + (
b*x^2)/a]^2 + 24*a^2*(-((b*x^2)/a))^(3/2)*Log[1 + (b*x^2)/a]*Log[(1 + Sqrt[-((b*x^2)/a)])/2] - 12*a^2*(-((b*x^
2)/a))^(3/2)*Log[(1 + Sqrt[-((b*x^2)/a)])/2]^2 + 24*a^2*(-((b*x^2)/a))^(3/2)*PolyLog[2, 1/2 - Sqrt[-((b*x^2)/a
)]/2]))/(3*a^2*x^3)

Integral number [158] \[ \int (f x)^m \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]

[B]   time = 2.27135 (sec), size = 994 ,normalized size = 13.08 \[ \frac{(f x)^m \left (\frac{6 p^3 \left (d \left (\left (-\frac{e x^2}{d}\right )^{\frac{m+1}{2}}-1\right ) \log ^2\left (e x^2+d\right )+(m+1) \left (e x^2+d\right ) \, _3F_2\left (1,1,\frac{1}{2}-\frac{m}{2};2,2;\frac{e x^2}{d}+1\right ) \log \left (e x^2+d\right )-(m+1) \left (e x^2+d\right ) \, _4F_3\left (1,1,1,\frac{1}{2}-\frac{m}{2};2,2,2;\frac{e x^2}{d}+1\right )\right ) \left (-\frac{e x^2}{d}\right )^{\frac{1}{2}-\frac{m}{2}}}{e}-\frac{3 m p^2 \left (d \left (\left (-\frac{e x^2}{d}\right )^{\frac{m+1}{2}}-1\right ) \log ^2\left (e x^2+d\right )+(m+1) \left (e x^2+d\right ) \, _3F_2\left (1,1,\frac{1}{2}-\frac{m}{2};2,2;\frac{e x^2}{d}+1\right ) \log \left (e x^2+d\right )-(m+1) \left (e x^2+d\right ) \, _4F_3\left (1,1,1,\frac{1}{2}-\frac{m}{2};2,2,2;\frac{e x^2}{d}+1\right )\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right ) \left (-\frac{e x^2}{d}\right )^{\frac{1}{2}-\frac{m}{2}}}{e}-\frac{3 p^2 \left (d \left (\left (-\frac{e x^2}{d}\right )^{\frac{m+1}{2}}-1\right ) \log ^2\left (e x^2+d\right )+(m+1) \left (e x^2+d\right ) \, _3F_2\left (1,1,\frac{1}{2}-\frac{m}{2};2,2;\frac{e x^2}{d}+1\right ) \log \left (e x^2+d\right )-(m+1) \left (e x^2+d\right ) \, _4F_3\left (1,1,1,\frac{1}{2}-\frac{m}{2};2,2,2;\frac{e x^2}{d}+1\right )\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right ) \left (-\frac{e x^2}{d}\right )^{\frac{1}{2}-\frac{m}{2}}}{e}+(m+1) p^3 x^2 \log ^3\left (e x^2+d\right )+m x^2 \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )^3+x^2 \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )^3+\frac{3 m p x^2 \left (d (m+3) \log \left (e x^2+d\right )-2 e x^2 \, _2F_1\left (1,\frac{m+3}{2};\frac{m+5}{2};-\frac{e x^2}{d}\right )\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )^2}{d (m+3)}+\frac{3 p x^2 \left (d (m+3) \log \left (e x^2+d\right )-2 e x^2 \, _2F_1\left (1,\frac{m+3}{2};\frac{m+5}{2};-\frac{e x^2}{d}\right )\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )^2}{d (m+3)}+\frac{6 d (m+1) p^3 \left (\frac{e x^2}{e x^2+d}\right )^{\frac{1}{2}-\frac{m}{2}} \left (8 \, _4F_3\left (\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2};\frac{3}{2}-\frac{m}{2},\frac{3}{2}-\frac{m}{2},\frac{3}{2}-\frac{m}{2};\frac{d}{e x^2+d}\right )+(m-1) \log \left (e x^2+d\right ) \left ((m-1) \, _2F_1\left (\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2};\frac{3}{2}-\frac{m}{2};\frac{d}{e x^2+d}\right ) \log \left (e x^2+d\right )-4 \, _3F_2\left (\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2};\frac{3}{2}-\frac{m}{2},\frac{3}{2}-\frac{m}{2};\frac{d}{e x^2+d}\right )\right )\right )}{e (m-1)^3}\right )}{(m+1)^2 x} \]

[In]

Integrate[(f*x)^m*Log[c*(d + e*x^2)^p]^3,x]

[Out]

((f*x)^m*((1 + m)*p^3*x^2*Log[d + e*x^2]^3 + (6*p^3*(-((e*x^2)/d))^(1/2 - m/2)*(-((1 + m)*(d + e*x^2)*Hypergeo
metricPFQ[{1, 1, 1, 1/2 - m/2}, {2, 2, 2}, 1 + (e*x^2)/d]) + (1 + m)*(d + e*x^2)*HypergeometricPFQ[{1, 1, 1/2
- m/2}, {2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2] + d*(-1 + (-((e*x^2)/d))^((1 + m)/2))*Log[d + e*x^2]^2))/e + (6*
d*(1 + m)*p^3*((e*x^2)/(d + e*x^2))^(1/2 - m/2)*(8*HypergeometricPFQ[{1/2 - m/2, 1/2 - m/2, 1/2 - m/2, 1/2 - m
/2}, {3/2 - m/2, 3/2 - m/2, 3/2 - m/2}, d/(d + e*x^2)] + (-1 + m)*Log[d + e*x^2]*(-4*HypergeometricPFQ[{1/2 -
m/2, 1/2 - m/2, 1/2 - m/2}, {3/2 - m/2, 3/2 - m/2}, d/(d + e*x^2)] + (-1 + m)*Hypergeometric2F1[1/2 - m/2, 1/2
 - m/2, 3/2 - m/2, d/(d + e*x^2)]*Log[d + e*x^2])))/(e*(-1 + m)^3) - (3*p^2*(-((e*x^2)/d))^(1/2 - m/2)*(-((1 +
 m)*(d + e*x^2)*HypergeometricPFQ[{1, 1, 1, 1/2 - m/2}, {2, 2, 2}, 1 + (e*x^2)/d]) + (1 + m)*(d + e*x^2)*Hyper
geometricPFQ[{1, 1, 1/2 - m/2}, {2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2] + d*(-1 + (-((e*x^2)/d))^((1 + m)/2))*Lo
g[d + e*x^2]^2)*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]))/e - (3*m*p^2*(-((e*x^2)/d))^(1/2 - m/2)*(-((1 +
m)*(d + e*x^2)*HypergeometricPFQ[{1, 1, 1, 1/2 - m/2}, {2, 2, 2}, 1 + (e*x^2)/d]) + (1 + m)*(d + e*x^2)*Hyperg
eometricPFQ[{1, 1, 1/2 - m/2}, {2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2] + d*(-1 + (-((e*x^2)/d))^((1 + m)/2))*Log
[d + e*x^2]^2)*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]))/e + (3*p*x^2*(-2*e*x^2*Hypergeometric2F1[1, (3 +
m)/2, (5 + m)/2, -((e*x^2)/d)] + d*(3 + m)*Log[d + e*x^2])*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(d*
(3 + m)) + (3*m*p*x^2*(-2*e*x^2*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((e*x^2)/d)] + d*(3 + m)*Log[d + e
*x^2])*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(d*(3 + m)) + x^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x
^2)^p])^3 + m*x^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^3))/((1 + m)^2*x)

Integral number [159] \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx \]

[B]   time = 1.02958 (sec), size = 466 ,normalized size = 6.3 \[ \frac{(f x)^m \left (\frac{4 d (m+1) p^2 \left (\frac{e x^2}{d+e x^2}\right )^{\frac{1}{2}-\frac{m}{2}} \left ((m-1) \log \left (d+e x^2\right ) \, _2F_1\left (\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2};\frac{3}{2}-\frac{m}{2};\frac{d}{e x^2+d}\right )-2 \, _3F_2\left (\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2};\frac{3}{2}-\frac{m}{2},\frac{3}{2}-\frac{m}{2};\frac{d}{e x^2+d}\right )\right )}{e (m-1)^2 x}+\frac{2 p \left (p \log \left (d+e x^2\right )-\log \left (c \left (d+e x^2\right )^p\right )\right ) \left (2 e x^3 \, _2F_1\left (1,\frac{m+3}{2};\frac{m+5}{2};-\frac{e x^2}{d}\right )-d (m+3) x \log \left (d+e x^2\right )\right )}{d (m+3)}-\frac{2 m p \left (p \log \left (d+e x^2\right )-\log \left (c \left (d+e x^2\right )^p\right )\right ) \left (d (m+3) x \log \left (d+e x^2\right )-2 e x^3 \, _2F_1\left (1,\frac{m+3}{2};\frac{m+5}{2};-\frac{e x^2}{d}\right )\right )}{d (m+3)}+m x \left (\log \left (c \left (d+e x^2\right )^p\right )-p \log \left (d+e x^2\right )\right )^2+x \left (\log \left (c \left (d+e x^2\right )^p\right )-p \log \left (d+e x^2\right )\right )^2+4 p^2 x \left (\frac{2 e x^2 \, _2F_1\left (1,\frac{m+3}{2};\frac{m+5}{2};-\frac{e x^2}{d}\right )}{d (m+3)}-\log \left (d+e x^2\right )\right )+(m+1) p^2 x \log ^2\left (d+e x^2\right )\right )}{(m+1)^2} \]

[In]

Integrate[(f*x)^m*Log[c*(d + e*x^2)^p]^2,x]

[Out]

((f*x)^m*(4*p^2*x*((2*e*x^2*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((e*x^2)/d)])/(d*(3 + m)) - Log[d + e*
x^2]) + (1 + m)*p^2*x*Log[d + e*x^2]^2 + (4*d*(1 + m)*p^2*((e*x^2)/(d + e*x^2))^(1/2 - m/2)*(-2*Hypergeometric
PFQ[{1/2 - m/2, 1/2 - m/2, 1/2 - m/2}, {3/2 - m/2, 3/2 - m/2}, d/(d + e*x^2)] + (-1 + m)*Hypergeometric2F1[1/2
 - m/2, 1/2 - m/2, 3/2 - m/2, d/(d + e*x^2)]*Log[d + e*x^2]))/(e*(-1 + m)^2*x) + (2*p*(2*e*x^3*Hypergeometric2
F1[1, (3 + m)/2, (5 + m)/2, -((e*x^2)/d)] - d*(3 + m)*x*Log[d + e*x^2])*(p*Log[d + e*x^2] - Log[c*(d + e*x^2)^
p]))/(d*(3 + m)) - (2*m*p*(-2*e*x^3*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((e*x^2)/d)] + d*(3 + m)*x*Log
[d + e*x^2])*(p*Log[d + e*x^2] - Log[c*(d + e*x^2)^p]))/(d*(3 + m)) + x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^
2)^p])^2 + m*x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2))/(1 + m)^2

Integral number [277] \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]

[B]   time = 4.48061 (sec), size = 1460 ,normalized size = 2.14 \[ \text{result too large to display} \]

[In]

Integrate[(f + g*x^2)*Log[c*(d + e*x^2)^p]^3,x]

[Out]

(g*p^3*x*(-18*(d + e*x^2)*HypergeometricPFQ[{-1/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e*x^2)/d] + 18*(d + e*x^2)*
HypergeometricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^2)/d]*Log[d + e*x^2] - 9*(d + e*x^2)*HypergeometricPFQ[
{-1/2, 1, 1}, {2, 2}, (d + e*x^2)/d]*Log[d + e*x^2]^2 + 2*d*Log[d + e*x^2]^3 - 2*d*Sqrt[1 - (d + e*x^2)/d]*Log
[d + e*x^2]^3 + 2*(d + e*x^2)*Sqrt[1 - (d + e*x^2)/d]*Log[d + e*x^2]^3))/(6*e*Sqrt[1 - (d + e*x^2)/d]) + (2*d*
g*p*x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/e + (6*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-(p*Log[
d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/Sqrt[e] - (2*d^(3/2)*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-(p*Log[d + e*x^2
]) + Log[c*(d + e*x^2)^p])^2)/e^(3/2) + 3*f*p*x*Log[d + e*x^2]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2
+ g*p*x^3*Log[d + e*x^2]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2 + f*x*(-(p*Log[d + e*x^2]) + Log[c*(d
+ e*x^2)^p])^2*(-6*p - p*Log[d + e*x^2] + Log[c*(d + e*x^2)^p]) + (g*x^3*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x
^2)^p])^2*(-2*p - p*Log[d + e*x^2] + Log[c*(d + e*x^2)^p]))/3 + 3*f*p^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^
2)^p])*(x*Log[d + e*x^2]^2 - (4*((-I)*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + Sqrt[e]*x*(-2 + Log[d + e*x^2])
- Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-2 + 2*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)] + Log[d + e*x^2]) - I*S
qrt[d]*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqrt[d] + Sqrt[e]*x)]))/Sqrt[e]) + 3*g*p^2*(-(p*Log[d + e*x^2]
) + Log[c*(d + e*x^2)^p])*((x^3*Log[d + e*x^2]^2)/3 - (4*((9*I)*d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + 3*d^(3
/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-8 + 6*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)] + 3*Log[d + e*x^2]) + Sqrt[e]
*x*(24*d - 2*e*x^2 + (-9*d + 3*e*x^2)*Log[d + e*x^2]) + (9*I)*d^(3/2)*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)
*Sqrt[d] + Sqrt[e]*x)]))/(27*e^(3/2))) + (f*p^3*(-48*Sqrt[-d^2]*Sqrt[d + e*x^2]*Sqrt[1 - d/(d + e*x^2)]*ArcSin
[Sqrt[d]/Sqrt[d + e*x^2]] - 6*Sqrt[-d^2]*Sqrt[1 - d/(d + e*x^2)]*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2,
1/2}, {3/2, 3/2, 3/2}, d/(d + e*x^2)] + 4*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^2)
]*Log[d + e*x^2] + Sqrt[d + e*x^2]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2]]*Log[d + e*x^2]^2) + Sqrt[-d]*e*x^2*(-48 + 2
4*Log[d + e*x^2] - 6*Log[d + e*x^2]^2 + Log[d + e*x^2]^3) + 24*d*Sqrt[e*x^2]*ArcTanh[Sqrt[e*x^2]/Sqrt[-d]]*(Lo
g[d + e*x^2] - Log[(d + e*x^2)/d]) + 6*(-d)^(3/2)*Sqrt[1 - (d + e*x^2)/d]*(Log[(d + e*x^2)/d]^2 - 4*Log[(d + e
*x^2)/d]*Log[(1 + Sqrt[1 - (d + e*x^2)/d])/2] + 2*Log[(1 + Sqrt[1 - (d + e*x^2)/d])/2]^2 - 4*PolyLog[2, 1/2 -
Sqrt[1 - (d + e*x^2)/d]/2])))/(Sqrt[-d]*e*x)

Integral number [298] \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]

[B]   time = 9.21294 (sec), size = 2539 ,normalized size = 2.26 \[ \text{Result too large to show} \]

[In]

Integrate[(f + g*x^3)^2*Log[c*(d + e*x^2)^p]^3,x]

[Out]

(f*g*p^3*(d + e*x^2)*(45*d - 3*e*x^2 + (-42*d + 6*e*x^2)*Log[d + e*x^2] + 6*(3*d - e*x^2)*Log[d + e*x^2]^2 - 4
*(d - e*x^2)*Log[d + e*x^2]^3))/(8*e^2) + (g^2*p^3*x*(-280*d^3*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, 1
 + (e*x^2)/d] - 280*d^2*e*x^2*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, 1 + (e*x^2)/d] - 112*d^3*Hypergeom
etricPFQ[{-5/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + (e*x^2)/d] - 112*d^2*e*x^2*HypergeometricPFQ[{-5/2, 1, 1, 1, 1}
, {2, 2, 2, 2}, 1 + (e*x^2)/d] + 280*d^3*HypergeometricPFQ[{-3/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + (e*x^2)/d] +
280*d^2*e*x^2*HypergeometricPFQ[{-3/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + (e*x^2)/d] - 210*d^3*HypergeometricPFQ[{
-1/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + (e*x^2)/d] - 210*d^2*e*x^2*HypergeometricPFQ[{-1/2, 1, 1, 1, 1}, {2, 2, 2
, 2}, 1 + (e*x^2)/d] + 16*d^3*Log[d + e*x^2] + 16*e^3*x^6*Sqrt[-((e*x^2)/d)]*Log[d + e*x^2] + 280*d^3*Hypergeo
metricPFQ[{-3/2, 1, 1}, {2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2] + 280*d^2*e*x^2*HypergeometricPFQ[{-3/2, 1, 1},
{2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2] - 280*d^3*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, 1 + (e*x^2)/d]*L
og[d + e*x^2] - 280*d^2*e*x^2*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2] + 21
0*d^3*HypergeometricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2] + 210*d^2*e*x^2*Hypergeometr
icPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2] - 32*d^3*Log[d + e*x^2]^2 + 28*d*e^2*x^4*Sqrt[
-((e*x^2)/d)]*Log[d + e*x^2]^2 - 4*e^3*x^6*Sqrt[-((e*x^2)/d)]*Log[d + e*x^2]^2 + 140*d^3*HypergeometricPFQ[{-3
/2, 1, 1}, {2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2]^2 + 140*d^2*e*x^2*HypergeometricPFQ[{-3/2, 1, 1}, {2, 2}, 1 +
 (e*x^2)/d]*Log[d + e*x^2]^2 - 105*d^3*HypergeometricPFQ[{-1/2, 1, 1}, {2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2]^2
 - 105*d^2*e*x^2*HypergeometricPFQ[{-1/2, 1, 1}, {2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2]^2 + 10*d^3*Log[d + e*x^
2]^3 + 10*e^3*x^6*Sqrt[-((e*x^2)/d)]*Log[d + e*x^2]^3 + 56*d^2*(d + e*x^2)*HypergeometricPFQ[{-5/2, 1, 1, 1},
{2, 2, 2}, 1 + (e*x^2)/d]*(3 + 2*Log[d + e*x^2]) - 56*d^2*(d + e*x^2)*HypergeometricPFQ[{-5/2, 1, 1}, {2, 2},
1 + (e*x^2)/d]*(1 + 3*Log[d + e*x^2] + Log[d + e*x^2]^2)))/(70*e^3*Sqrt[-((e*x^2)/d)]) - (3*f*g*p^2*(e*x^2*(-6
*d + e*x^2) + (6*d^2 + 4*d*e*x^2 - 2*e^2*x^4)*Log[d + e*x^2] - 2*(d^2 - e^2*x^4)*Log[d + e*x^2]^2)*(p*Log[d +
e*x^2] - Log[c*(d + e*x^2)^p]))/(4*e^2) + (3*d*f*g*p*x^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(2*e)
 - (2*d^2*g^2*p*x^3*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(7*e^2) + (6*d*g^2*p*x^5*(-(p*Log[d + e*x^
2]) + Log[c*(d + e*x^2)^p])^2)/(35*e) - (6*Sqrt[d]*(-7*e^3*f^2 + d^3*g^2)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-(p*L
og[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(7*e^(7/2)) - (3*d^2*f*g*p*Log[d + e*x^2]*(-(p*Log[d + e*x^2]) + Log
[c*(d + e*x^2)^p])^2)/(2*e^2) + (3*p*x*(14*f^2 + 7*f*g*x^3 + 2*g^2*x^6)*Log[d + e*x^2]*(-(p*Log[d + e*x^2]) +
Log[c*(d + e*x^2)^p])^2)/14 - (g^2*x^7*(6*p + 7*p*Log[d + e*x^2] - 7*Log[c*(d + e*x^2)^p])*(-(p*Log[d + e*x^2]
) + Log[c*(d + e*x^2)^p])^2)/49 - (f*g*x^4*(3*p + 2*p*Log[d + e*x^2] - 2*Log[c*(d + e*x^2)^p])*(-(p*Log[d + e*
x^2]) + Log[c*(d + e*x^2)^p])^2)/4 + (x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2*(-42*e^3*f^2*p + 6*d^3*
g^2*p + 7*e^3*f^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])))/(7*e^3) - (3*f^2*p^2*(p*Log[d + e*x^2] - Log[
c*(d + e*x^2)^p])*((4*I)*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + 4*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-2 + 2
*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)] + Log[d + e*x^2]) + Sqrt[e]*x*(8 - 4*Log[d + e*x^2] + Log[d + e*x^2]
^2) + (4*I)*Sqrt[d]*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqrt[d] + Sqrt[e]*x)]))/Sqrt[e] + 3*g^2*p^2*(-(p*
Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])*((x^7*Log[d + e*x^2]^2)/7 - (4*((11025*I)*d^(7/2)*ArcTan[(Sqrt[e]*x)/S
qrt[d]]^2 + 105*d^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-352 + 210*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)] + 105
*Log[d + e*x^2]) + Sqrt[e]*x*(36960*d^3 - 4970*d^2*e*x^2 + 1512*d*e^2*x^4 - 450*e^3*x^6 - 105*(105*d^3 - 35*d^
2*e*x^2 + 21*d*e^2*x^4 - 15*e^3*x^6)*Log[d + e*x^2]) + (11025*I)*d^(7/2)*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((
-I)*Sqrt[d] + Sqrt[e]*x)]))/(77175*e^(7/2))) + (f^2*p^3*(-48*Sqrt[-d^2]*Sqrt[(e*x^2)/(d + e*x^2)]*Sqrt[d + e*x
^2]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2]] + Sqrt[-d]*e*x^2*(-48 + 24*Log[d + e*x^2] - 6*Log[d + e*x^2]^2 + Log[d + e
*x^2]^3) - 6*Sqrt[-d^2]*Sqrt[(e*x^2)/(d + e*x^2)]*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2
, 3/2}, d/(d + e*x^2)] + Log[d + e*x^2]*(4*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^2
)] + Sqrt[d + e*x^2]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2]]*Log[d + e*x^2])) + 24*d*Sqrt[e*x^2]*ArcTanh[Sqrt[e*x^2]/S
qrt[-d]]*(Log[d + e*x^2] - Log[1 + (e*x^2)/d]) + 6*(-d)^(3/2)*Sqrt[-((e*x^2)/d)]*(Log[1 + (e*x^2)/d]^2 - 4*Log
[1 + (e*x^2)/d]*Log[(1 + Sqrt[-((e*x^2)/d)])/2] + 2*Log[(1 + Sqrt[-((e*x^2)/d)])/2]^2 - 4*PolyLog[2, 1/2 - Sqr
t[-((e*x^2)/d)]/2])))/(Sqrt[-d]*e*x)

Integral number [299] \[ \int \left (f+g x^3\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]

[B]   time = 2.38113 (sec), size = 1066 ,normalized size = 2.06 \[ -\frac{3}{16} g p^3 x^4+\frac{1}{4} g \log ^3\left (c \left (e x^2+d\right )^p\right ) x^4-\frac{3}{8} g p \log ^2\left (c \left (e x^2+d\right )^p\right ) x^4+\frac{3}{8} g p^2 \log \left (c \left (e x^2+d\right )^p\right ) x^4+\frac{21 d g p^3 x^2}{8 e}+\frac{3 d g p \log ^2\left (c \left (e x^2+d\right )^p\right ) x^2}{4 e}-\frac{9 d g p^2 \log \left (c \left (e x^2+d\right )^p\right ) x^2}{4 e}+3 f p \log \left (e x^2+d\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )^2 x+f \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )^2 \left (-\log \left (e x^2+d\right ) p-6 p+\log \left (c \left (e x^2+d\right )^p\right )\right ) x-\frac{d^2 g \log ^3\left (c \left (e x^2+d\right )^p\right )}{4 e^2}+\frac{9 d^2 g p \log ^2\left (c \left (e x^2+d\right )^p\right )}{8 e^2}+\frac{6 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )^2}{\sqrt{e}}-\frac{3 d^2 g p^3 \log \left (e x^2+d\right )}{8 e^2}-\frac{9 d^2 g p^2 \log \left (c \left (e x^2+d\right )^p\right )}{4 e^2}+3 f p^2 \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right ) \left (x \log ^2\left (e x^2+d\right )-\frac{4 \left (-i \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2-\sqrt{d} \left (2 \log \left (\frac{2 \sqrt{d}}{i \sqrt{e} x+\sqrt{d}}\right )+\log \left (e x^2+d\right )-2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )+\sqrt{e} x \left (\log \left (e x^2+d\right )-2\right )-i \sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{e} x+i \sqrt{d}}{\sqrt{e} x-i \sqrt{d}}\right )\right )}{\sqrt{e}}\right )+\frac{f p^3 \left (\sqrt{-d} e \left (\log ^3\left (e x^2+d\right )-6 \log ^2\left (e x^2+d\right )+24 \log \left (e x^2+d\right )-48\right ) x^2-48 \sqrt{-d^2} \sqrt{e x^2+d} \sqrt{1-\frac{d}{e x^2+d}} \sin ^{-1}\left (\frac{\sqrt{d}}{\sqrt{e x^2+d}}\right )-6 \sqrt{-d^2} \sqrt{1-\frac{d}{e x^2+d}} \left (\sqrt{e x^2+d} \sin ^{-1}\left (\frac{\sqrt{d}}{\sqrt{e x^2+d}}\right ) \log ^2\left (e x^2+d\right )+4 \sqrt{d} \, _3F_2\left (\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{d}{e x^2+d}\right ) \log \left (e x^2+d\right )+8 \sqrt{d} \, _4F_3\left (\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{d}{e x^2+d}\right )\right )+24 d \sqrt{e x^2} \tanh ^{-1}\left (\frac{\sqrt{e x^2}}{\sqrt{-d}}\right ) \left (\log \left (e x^2+d\right )-\log \left (\frac{e x^2+d}{d}\right )\right )+6 (-d)^{3/2} \sqrt{1-\frac{e x^2+d}{d}} \left (\log ^2\left (\frac{e x^2+d}{d}\right )-4 \log \left (\frac{1}{2} \left (\sqrt{1-\frac{e x^2+d}{d}}+1\right )\right ) \log \left (\frac{e x^2+d}{d}\right )+2 \log ^2\left (\frac{1}{2} \left (\sqrt{1-\frac{e x^2+d}{d}}+1\right )\right )-4 \text{PolyLog}\left (2,\frac{1}{2}-\frac{1}{2} \sqrt{1-\frac{e x^2+d}{d}}\right )\right )\right )}{\sqrt{-d} e x} \]

[In]

Integrate[(f + g*x^3)*Log[c*(d + e*x^2)^p]^3,x]

[Out]

(21*d*g*p^3*x^2)/(8*e) - (3*g*p^3*x^4)/16 - (3*d^2*g*p^3*Log[d + e*x^2])/(8*e^2) - (9*d^2*g*p^2*Log[c*(d + e*x
^2)^p])/(4*e^2) - (9*d*g*p^2*x^2*Log[c*(d + e*x^2)^p])/(4*e) + (3*g*p^2*x^4*Log[c*(d + e*x^2)^p])/8 + (9*d^2*g
*p*Log[c*(d + e*x^2)^p]^2)/(8*e^2) + (3*d*g*p*x^2*Log[c*(d + e*x^2)^p]^2)/(4*e) - (3*g*p*x^4*Log[c*(d + e*x^2)
^p]^2)/8 - (d^2*g*Log[c*(d + e*x^2)^p]^3)/(4*e^2) + (g*x^4*Log[c*(d + e*x^2)^p]^3)/4 + (6*Sqrt[d]*f*p*ArcTan[(
Sqrt[e]*x)/Sqrt[d]]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/Sqrt[e] + 3*f*p*x*Log[d + e*x^2]*(-(p*Log[
d + e*x^2]) + Log[c*(d + e*x^2)^p])^2 + f*x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2*(-6*p - p*Log[d + e
*x^2] + Log[c*(d + e*x^2)^p]) + 3*f*p^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])*(x*Log[d + e*x^2]^2 - (4*
((-I)*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + Sqrt[e]*x*(-2 + Log[d + e*x^2]) - Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqr
t[d]]*(-2 + 2*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)] + Log[d + e*x^2]) - I*Sqrt[d]*PolyLog[2, (I*Sqrt[d] + S
qrt[e]*x)/((-I)*Sqrt[d] + Sqrt[e]*x)]))/Sqrt[e]) + (f*p^3*(-48*Sqrt[-d^2]*Sqrt[d + e*x^2]*Sqrt[1 - d/(d + e*x^
2)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2]] - 6*Sqrt[-d^2]*Sqrt[1 - d/(d + e*x^2)]*(8*Sqrt[d]*HypergeometricPFQ[{1/2,
1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e*x^2)] + 4*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(
d + e*x^2)]*Log[d + e*x^2] + Sqrt[d + e*x^2]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2]]*Log[d + e*x^2]^2) + Sqrt[-d]*e*x^
2*(-48 + 24*Log[d + e*x^2] - 6*Log[d + e*x^2]^2 + Log[d + e*x^2]^3) + 24*d*Sqrt[e*x^2]*ArcTanh[Sqrt[e*x^2]/Sqr
t[-d]]*(Log[d + e*x^2] - Log[(d + e*x^2)/d]) + 6*(-d)^(3/2)*Sqrt[1 - (d + e*x^2)/d]*(Log[(d + e*x^2)/d]^2 - 4*
Log[(d + e*x^2)/d]*Log[(1 + Sqrt[1 - (d + e*x^2)/d])/2] + 2*Log[(1 + Sqrt[1 - (d + e*x^2)/d])/2]^2 - 4*PolyLog
[2, 1/2 - Sqrt[1 - (d + e*x^2)/d]/2])))/(Sqrt[-d]*e*x)

Integral number [485] \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx \]

[B]   time = 9.0109 (sec), size = 3146 ,normalized size = 3.97 \[ \text{Result too large to show} \]

[In]

Integrate[x^2*(a + b*Log[c*(d + e*x^(2/3))^n])^3,x]

[Out]

(b^3*n^3*x^(1/3)*(32*d^4 - 32*d^4*Sqrt[1 - (d + e*x^(2/3))/d] + 128*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(
2/3)) - 192*d^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2 + 128*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/
3))^3 - 32*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^4 + 1584*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1
, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d] - 4536*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1, 1}, {2, 2, 2}
, (d + e*x^(2/3))/d] + 3780*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/
d] - 864*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e*x^(2/3))/d] + 3024*d^3
*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e*x^(2/3))/d] - 3780*d^3*(d + e*x^(2
/3))*HypergeometricPFQ[{-3/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e*x^(2/3))/d] + 1890*d^3*(d + e*x^(2/3))*Hyperge
ometricPFQ[{-1/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e*x^(2/3))/d] - 240*d^4*Log[d + e*x^(2/3)] + 240*d^4*Sqrt[1
- (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] - 672*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))*Log[d + e*x^(2/3
)] + 576*d^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2*Log[d + e*x^(2/3)] - 96*d*Sqrt[1 - (d + e*x^(2/3))/
d]*(d + e*x^(2/3))^3*Log[d + e*x^(2/3)] - 48*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^4*Log[d + e*x^(2/3)]
- 3780*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1}, {2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] + 864
*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] - 302
4*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] + 37
80*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] - 1
890*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] +
284*d^4*Log[d + e*x^(2/3)]^2 - 284*d^4*Sqrt[1 - (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)]^2 + 668*d^3*Sqrt[1 - (d
+ e*x^(2/3))/d]*(d + e*x^(2/3))*Log[d + e*x^(2/3)]^2 - 552*d^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2*L
og[d + e*x^(2/3)]^2 + 236*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^3*Log[d + e*x^(2/3)]^2 - 68*Sqrt[1 - (
d + e*x^(2/3))/d]*(d + e*x^(2/3))^4*Log[d + e*x^(2/3)]^2 - 1890*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1
, 1}, {2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)]^2 + 945*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1}
, {2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)]^2 - 70*d^4*Log[d + e*x^(2/3)]^3 + 70*d^4*Sqrt[1 - (d + e*x^(2/
3))/d]*Log[d + e*x^(2/3)]^3 - 280*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))*Log[d + e*x^(2/3)]^3 + 420*d
^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2*Log[d + e*x^(2/3)]^3 - 280*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d +
 e*x^(2/3))^3*Log[d + e*x^(2/3)]^3 + 70*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^4*Log[d + e*x^(2/3)]^3 + 1
512*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1}, {2, 2}, (d + e*x^(2/3))/d]*(1 + 3*Log[d + e*x^(2/3)] +
 Log[d + e*x^(2/3)]^2) - 144*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1}, {2, 2}, (d + e*x^(2/3))/d]*(6
 + 11*Log[d + e*x^(2/3)] + 3*Log[d + e*x^(2/3)]^2)))/(210*e^4*Sqrt[1 - (d + e*x^(2/3))/d]) + (b^2*n^2*x^(1/3)*
(-120*d^4 + 120*d^4*Sqrt[1 - (d + e*x^(2/3))/d] - 336*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3)) + 288*d^
2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2 - 48*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^3 - 24*Sqrt
[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^4 - 1890*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1}, {2, 2}, (
d + e*x^(2/3))/d] + 432*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d] -
 1512*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d] + 1890*d^3*(d + e*x
^(2/3))*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d] - 945*d^3*(d + e*x^(2/3))*Hypergeomet
ricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d] + 284*d^4*Log[d + e*x^(2/3)] - 284*d^4*Sqrt[1 - (d + e*x
^(2/3))/d]*Log[d + e*x^(2/3)] + 668*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))*Log[d + e*x^(2/3)] - 552*d
^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2*Log[d + e*x^(2/3)] + 236*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e
*x^(2/3))^3*Log[d + e*x^(2/3)] - 68*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^4*Log[d + e*x^(2/3)] - 1890*d^
3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1}, {2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] + 945*d^3*(d +
 e*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1}, {2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] - 105*d^4*Log[d + e*
x^(2/3)]^2 + 105*d^4*Sqrt[1 - (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)]^2 - 420*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d
 + e*x^(2/3))*Log[d + e*x^(2/3)]^2 + 630*d^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2*Log[d + e*x^(2/3)]^
2 - 420*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^3*Log[d + e*x^(2/3)]^2 + 105*Sqrt[1 - (d + e*x^(2/3))/d]
*(d + e*x^(2/3))^4*Log[d + e*x^(2/3)]^2 + 756*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1}, {2, 2}, (d +
 e*x^(2/3))/d]*(3 + 2*Log[d + e*x^(2/3)]) - 72*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1}, {2, 2}, (d
+ e*x^(2/3))/d]*(11 + 6*Log[d + e*x^(2/3)]))*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])))/(10
5*e^4*Sqrt[1 - (d + e*x^(2/3))/d]) - (2*b*d^4*n*x^(1/3)*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3)
)^n]))^2)/e^4 + (2*b*d^3*n*x*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^2)/(3*e^3) - (2*b*d^
2*n*x^(5/3)*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^2)/(5*e^2) + (2*b*d*n*x^(7/3)*(a + b*
(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^2)/(7*e) + (2*b*d^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[
d]]*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^2)/e^(9/2) + b*n*x^3*Log[d + e*x^(2/3)]*(a +
b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^2 + (x^3*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d +
e*x^(2/3))^n]))^2*(3*a - 2*b*n + 3*b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])))/9

Integral number [486] \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx \]

[A]   time = 1.24265 (sec), size = 598 ,normalized size = 1.23 \[ \frac{3 b^2 n^2 x \left (-a-b \log \left (c \left (d+e x^{2/3}\right )^n\right )+b n \log \left (d+e x^{2/3}\right )\right ) \left (3 \left (d+e x^{2/3}\right ) \, _4F_3\left (-\frac{1}{2},1,1,1;2,2,2;\frac{x^{2/3} e}{d}+1\right )+\log \left (d+e x^{2/3}\right ) \left (\left (d-d \left (-\frac{e x^{2/3}}{d}\right )^{3/2}\right ) \log \left (d+e x^{2/3}\right )-3 \left (d+e x^{2/3}\right ) \, _3F_2\left (-\frac{1}{2},1,1;2,2;\frac{x^{2/3} e}{d}+1\right )\right )\right )}{d \left (-\frac{e x^{2/3}}{d}\right )^{3/2}}-\frac{b^3 n^3 x \left (\log \left (d+e x^{2/3}\right ) \left (18 \left (d+e x^{2/3}\right ) \, _4F_3\left (-\frac{1}{2},1,1,1;2,2,2;\frac{x^{2/3} e}{d}+1\right )+\log \left (d+e x^{2/3}\right ) \left (2 \left (d-d \left (-\frac{e x^{2/3}}{d}\right )^{3/2}\right ) \log \left (d+e x^{2/3}\right )-9 \left (d+e x^{2/3}\right ) \, _3F_2\left (-\frac{1}{2},1,1;2,2;\frac{x^{2/3} e}{d}+1\right )\right )\right )-18 \left (d+e x^{2/3}\right ) \, _5F_4\left (-\frac{1}{2},1,1,1,1;2,2,2,2;\frac{x^{2/3} e}{d}+1\right )\right )}{2 d \left (-\frac{e x^{2/3}}{d}\right )^{3/2}}-\frac{6 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2}{e^{3/2}}+3 b n x \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2+\frac{6 b d n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2}{e}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )-2 b n\right ) \]

[In]

Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])^3,x]

[Out]

-(b^3*n^3*x*(-18*(d + e*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + (e*x^(2/3))/d] + Log[
d + e*x^(2/3)]*(18*(d + e*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, 1 + (e*x^(2/3))/d] + Log[d +
e*x^(2/3)]*(-9*(d + e*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1}, {2, 2}, 1 + (e*x^(2/3))/d] + 2*(d - d*(-((e*x^(
2/3))/d))^(3/2))*Log[d + e*x^(2/3)]))))/(2*d*(-((e*x^(2/3))/d))^(3/2)) + (3*b^2*n^2*x*(3*(d + e*x^(2/3))*Hyper
geometricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, 1 + (e*x^(2/3))/d] + Log[d + e*x^(2/3)]*(-3*(d + e*x^(2/3))*Hypergeom
etricPFQ[{-1/2, 1, 1}, {2, 2}, 1 + (e*x^(2/3))/d] + (d - d*(-((e*x^(2/3))/d))^(3/2))*Log[d + e*x^(2/3)]))*(-a
+ b*n*Log[d + e*x^(2/3)] - b*Log[c*(d + e*x^(2/3))^n]))/(d*(-((e*x^(2/3))/d))^(3/2)) + (6*b*d*n*x^(1/3)*(a - b
*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/e - (6*b*d^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a
 - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/e^(3/2) + 3*b*n*x*Log[d + e*x^(2/3)]*(a - b*n*Log[d
 + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2 + x*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2*
(a - 2*b*n - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])

Integral number [487] \[ \int \frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx \]

[B]   time = 7.16765 (sec), size = 1028 ,normalized size = 3.23 \[ \frac{b^3 \left (\frac{d^{5/2} \log ^3\left (d+e x^{2/3}\right )}{\sqrt{-d}}+6 \sqrt{-d} \left (d+e x^{2/3}\right )^{3/2} \left (\frac{e x^{2/3}}{d+e x^{2/3}}\right )^{3/2} \sin ^{-1}\left (\frac{\sqrt{d}}{\sqrt{d+e x^{2/3}}}\right ) \log ^2\left (d+e x^{2/3}\right )-6 \sqrt{-d^2} e x^{2/3} \log ^2\left (d+e x^{2/3}\right )-24 \sqrt{d} \left (e x^{2/3}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e x^{2/3}}}{\sqrt{-d}}\right ) \log \left (d+e x^{2/3}\right )+24 \sqrt{-d^2} e \sqrt{\frac{e x^{2/3}}{d+e x^{2/3}}} x^{2/3} \, _3F_2\left (\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{d}{d+e x^{2/3}}\right ) \log \left (d+e x^{2/3}\right )-12 d \sqrt{-d^2} \left (-\frac{e x^{2/3}}{d}\right )^{3/2} \log ^2\left (\frac{1}{2} \left (\sqrt{-\frac{e x^{2/3}}{d}}+1\right )\right )-6 d \sqrt{-d^2} \left (-\frac{e x^{2/3}}{d}\right )^{3/2} \log ^2\left (\frac{x^{2/3} e}{d}+1\right )+48 \sqrt{-d^2} e \sqrt{\frac{e x^{2/3}}{d+e x^{2/3}}} x^{2/3} \, _4F_3\left (\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{d}{d+e x^{2/3}}\right )+24 \sqrt{d} \left (e x^{2/3}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e x^{2/3}}}{\sqrt{-d}}\right ) \log \left (\frac{x^{2/3} e}{d}+1\right )+24 d \sqrt{-d^2} \left (-\frac{e x^{2/3}}{d}\right )^{3/2} \log \left (\frac{1}{2} \left (\sqrt{-\frac{e x^{2/3}}{d}}+1\right )\right ) \log \left (\frac{x^{2/3} e}{d}+1\right )+24 d \sqrt{-d^2} \left (-\frac{e x^{2/3}}{d}\right )^{3/2} \text{PolyLog}\left (2,\frac{1}{2}-\frac{1}{2} \sqrt{-\frac{e x^{2/3}}{d}}\right )\right ) n^3}{\sqrt{-d} d^{3/2} x}-\frac{3 b^2 \left (-3 d \left (d+e x^{2/3}\right ) \, _4F_3\left (1,1,1,\frac{5}{2};2,2,2;\frac{x^{2/3} e}{d}+1\right ) \left (-\frac{e x^{2/3}}{d}\right )^{3/2}-d \log \left (d+e x^{2/3}\right ) \left (4 d \log \left (\frac{1}{2} \left (\sqrt{-\frac{e x^{2/3}}{d}}+1\right )\right ) \left (-\frac{e x^{2/3}}{d}\right )^{3/2}+\left (d-d \left (-\frac{e x^{2/3}}{d}\right )^{3/2}\right ) \log \left (d+e x^{2/3}\right )-4 e \left (\sqrt{-\frac{e x^{2/3}}{d}}-1\right ) x^{2/3}\right )\right ) \left (-a+b n \log \left (d+e x^{2/3}\right )-b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) n^2}{d^2 x}-\frac{6 b e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 n}{d^{3/2}}-\frac{3 b \log \left (d+e x^{2/3}\right ) \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 n}{x}-\frac{6 b e \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 n}{d \sqrt [3]{x}}-\frac{\left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x} \]

[In]

Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])^3/x^2,x]

[Out]

(-3*b^2*n^2*(-3*d*(d + e*x^(2/3))*(-((e*x^(2/3))/d))^(3/2)*HypergeometricPFQ[{1, 1, 1, 5/2}, {2, 2, 2}, 1 + (e
*x^(2/3))/d] - d*Log[d + e*x^(2/3)]*(-4*e*(-1 + Sqrt[-((e*x^(2/3))/d)])*x^(2/3) + 4*d*(-((e*x^(2/3))/d))^(3/2)
*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2] + (d - d*(-((e*x^(2/3))/d))^(3/2))*Log[d + e*x^(2/3)]))*(-a + b*n*Log[d +
 e*x^(2/3)] - b*Log[c*(d + e*x^(2/3))^n]))/(d^2*x) - (6*b*e*n*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(
2/3))^n])^2)/(d*x^(1/3)) - (6*b*e^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a - b*n*Log[d + e*x^(2/3)] + b*Lo
g[c*(d + e*x^(2/3))^n])^2)/d^(3/2) - (3*b*n*Log[d + e*x^(2/3)]*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^
(2/3))^n])^2)/x - (a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^3/x + (b^3*n^3*(48*Sqrt[-d^2]*e*Sq
rt[(e*x^(2/3))/(d + e*x^(2/3))]*x^(2/3)*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e*x^(2
/3))] - 12*d*Sqrt[-d^2]*(-((e*x^(2/3))/d))^(3/2)*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]^2 - 24*Sqrt[d]*(e*x^(2/3)
)^(3/2)*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*Log[d + e*x^(2/3)] + 24*Sqrt[-d^2]*e*Sqrt[(e*x^(2/3))/(d + e*x^(2/3)
)]*x^(2/3)*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^(2/3))]*Log[d + e*x^(2/3)] - 6*Sqrt[-d^2]
*e*x^(2/3)*Log[d + e*x^(2/3)]^2 + 6*Sqrt[-d]*(d + e*x^(2/3))^(3/2)*((e*x^(2/3))/(d + e*x^(2/3)))^(3/2)*ArcSin[
Sqrt[d]/Sqrt[d + e*x^(2/3)]]*Log[d + e*x^(2/3)]^2 + (d^(5/2)*Log[d + e*x^(2/3)]^3)/Sqrt[-d] + 24*Sqrt[d]*(e*x^
(2/3))^(3/2)*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*Log[1 + (e*x^(2/3))/d] + 24*d*Sqrt[-d^2]*(-((e*x^(2/3))/d))^(3/
2)*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]*Log[1 + (e*x^(2/3))/d] - 6*d*Sqrt[-d^2]*(-((e*x^(2/3))/d))^(3/2)*Log[1
+ (e*x^(2/3))/d]^2 + 24*d*Sqrt[-d^2]*(-((e*x^(2/3))/d))^(3/2)*PolyLog[2, 1/2 - Sqrt[-((e*x^(2/3))/d)]/2]))/(Sq
rt[-d]*d^(3/2)*x)

Integral number [488] \[ \int \frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^4} \, dx \]

[A]   time = 2.8917 (sec), size = 803 ,normalized size = 1.27 \[ \frac{-70 \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 d^5-210 b n \log \left (d+e x^{2/3}\right ) \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d^5-60 b e n x^{2/3} \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d^4+84 b e^2 n x^{4/3} \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d^3-140 b e^3 n x^2 \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d^2+420 b e^4 n x^{8/3} \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d+420 b e^{9/2} n x^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \sqrt{d}+35 b^3 n^3 \left (54 \left (d+e x^{2/3}\right ) \sqrt{-\frac{e x^{2/3}}{d}} x^{8/3} \, _5F_4\left (1,1,1,1,\frac{11}{2};2,2,2,2;\frac{x^{2/3} e}{d}+1\right ) e^4+\log \left (d+e x^{2/3}\right ) \left (54 d \left (d+e x^{2/3}\right ) \left (-\frac{e x^{2/3}}{d}\right )^{3/2} x^2 \, _4F_3\left (1,1,1,\frac{11}{2};2,2,2;\frac{x^{2/3} e}{d}+1\right ) e^3+\log \left (d+e x^{2/3}\right ) \left (27 e^4 \left (d+e x^{2/3}\right ) \sqrt{-\frac{e x^{2/3}}{d}} x^{8/3} \, _3F_2\left (1,1,\frac{11}{2};2,2;\frac{x^{2/3} e}{d}+1\right )-2 d \left (d^4+e^3 \left (-\frac{e x^{2/3}}{d}\right )^{3/2} x^2 d\right ) \log \left (d+e x^{2/3}\right )\right )\right )\right )+\frac{210 b^2 n^2 \left (\log \left (d+e x^{2/3}\right ) \left (9 \left (d+e x^{2/3}\right ) x^{10/3} \, _3F_2\left (1,1,\frac{11}{2};2,2;\frac{x^{2/3} e}{d}+1\right ) e^5+d \left (\sqrt{-\frac{e x^{2/3}}{d}} d^5+e^5 x^{10/3}\right ) \log \left (d+e x^{2/3}\right )\right )-9 e^5 \left (d+e x^{2/3}\right ) x^{10/3} \, _4F_3\left (1,1,1,\frac{11}{2};2,2,2;\frac{x^{2/3} e}{d}+1\right )\right ) \left (-a+b n \log \left (d+e x^{2/3}\right )-b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{\sqrt{-\frac{e x^{2/3}}{d}} d}}{210 d^5 x^3} \]

[In]

Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])^3/x^4,x]

[Out]

(35*b^3*n^3*(54*e^4*(d + e*x^(2/3))*Sqrt[-((e*x^(2/3))/d)]*x^(8/3)*HypergeometricPFQ[{1, 1, 1, 1, 11/2}, {2, 2
, 2, 2}, 1 + (e*x^(2/3))/d] + Log[d + e*x^(2/3)]*(54*d*e^3*(d + e*x^(2/3))*(-((e*x^(2/3))/d))^(3/2)*x^2*Hyperg
eometricPFQ[{1, 1, 1, 11/2}, {2, 2, 2}, 1 + (e*x^(2/3))/d] + Log[d + e*x^(2/3)]*(27*e^4*(d + e*x^(2/3))*Sqrt[-
((e*x^(2/3))/d)]*x^(8/3)*HypergeometricPFQ[{1, 1, 11/2}, {2, 2}, 1 + (e*x^(2/3))/d] - 2*d*(d^4 + d*e^3*(-((e*x
^(2/3))/d))^(3/2)*x^2)*Log[d + e*x^(2/3)]))) + (210*b^2*n^2*(-9*e^5*(d + e*x^(2/3))*x^(10/3)*HypergeometricPFQ
[{1, 1, 1, 11/2}, {2, 2, 2}, 1 + (e*x^(2/3))/d] + Log[d + e*x^(2/3)]*(9*e^5*(d + e*x^(2/3))*x^(10/3)*Hypergeom
etricPFQ[{1, 1, 11/2}, {2, 2}, 1 + (e*x^(2/3))/d] + d*(d^5*Sqrt[-((e*x^(2/3))/d)] + e^5*x^(10/3))*Log[d + e*x^
(2/3)]))*(-a + b*n*Log[d + e*x^(2/3)] - b*Log[c*(d + e*x^(2/3))^n]))/(d*Sqrt[-((e*x^(2/3))/d)]) - 60*b*d^4*e*n
*x^(2/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2 + 84*b*d^3*e^2*n*x^(4/3)*(a - b*n*Log[d +
 e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2 - 140*b*d^2*e^3*n*x^2*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e
*x^(2/3))^n])^2 + 420*b*d*e^4*n*x^(8/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2 + 420*b*Sq
rt[d]*e^(9/2)*n*x^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n]
)^2 - 210*b*d^5*n*Log[d + e*x^(2/3)]*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2 - 70*d^5*(a -
 b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^3)/(210*d^5*x^3)

Integral number [528] \[ \int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3 \, dx \]

[A]   time = 4.8286 (sec), size = 764 ,normalized size = 0.6 \[ -\frac{b^2 n^2 \left (-a-b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )+b n \log \left (d+\frac{e}{x^{2/3}}\right )\right ) \left (\log \left (d+\frac{e}{x^{2/3}}\right ) \left (9 e^5 \left (d x^{2/3}+e\right ) \, _3F_2\left (1,1,\frac{11}{2};2,2;\frac{e}{d x^{2/3}}+1\right )+d x^{2/3} \left (d^5 x^{10/3} \sqrt{-\frac{e}{d x^{2/3}}}+e^5\right ) \log \left (d+\frac{e}{x^{2/3}}\right )\right )-9 e^5 \left (d x^{2/3}+e\right ) \, _4F_3\left (1,1,1,\frac{11}{2};2,2,2;\frac{e}{d x^{2/3}}+1\right )\right )}{d^6 x \sqrt{-\frac{e}{d x^{2/3}}}}+\frac{b^3 n^3 \left (\log \left (d+\frac{e}{x^{2/3}}\right ) \left (\log \left (d+\frac{e}{x^{2/3}}\right ) \left (27 e^5 \left (d x^{2/3}+e\right ) \, _3F_2\left (1,1,\frac{11}{2};2,2;\frac{e}{d x^{2/3}}+1\right )+2 d x^{2/3} \left (d^5 x^{10/3} \sqrt{-\frac{e}{d x^{2/3}}}+e^5\right ) \log \left (d+\frac{e}{x^{2/3}}\right )\right )-54 e^5 \left (d x^{2/3}+e\right ) \, _4F_3\left (1,1,1,\frac{11}{2};2,2,2;\frac{e}{d x^{2/3}}+1\right )\right )+54 e^5 \left (d x^{2/3}+e\right ) \, _5F_4\left (1,1,1,1,\frac{11}{2};2,2,2,2;\frac{e}{d x^{2/3}}+1\right )\right )}{6 d^6 x \sqrt{-\frac{e}{d x^{2/3}}}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2}{5 d^2}+\frac{2 b e^{9/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2}{d^{9/2}}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^3+b n x^3 \log \left (d+\frac{e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2 \]

[In]

Integrate[x^2*(a + b*Log[c*(d + e/x^(2/3))^n])^3,x]

[Out]

(b^3*n^3*(54*e^5*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 1, 1, 11/2}, {2, 2, 2, 2}, 1 + e/(d*x^(2/3))] + Log[
d + e/x^(2/3)]*(-54*e^5*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 1, 11/2}, {2, 2, 2}, 1 + e/(d*x^(2/3))] + Log
[d + e/x^(2/3)]*(27*e^5*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 11/2}, {2, 2}, 1 + e/(d*x^(2/3))] + 2*d*x^(2/
3)*(e^5 + d^5*Sqrt[-(e/(d*x^(2/3)))]*x^(10/3))*Log[d + e/x^(2/3)]))))/(6*d^6*Sqrt[-(e/(d*x^(2/3)))]*x) - (b^2*
n^2*(-9*e^5*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 1, 11/2}, {2, 2, 2}, 1 + e/(d*x^(2/3))] + Log[d + e/x^(2/
3)]*(9*e^5*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 11/2}, {2, 2}, 1 + e/(d*x^(2/3))] + d*x^(2/3)*(e^5 + d^5*S
qrt[-(e/(d*x^(2/3)))]*x^(10/3))*Log[d + e/x^(2/3)]))*(-a + b*n*Log[d + e/x^(2/3)] - b*Log[c*(d + e/x^(2/3))^n]
))/(d^6*Sqrt[-(e/(d*x^(2/3)))]*x) - (2*b*e^4*n*x^(1/3)*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n
])^2)/d^4 + (2*b*e^3*n*x*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2)/(3*d^3) - (2*b*e^2*n*x^(
5/3)*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2)/(5*d^2) + (2*b*e*n*x^(7/3)*(a - b*n*Log[d +
e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2)/(7*d) + (2*b*e^(9/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a - b*n*
Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2)/d^(9/2) + b*n*x^3*Log[d + e/x^(2/3)]*(a - b*n*Log[d + e/x^
(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2 + (x^3*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^3)/3

Integral number [529] \[ \int \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3 \, dx \]

[A]   time = 5.68845 (sec), size = 824 ,normalized size = 1.12 \[ \frac{b^3 \sqrt [3]{x} \left (\sqrt{d} \left (6 e+d x^{2/3} \log \left (d+\frac{e}{x^{2/3}}\right )\right ) \log ^2\left (d+\frac{e}{x^{2/3}}\right )-6 e \sqrt{\frac{e}{x^{2/3} d+e}} \left (8 \sqrt{d} \, _4F_3\left (\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{d}{d+\frac{e}{x^{2/3}}}\right )+\log \left (d+\frac{e}{x^{2/3}}\right ) \left (4 \sqrt{d} \, _3F_2\left (\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{d}{d+\frac{e}{x^{2/3}}}\right )+\sqrt{d+\frac{e}{x^{2/3}}} \sin ^{-1}\left (\frac{\sqrt{d}}{\sqrt{d+\frac{e}{x^{2/3}}}}\right ) \log \left (d+\frac{e}{x^{2/3}}\right )\right )\right )+6 \sqrt{d} e \left (\frac{4 \sqrt{\frac{e}{x^{2/3}}} \tanh ^{-1}\left (\frac{\sqrt{\frac{e}{x^{2/3}}}}{\sqrt{-d}}\right ) \left (\log \left (d+\frac{e}{x^{2/3}}\right )-\log \left (\frac{e}{d x^{2/3}}+1\right )\right )}{\sqrt{-d}}-\sqrt{-\frac{e}{d x^{2/3}}} \left (2 \log ^2\left (\frac{1}{2} \left (\sqrt{-\frac{e}{d x^{2/3}}}+1\right )\right )-4 \log \left (\frac{e}{d x^{2/3}}+1\right ) \log \left (\frac{1}{2} \left (\sqrt{-\frac{e}{d x^{2/3}}}+1\right )\right )+\log ^2\left (\frac{e}{d x^{2/3}}+1\right )-4 \text{PolyLog}\left (2,\frac{1}{2}-\frac{1}{2} \sqrt{-\frac{e}{d x^{2/3}}}\right )\right )\right )\right ) n^3}{d^{3/2}}+\frac{3 b^2 \left (-3 \left (x^{2/3} d+e\right ) \, _4F_3\left (1,1,1,\frac{5}{2};2,2,2;\frac{e}{d x^{2/3}}+1\right ) e^2-d x^{2/3} \log \left (d+\frac{e}{x^{2/3}}\right ) \left (4 \log \left (\frac{1}{2} \left (\sqrt{-\frac{e}{d x^{2/3}}}+1\right )\right ) e^2+4 \left (e-\frac{e}{\sqrt{-\frac{e}{d x^{2/3}}}}\right ) e+\left (d^2 \sqrt{-\frac{e}{d x^{2/3}}} x^{4/3}-e^2\right ) \log \left (d+\frac{e}{x^{2/3}}\right )\right )\right ) \left (-a+b n \log \left (d+\frac{e}{x^{2/3}}\right )-b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) n^2}{d^3 \sqrt{-\frac{e}{d x^{2/3}}} x}-\frac{6 b e^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 n}{d^{3/2}}+3 b x \log \left (d+\frac{e}{x^{2/3}}\right ) \left (a-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 n+\frac{6 b e \sqrt [3]{x} \left (a-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 n}{d}+x \left (a-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3 \]

[In]

Integrate[(a + b*Log[c*(d + e/x^(2/3))^n])^3,x]

[Out]

(3*b^2*n^2*(-3*e^2*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 1, 5/2}, {2, 2, 2}, 1 + e/(d*x^(2/3))] - d*x^(2/3)
*Log[d + e/x^(2/3)]*(4*e*(e - e/Sqrt[-(e/(d*x^(2/3)))]) + 4*e^2*Log[(1 + Sqrt[-(e/(d*x^(2/3)))])/2] + (-e^2 +
d^2*Sqrt[-(e/(d*x^(2/3)))]*x^(4/3))*Log[d + e/x^(2/3)]))*(-a + b*n*Log[d + e/x^(2/3)] - b*Log[c*(d + e/x^(2/3)
)^n]))/(d^3*Sqrt[-(e/(d*x^(2/3)))]*x) + (6*b*e*n*x^(1/3)*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))
^n])^2)/d - (6*b*e^(3/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2
/3))^n])^2)/d^(3/2) + 3*b*n*x*Log[d + e/x^(2/3)]*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2 +
 x*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^3 + (b^3*n^3*x^(1/3)*(Sqrt[d]*Log[d + e/x^(2/3)]^
2*(6*e + d*x^(2/3)*Log[d + e/x^(2/3)]) - 6*e*Sqrt[e/(e + d*x^(2/3))]*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1
/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e/x^(2/3))] + Log[d + e/x^(2/3)]*(4*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2
}, {3/2, 3/2}, d/(d + e/x^(2/3))] + Sqrt[d + e/x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e/x^(2/3)]]*Log[d + e/x^(2/3)]
)) + 6*Sqrt[d]*e*((4*Sqrt[e/x^(2/3)]*ArcTanh[Sqrt[e/x^(2/3)]/Sqrt[-d]]*(Log[d + e/x^(2/3)] - Log[1 + e/(d*x^(2
/3))]))/Sqrt[-d] - Sqrt[-(e/(d*x^(2/3)))]*(2*Log[(1 + Sqrt[-(e/(d*x^(2/3)))])/2]^2 - 4*Log[(1 + Sqrt[-(e/(d*x^
(2/3)))])/2]*Log[1 + e/(d*x^(2/3))] + Log[1 + e/(d*x^(2/3))]^2 - 4*PolyLog[2, 1/2 - Sqrt[-(e/(d*x^(2/3)))]/2])
)))/d^(3/2)

Integral number [530] \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x^2} \, dx \]

[B]   time = 2.23848 (sec), size = 1097 ,normalized size = 2.28 \[ \frac{b^3 \left (18 \left (x^{2/3} d+e\right ) \, _5F_4\left (-\frac{1}{2},1,1,1,1;2,2,2,2;\frac{e}{d x^{2/3}}+1\right )-\log \left (d+\frac{e}{x^{2/3}}\right ) \left (18 \left (x^{2/3} d+e\right ) \, _4F_3\left (-\frac{1}{2},1,1,1;2,2,2;\frac{e}{d x^{2/3}}+1\right )+\log \left (d+\frac{e}{x^{2/3}}\right ) \left (2 \left (x^{2/3} d+e \sqrt{-\frac{e}{d x^{2/3}}}\right ) \log \left (d+\frac{e}{x^{2/3}}\right )-9 \left (x^{2/3} d+e\right ) \, _3F_2\left (-\frac{1}{2},1,1;2,2;\frac{e}{d x^{2/3}}+1\right )\right )\right )\right ) n^3}{2 e \sqrt{-\frac{e}{d x^{2/3}}} x}+\frac{b^2 \left (-a+b n \log \left (d+\frac{e}{x^{2/3}}\right )-b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \left (9 e^{3/2} \log ^2\left (d+\frac{e}{x^{2/3}}\right )-12 e^{3/2} \log \left (d+\frac{e}{x^{2/3}}\right )+18 \sqrt{-d} d x \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (d+\frac{e}{x^{2/3}}\right )+18 (-d)^{3/2} x \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right ) \log \left (d+\frac{e}{x^{2/3}}\right )+36 d \sqrt{e} x^{2/3} \log \left (d+\frac{e}{x^{2/3}}\right )+9 (-d)^{3/2} x \log ^2\left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )+9 \sqrt{-d} d x \log ^2\left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right )+8 e^{3/2}+96 d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{e}}{\sqrt{d} \sqrt [3]{x}}\right )+18 \sqrt{-d} d x \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )+18 (-d)^{3/2} x \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2} \left (\frac{\sqrt [3]{x} \sqrt{-d}}{\sqrt{e}}+1\right )\right )+36 (-d)^{3/2} x \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right ) \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )+36 \sqrt{-d} d x \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )+36 \sqrt{-d} d x \text{PolyLog}\left (2,1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )+18 (-d)^{3/2} x \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )+18 \sqrt{-d} d x \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt [3]{x} \sqrt{-d}}{\sqrt{e}}+1\right )\right )+36 (-d)^{3/2} x \text{PolyLog}\left (2,\frac{\sqrt [3]{x} \sqrt{-d}}{\sqrt{e}}+1\right )-96 d \sqrt{e} x^{2/3}\right ) n^2}{3 e^{3/2} x}-\frac{6 b d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 n}{e^{3/2}}-\frac{3 b \log \left (d+\frac{e}{x^{2/3}}\right ) \left (a-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 n}{x}-\frac{6 b d \left (a-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 n}{e \sqrt [3]{x}}-\frac{\left (a-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 \left (a-2 b n-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{x} \]

[In]

Integrate[(a + b*Log[c*(d + e/x^(2/3))^n])^3/x^2,x]

[Out]

(b^3*n^3*(18*(e + d*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + e/(d*x^(2/3))] - Log[d +
e/x^(2/3)]*(18*(e + d*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, 1 + e/(d*x^(2/3))] + Log[d + e/x^
(2/3)]*(-9*(e + d*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1}, {2, 2}, 1 + e/(d*x^(2/3))] + 2*(e*Sqrt[-(e/(d*x^(2/
3)))] + d*x^(2/3))*Log[d + e/x^(2/3)]))))/(2*e*Sqrt[-(e/(d*x^(2/3)))]*x) - (6*b*d*n*(a - b*n*Log[d + e/x^(2/3)
] + b*Log[c*(d + e/x^(2/3))^n])^2)/(e*x^(1/3)) - (6*b*d^(3/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a - b*n*Log
[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2)/e^(3/2) - (3*b*n*Log[d + e/x^(2/3)]*(a - b*n*Log[d + e/x^(2/3
)] + b*Log[c*(d + e/x^(2/3))^n])^2)/x - ((a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2*(a - 2*b*
n - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n]))/x + (b^2*n^2*(-a + b*n*Log[d + e/x^(2/3)] - b*Log[c*
(d + e/x^(2/3))^n])*(8*e^(3/2) - 96*d*Sqrt[e]*x^(2/3) + 96*d^(3/2)*x*ArcTan[Sqrt[e]/(Sqrt[d]*x^(1/3))] - 12*e^
(3/2)*Log[d + e/x^(2/3)] + 36*d*Sqrt[e]*x^(2/3)*Log[d + e/x^(2/3)] + 9*e^(3/2)*Log[d + e/x^(2/3)]^2 + 18*Sqrt[
-d]*d*x*Log[d + e/x^(2/3)]*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)] + 9*(-d)^(3/2)*x*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]^2
+ 18*(-d)^(3/2)*x*Log[d + e/x^(2/3)]*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] + 9*Sqrt[-d]*d*x*Log[Sqrt[e] + Sqrt[-d]*x
^(1/3)]^2 + 18*Sqrt[-d]*d*x*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])] + 18*(-d
)^(3/2)*x*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2] + 36*(-d)^(3/2)*x*Log[Sqrt[e
] + Sqrt[-d]*x^(1/3)]*Log[-((Sqrt[-d]*x^(1/3))/Sqrt[e])] + 36*Sqrt[-d]*d*x*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log
[(Sqrt[-d]*x^(1/3))/Sqrt[e]] + 36*Sqrt[-d]*d*x*PolyLog[2, 1 - (Sqrt[-d]*x^(1/3))/Sqrt[e]] + 18*(-d)^(3/2)*x*Po
lyLog[2, 1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])] + 18*Sqrt[-d]*d*x*PolyLog[2, (1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2
] + 36*(-d)^(3/2)*x*PolyLog[2, 1 + (Sqrt[-d]*x^(1/3))/Sqrt[e]]))/(3*e^(3/2)*x)

Integral number [531] \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x^4} \, dx \]

[B]   time = 9.0079 (sec), size = 2858 ,normalized size = 3.65 \[ \text{Result too large to show} \]

[In]

Integrate[(a + b*Log[c*(d + e/x^(2/3))^n])^3/x^4,x]

[Out]

-(b^3*n^3*(32*d^4 - 32*d^4*Sqrt[1 - (d + e/x^(2/3))/d] + 128*d^3*Sqrt[1 - (d + e/x^(2/3))/d]*(d + e/x^(2/3)) -
 192*d^2*Sqrt[1 - (d + e/x^(2/3))/d]*(d + e/x^(2/3))^2 + 128*d*Sqrt[1 - (d + e/x^(2/3))/d]*(d + e/x^(2/3))^3 -
 32*Sqrt[1 - (d + e/x^(2/3))/d]*(d + e/x^(2/3))^4 + 1584*d^3*(d + e/x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1, 1}
, {2, 2, 2}, (d + e/x^(2/3))/d] - 4536*d^3*(d + e/x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1, 1}, {2, 2, 2}, (d +
e/x^(2/3))/d] + 3780*d^3*(d + e/x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, (d + e/x^(2/3))/d] - 86
4*d^3*(d + e/x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e/x^(2/3))/d] + 3024*d^3*(d + e
/x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e/x^(2/3))/d] - 3780*d^3*(d + e/x^(2/3))*Hy
pergeometricPFQ[{-3/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e/x^(2/3))/d] + 1890*d^3*(d + e/x^(2/3))*Hypergeometric
PFQ[{-1/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e/x^(2/3))/d] - 240*d^4*Log[d + e/x^(2/3)] + 240*d^4*Sqrt[1 - (d +
e/x^(2/3))/d]*Log[d + e/x^(2/3)] - 672*d^3*Sqrt[1 - (d + e/x^(2/3))/d]*(d + e/x^(2/3))*Log[d + e/x^(2/3)] + 57
6*d^2*Sqrt[1 - (d + e/x^(2/3))/d]*(d + e/x^(2/3))^2*Log[d + e/x^(2/3)] - 96*d*Sqrt[1 - (d + e/x^(2/3))/d]*(d +
 e/x^(2/3))^3*Log[d + e/x^(2/3)] - 48*Sqrt[1 - (d + e/x^(2/3))/d]*(d + e/x^(2/3))^4*Log[d + e/x^(2/3)] - 3780*
d^3*(d + e/x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1}, {2, 2}, (d + e/x^(2/3))/d]*Log[d + e/x^(2/3)] + 864*d^3*(d
 + e/x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1, 1}, {2, 2, 2}, (d + e/x^(2/3))/d]*Log[d + e/x^(2/3)] - 3024*d^3*(
d + e/x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1, 1}, {2, 2, 2}, (d + e/x^(2/3))/d]*Log[d + e/x^(2/3)] + 3780*d^3*
(d + e/x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, (d + e/x^(2/3))/d]*Log[d + e/x^(2/3)] - 1890*d^3
*(d + e/x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, (d + e/x^(2/3))/d]*Log[d + e/x^(2/3)] + 284*d^4
*Log[d + e/x^(2/3)]^2 - 284*d^4*Sqrt[1 - (d + e/x^(2/3))/d]*Log[d + e/x^(2/3)]^2 + 668*d^3*Sqrt[1 - (d + e/x^(
2/3))/d]*(d + e/x^(2/3))*Log[d + e/x^(2/3)]^2 - 552*d^2*Sqrt[1 - (d + e/x^(2/3))/d]*(d + e/x^(2/3))^2*Log[d +
e/x^(2/3)]^2 + 236*d*Sqrt[1 - (d + e/x^(2/3))/d]*(d + e/x^(2/3))^3*Log[d + e/x^(2/3)]^2 - 68*Sqrt[1 - (d + e/x
^(2/3))/d]*(d + e/x^(2/3))^4*Log[d + e/x^(2/3)]^2 - 1890*d^3*(d + e/x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1}, {
2, 2}, (d + e/x^(2/3))/d]*Log[d + e/x^(2/3)]^2 + 945*d^3*(d + e/x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1}, {2, 2
}, (d + e/x^(2/3))/d]*Log[d + e/x^(2/3)]^2 - 70*d^4*Log[d + e/x^(2/3)]^3 + 70*d^4*Sqrt[1 - (d + e/x^(2/3))/d]*
Log[d + e/x^(2/3)]^3 - 280*d^3*Sqrt[1 - (d + e/x^(2/3))/d]*(d + e/x^(2/3))*Log[d + e/x^(2/3)]^3 + 420*d^2*Sqrt
[1 - (d + e/x^(2/3))/d]*(d + e/x^(2/3))^2*Log[d + e/x^(2/3)]^3 - 280*d*Sqrt[1 - (d + e/x^(2/3))/d]*(d + e/x^(2
/3))^3*Log[d + e/x^(2/3)]^3 + 70*Sqrt[1 - (d + e/x^(2/3))/d]*(d + e/x^(2/3))^4*Log[d + e/x^(2/3)]^3 + 1512*d^3
*(d + e/x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1}, {2, 2}, (d + e/x^(2/3))/d]*(1 + 3*Log[d + e/x^(2/3)] + Log[d
+ e/x^(2/3)]^2) - 144*d^3*(d + e/x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1}, {2, 2}, (d + e/x^(2/3))/d]*(6 + 11*L
og[d + e/x^(2/3)] + 3*Log[d + e/x^(2/3)]^2)))/(210*e^4*Sqrt[1 - (d + e/x^(2/3))/d]*x^(1/3)) - (2*b*d*n*(a + b*
(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(7*e*x^(7/3)) + (2*b*d^2*n*(a + b*(-(n*Log[d + e/x^(2
/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(5*e^2*x^(5/3)) - (2*b*d^3*n*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d
+ e/x^(2/3))^n]))^2)/(3*e^3*x) + (2*b*d^4*n*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(e
^4*x^(1/3)) + (2*b*d^(9/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/
x^(2/3))^n]))^2)/e^(9/2) - (b*n*Log[d + e/x^(2/3)]*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n])
)^2)/x^3 - ((a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2*(3*a - 2*b*n + 3*b*(-(n*Log[d + e/x
^(2/3)]) + Log[c*(d + e/x^(2/3))^n])))/(9*x^3) + 9*b^2*n^2*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2
/3))^n]))*(-Log[(e + d*x^(2/3))/x^(2/3)]^2/(9*x^3) + (-9800*e^(9/2) + 28800*d*e^(7/2)*x^(2/3) - 72072*d^2*e^(5
/2)*x^(4/3) + 208320*d^3*e^(3/2)*x^2 - 1418760*d^4*Sqrt[e]*x^(8/3) + 1418760*d^(9/2)*x^3*ArcTan[Sqrt[e]/(Sqrt[
d]*x^(1/3))] + 44100*e^(9/2)*Log[d + e/x^(2/3)] - 56700*d*e^(7/2)*x^(2/3)*Log[d + e/x^(2/3)] + 79380*d^2*e^(5/
2)*x^(4/3)*Log[d + e/x^(2/3)] - 132300*d^3*e^(3/2)*x^2*Log[d + e/x^(2/3)] + 396900*d^4*Sqrt[e]*x^(8/3)*Log[d +
 e/x^(2/3)] + 198450*(-d)^(9/2)*x^3*Log[d + e/x^(2/3)]*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)] - 99225*(-d)^(9/2)*x^3*
Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]^2 - 198450*(-d)^(9/2)*x^3*Log[d + e/x^(2/3)]*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] +
 99225*(-d)^(9/2)*x^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]^2 + 198450*(-d)^(9/2)*x^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)
]*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])] - 198450*(-d)^(9/2)*x^3*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(1 + (
Sqrt[-d]*x^(1/3))/Sqrt[e])/2] - 396900*(-d)^(9/2)*x^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[-((Sqrt[-d]*x^(1/3))
/Sqrt[e])] + 396900*(-d)^(9/2)*x^3*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(Sqrt[-d]*x^(1/3))/Sqrt[e]] + 396900*(-
d)^(9/2)*x^3*PolyLog[2, 1 - (Sqrt[-d]*x^(1/3))/Sqrt[e]] - 198450*(-d)^(9/2)*x^3*PolyLog[2, 1/2 - (Sqrt[-d]*x^(
1/3))/(2*Sqrt[e])] + 198450*(-d)^(9/2)*x^3*PolyLog[2, (1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2] - 396900*(-d)^(9/2)*
x^3*PolyLog[2, 1 + (Sqrt[-d]*x^(1/3))/Sqrt[e]])/(893025*e^(9/2)*x^3))

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