3.98 \(\int x^2 \log ^3(c (a+b x^2)^p) \, dx\)

Optimal. Leaf size=379 \[ \frac{32 i a^{3/2} p^3 \text{PolyLog}\left (2,1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{3 b^{3/2}}-\frac{2 a^2 p \text{Unintegrable}\left (\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2},x\right )}{b}+\frac{32 a^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}+\frac{32 i a^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{3 b^{3/2}}-\frac{208 a^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{9 b^{3/2}}+\frac{64 a^{3/2} p^3 \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 b^{3/2}}+\frac{8}{9} p^2 x^3 \log \left (c \left (a+b x^2\right )^p\right )-\frac{32 a p^2 x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}-\frac{2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}+\frac{1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )+\frac{208 a p^3 x}{9 b}-\frac{16}{27} p^3 x^3 \]

[Out]

(208*a*p^3*x)/(9*b) - (16*p^3*x^3)/27 - (208*a^(3/2)*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(9*b^(3/2)) + (((32*I)/3
)*a^(3/2)*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/b^(3/2) + (64*a^(3/2)*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt
[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(3*b^(3/2)) - (32*a*p^2*x*Log[c*(a + b*x^2)^p])/(3*b) + (8*p^2*x^3*Log[c*(a + b
*x^2)^p])/9 + (32*a^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/(3*b^(3/2)) + (2*a*p*x*Log[c*(
a + b*x^2)^p]^2)/b - (2*p*x^3*Log[c*(a + b*x^2)^p]^2)/3 + (x^3*Log[c*(a + b*x^2)^p]^3)/3 + (((32*I)/3)*a^(3/2)
*p^3*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/b^(3/2) - (2*a^2*p*Unintegrable[Log[c*(a + b*x^2)^p]
^2/(a + b*x^2), x])/b

________________________________________________________________________________________

Rubi [A]  time = 0.833947, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*Log[c*(a + b*x^2)^p]^3,x]

[Out]

(208*a*p^3*x)/(9*b) - (16*p^3*x^3)/27 - (208*a^(3/2)*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(9*b^(3/2)) + (((32*I)/3
)*a^(3/2)*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/b^(3/2) + (64*a^(3/2)*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt
[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(3*b^(3/2)) - (32*a*p^2*x*Log[c*(a + b*x^2)^p])/(3*b) + (8*p^2*x^3*Log[c*(a + b
*x^2)^p])/9 + (32*a^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/(3*b^(3/2)) + (2*a*p*x*Log[c*(
a + b*x^2)^p]^2)/b - (2*p*x^3*Log[c*(a + b*x^2)^p]^2)/3 + (x^3*Log[c*(a + b*x^2)^p]^3)/3 + (((32*I)/3)*a^(3/2)
*p^3*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/b^(3/2) - (2*a^2*p*Defer[Int][Log[c*(a + b*x^2)^p]^2
/(a + b*x^2), x])/b

Rubi steps

\begin{align*} \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac{1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-(2 b p) \int \frac{x^4 \log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ &=\frac{1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-(2 b p) \int \left (-\frac{a \log ^2\left (c \left (a+b x^2\right )^p\right )}{b^2}+\frac{x^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}+\frac{a^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{b^2 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac{1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-(2 p) \int x^2 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx+\frac{(2 a p) \int \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx}{b}-\frac{\left (2 a^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}\\ &=\frac{2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac{2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac{\left (2 a^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}-\left (8 a p^2\right ) \int \frac{x^2 \log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\frac{1}{3} \left (8 b p^2\right ) \int \frac{x^4 \log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ &=\frac{2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac{2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac{\left (2 a^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}-\left (8 a p^2\right ) \int \left (\frac{\log \left (c \left (a+b x^2\right )^p\right )}{b}-\frac{a \log \left (c \left (a+b x^2\right )^p\right )}{b \left (a+b x^2\right )}\right ) \, dx+\frac{1}{3} \left (8 b p^2\right ) \int \left (-\frac{a \log \left (c \left (a+b x^2\right )^p\right )}{b^2}+\frac{x^2 \log \left (c \left (a+b x^2\right )^p\right )}{b}+\frac{a^2 \log \left (c \left (a+b x^2\right )^p\right )}{b^2 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac{2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac{2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac{\left (2 a^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}+\frac{1}{3} \left (8 p^2\right ) \int x^2 \log \left (c \left (a+b x^2\right )^p\right ) \, dx-\frac{\left (8 a p^2\right ) \int \log \left (c \left (a+b x^2\right )^p\right ) \, dx}{3 b}-\frac{\left (8 a p^2\right ) \int \log \left (c \left (a+b x^2\right )^p\right ) \, dx}{b}+\frac{\left (8 a^2 p^2\right ) \int \frac{\log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{3 b}+\frac{\left (8 a^2 p^2\right ) \int \frac{\log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}\\ &=-\frac{32 a p^2 x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}+\frac{8}{9} p^2 x^3 \log \left (c \left (a+b x^2\right )^p\right )+\frac{32 a^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}+\frac{2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac{2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac{\left (2 a^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}+\frac{1}{3} \left (16 a p^3\right ) \int \frac{x^2}{a+b x^2} \, dx+\left (16 a p^3\right ) \int \frac{x^2}{a+b x^2} \, dx-\frac{1}{3} \left (16 a^2 p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \left (a+b x^2\right )} \, dx-\left (16 a^2 p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \left (a+b x^2\right )} \, dx-\frac{1}{9} \left (16 b p^3\right ) \int \frac{x^4}{a+b x^2} \, dx\\ &=\frac{64 a p^3 x}{3 b}-\frac{32 a p^2 x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}+\frac{8}{9} p^2 x^3 \log \left (c \left (a+b x^2\right )^p\right )+\frac{32 a^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}+\frac{2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac{2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac{\left (2 a^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}-\frac{\left (16 a^2 p^3\right ) \int \frac{1}{a+b x^2} \, dx}{3 b}-\frac{\left (16 a^2 p^3\right ) \int \frac{1}{a+b x^2} \, dx}{b}-\frac{\left (16 a^{3/2} p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a+b x^2} \, dx}{3 \sqrt{b}}-\frac{\left (16 a^{3/2} p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a+b x^2} \, dx}{\sqrt{b}}-\frac{1}{9} \left (16 b p^3\right ) \int \left (-\frac{a}{b^2}+\frac{x^2}{b}+\frac{a^2}{b^2 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac{208 a p^3 x}{9 b}-\frac{16 p^3 x^3}{27}-\frac{64 a^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 b^{3/2}}+\frac{32 i a^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{3 b^{3/2}}-\frac{32 a p^2 x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}+\frac{8}{9} p^2 x^3 \log \left (c \left (a+b x^2\right )^p\right )+\frac{32 a^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}+\frac{2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac{2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac{\left (2 a^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}+\frac{\left (16 a p^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{i-\frac{\sqrt{b} x}{\sqrt{a}}} \, dx}{3 b}+\frac{\left (16 a p^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{i-\frac{\sqrt{b} x}{\sqrt{a}}} \, dx}{b}-\frac{\left (16 a^2 p^3\right ) \int \frac{1}{a+b x^2} \, dx}{9 b}\\ &=\frac{208 a p^3 x}{9 b}-\frac{16 p^3 x^3}{27}-\frac{208 a^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{9 b^{3/2}}+\frac{32 i a^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{3 b^{3/2}}+\frac{64 a^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{3 b^{3/2}}-\frac{32 a p^2 x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}+\frac{8}{9} p^2 x^3 \log \left (c \left (a+b x^2\right )^p\right )+\frac{32 a^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}+\frac{2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac{2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac{\left (2 a^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}-\frac{\left (16 a p^3\right ) \int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{b} x}{\sqrt{a}}}\right )}{1+\frac{b x^2}{a}} \, dx}{3 b}-\frac{\left (16 a p^3\right ) \int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{b} x}{\sqrt{a}}}\right )}{1+\frac{b x^2}{a}} \, dx}{b}\\ &=\frac{208 a p^3 x}{9 b}-\frac{16 p^3 x^3}{27}-\frac{208 a^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{9 b^{3/2}}+\frac{32 i a^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{3 b^{3/2}}+\frac{64 a^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{3 b^{3/2}}-\frac{32 a p^2 x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}+\frac{8}{9} p^2 x^3 \log \left (c \left (a+b x^2\right )^p\right )+\frac{32 a^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}+\frac{2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac{2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac{\left (2 a^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}+\frac{\left (16 i a^{3/2} p^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{b} x}{\sqrt{a}}}\right )}{3 b^{3/2}}+\frac{\left (16 i a^{3/2} p^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{b} x}{\sqrt{a}}}\right )}{b^{3/2}}\\ &=\frac{208 a p^3 x}{9 b}-\frac{16 p^3 x^3}{27}-\frac{208 a^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{9 b^{3/2}}+\frac{32 i a^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{3 b^{3/2}}+\frac{64 a^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{3 b^{3/2}}-\frac{32 a p^2 x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}+\frac{8}{9} p^2 x^3 \log \left (c \left (a+b x^2\right )^p\right )+\frac{32 a^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}+\frac{2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac{2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )+\frac{32 i a^{3/2} p^3 \text{Li}_2\left (1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{3 b^{3/2}}-\frac{\left (2 a^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}\\ \end{align*}

Mathematica [A]  time = 3.753, 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 ) \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 13.369, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( \ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*ln(c*(b*x^2+a)^p)^3,x)

[Out]

int(x^2*ln(c*(b*x^2+a)^p)^3,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*log(c*(b*x^2+a)^p)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*log(c*(b*x^2+a)^p)^3,x, algorithm="fricas")

[Out]

integral(x^2*log((b*x^2 + a)^p*c)^3, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \log{\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*ln(c*(b*x**2+a)**p)**3,x)

[Out]

Integral(x**2*log(c*(a + b*x**2)**p)**3, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*log(c*(b*x^2+a)^p)^3,x, algorithm="giac")

[Out]

integrate(x^2*log((b*x^2 + a)^p*c)^3, x)