3.266 \(\int \frac{(d+e x^2)^{3/2} (a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=260 \[ -\frac{1}{2} b d^{3/2} n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )+\frac{1}{3} \left (-3 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )+3 d \sqrt{d+e x^2}+\left (d+e x^2\right )^{3/2}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{1}{2} b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2+\frac{4}{3} b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )-b d^{3/2} n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )-\frac{1}{9} b n \left (d+e x^2\right )^{3/2}-\frac{4}{3} b d n \sqrt{d+e x^2} \]

[Out]

(-4*b*d*n*Sqrt[d + e*x^2])/3 - (b*n*(d + e*x^2)^(3/2))/9 + (4*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/3
+ (b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]^2)/2 + ((3*d*Sqrt[d + e*x^2] + (d + e*x^2)^(3/2) - 3*d^(3/2)*A
rcTanh[Sqrt[d + e*x^2]/Sqrt[d]])*(a + b*Log[c*x^n]))/3 - b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*Log[(2*S
qrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])] - (b*d^(3/2)*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/2

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Rubi [A]  time = 0.392062, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {266, 50, 63, 208, 2348, 5984, 5918, 2402, 2315} \[ -\frac{1}{2} b d^{3/2} n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )+\frac{1}{3} \left (-3 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )+3 d \sqrt{d+e x^2}+\left (d+e x^2\right )^{3/2}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{1}{2} b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2+\frac{4}{3} b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )-b d^{3/2} n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )-\frac{1}{9} b n \left (d+e x^2\right )^{3/2}-\frac{4}{3} b d n \sqrt{d+e x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b*d*n*Sqrt[d + e*x^2])/3 - (b*n*(d + e*x^2)^(3/2))/9 + (4*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/3
+ (b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]^2)/2 + ((3*d*Sqrt[d + e*x^2] + (d + e*x^2)^(3/2) - 3*d^(3/2)*A
rcTanh[Sqrt[d + e*x^2]/Sqrt[d]])*(a + b*Log[c*x^n]))/3 - b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*Log[(2*S
qrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])] - (b*d^(3/2)*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/2

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 2348

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.))/(x_), x_Symbol] :> With[{u = IntHi
de[(d + e*x^r)^q/x, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[Dist[1/x, u, x], x], x]] /; FreeQ[{a, b
, c, d, e, n, r}, x] && IntegerQ[q - 1/2]

Rule 5984

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 5918

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTanh[c*x])^p*
Log[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTanh[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 - c^2
*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{1}{3} \left (3 d \sqrt{d+e x^2}+\left (d+e x^2\right )^{3/2}-3 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac{d \sqrt{d+e x^2}}{x}+\frac{\left (d+e x^2\right )^{3/2}}{3 x}-\frac{d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{x}\right ) \, dx\\ &=\frac{1}{3} \left (3 d \sqrt{d+e x^2}+\left (d+e x^2\right )^{3/2}-3 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{3} (b n) \int \frac{\left (d+e x^2\right )^{3/2}}{x} \, dx-(b d n) \int \frac{\sqrt{d+e x^2}}{x} \, dx+\left (b d^{3/2} n\right ) \int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{x} \, dx\\ &=\frac{1}{3} \left (3 d \sqrt{d+e x^2}+\left (d+e x^2\right )^{3/2}-3 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{6} (b n) \operatorname{Subst}\left (\int \frac{(d+e x)^{3/2}}{x} \, dx,x,x^2\right )-\frac{1}{2} (b d n) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x}}{x} \, dx,x,x^2\right )+\frac{1}{2} \left (b d^{3/2} n\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{x} \, dx,x,x^2\right )\\ &=-b d n \sqrt{d+e x^2}-\frac{1}{9} b n \left (d+e x^2\right )^{3/2}+\frac{1}{3} \left (3 d \sqrt{d+e x^2}+\left (d+e x^2\right )^{3/2}-3 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{6} (b d n) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x}}{x} \, dx,x,x^2\right )+\left (b d^{3/2} n\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{x}{\sqrt{d}}\right )}{-d+x^2} \, dx,x,\sqrt{d+e x^2}\right )-\frac{1}{2} \left (b d^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )\\ &=-\frac{4}{3} b d n \sqrt{d+e x^2}-\frac{1}{9} b n \left (d+e x^2\right )^{3/2}+\frac{1}{2} b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2+\frac{1}{3} \left (3 d \sqrt{d+e x^2}+\left (d+e x^2\right )^{3/2}-3 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-(b d n) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{d}}\right )}{1-\frac{x}{\sqrt{d}}} \, dx,x,\sqrt{d+e x^2}\right )-\frac{1}{6} \left (b d^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )-\frac{\left (b d^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{e}\\ &=-\frac{4}{3} b d n \sqrt{d+e x^2}-\frac{1}{9} b n \left (d+e x^2\right )^{3/2}+b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )+\frac{1}{2} b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2+\frac{1}{3} \left (3 d \sqrt{d+e x^2}+\left (d+e x^2\right )^{3/2}-3 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )+(b d n) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-\frac{x}{\sqrt{d}}}\right )}{1-\frac{x^2}{d}} \, dx,x,\sqrt{d+e x^2}\right )-\frac{\left (b d^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{3 e}\\ &=-\frac{4}{3} b d n \sqrt{d+e x^2}-\frac{1}{9} b n \left (d+e x^2\right )^{3/2}+\frac{4}{3} b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )+\frac{1}{2} b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2+\frac{1}{3} \left (3 d \sqrt{d+e x^2}+\left (d+e x^2\right )^{3/2}-3 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )-\left (b d^{3/2} n\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{\sqrt{d+e x^2}}{\sqrt{d}}}\right )\\ &=-\frac{4}{3} b d n \sqrt{d+e x^2}-\frac{1}{9} b n \left (d+e x^2\right )^{3/2}+\frac{4}{3} b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )+\frac{1}{2} b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2+\frac{1}{3} \left (3 d \sqrt{d+e x^2}+\left (d+e x^2\right )^{3/2}-3 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )-\frac{1}{2} b d^{3/2} n \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{d+e x^2}}{\sqrt{d}}}\right )\\ \end{align*}

Mathematica [C]  time = 0.767813, size = 301, normalized size = 1.16 \[ \frac{b e n x^2 \sqrt{d+e x^2} \left (\frac{d \log (x) \left (\left (\frac{e x^2}{d}+1\right )^{3/2}-1\right )}{3 e x^2}-\frac{1}{4} \, _3F_2\left (-\frac{1}{2},1,1;2,2;-\frac{e x^2}{d}\right )\right )}{\sqrt{\frac{e x^2}{d}+1}}+\frac{b d n \sqrt{d+e x^2} \left (-\, _3F_2\left (-\frac{1}{2},-\frac{1}{2},-\frac{1}{2};\frac{1}{2},\frac{1}{2};-\frac{d}{e x^2}\right )+\log (x) \sqrt{\frac{d}{e x^2}+1}-\frac{\sqrt{d} \log (x) \sinh ^{-1}\left (\frac{\sqrt{d}}{\sqrt{e} x}\right )}{\sqrt{e} x}\right )}{\sqrt{\frac{d}{e x^2}+1}}-d^{3/2} \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )+d^{3/2} \log (x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )+\frac{1}{3} \sqrt{d+e x^2} \left (4 d+e x^2\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*n*x^2*Sqrt[d + e*x^2]*(-HypergeometricPFQ[{-1/2, 1, 1}, {2, 2}, -((e*x^2)/d)]/4 + (d*(-1 + (1 + (e*x^2)/d
)^(3/2))*Log[x])/(3*e*x^2)))/Sqrt[1 + (e*x^2)/d] + (b*d*n*Sqrt[d + e*x^2]*(-HypergeometricPFQ[{-1/2, -1/2, -1/
2}, {1/2, 1/2}, -(d/(e*x^2))] + Sqrt[1 + d/(e*x^2)]*Log[x] - (Sqrt[d]*ArcSinh[Sqrt[d]/(Sqrt[e]*x)]*Log[x])/(Sq
rt[e]*x)))/Sqrt[1 + d/(e*x^2)] + (Sqrt[d + e*x^2]*(4*d + e*x^2)*(a - b*n*Log[x] + b*Log[c*x^n]))/3 + d^(3/2)*L
og[x]*(a - b*n*Log[x] + b*Log[c*x^n]) - d^(3/2)*(a - b*n*Log[x] + b*Log[c*x^n])*Log[d + Sqrt[d]*Sqrt[d + e*x^2
]]

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Maple [F]  time = 0.406, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{x} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^(3/2)*(a+b*ln(c*x^n))/x,x)

[Out]

int((e*x^2+d)^(3/2)*(a+b*ln(c*x^n))/x,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((e*x^2+d)^(3/2)*(a+b*log(c*x^n))/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b e x^{2} + b d\right )} \sqrt{e x^{2} + d} \log \left (c x^{n}\right ) +{\left (a e x^{2} + a d\right )} \sqrt{e x^{2} + d}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*log(c*x^n))/x,x, algorithm="fricas")

[Out]

integral(((b*e*x^2 + b*d)*sqrt(e*x^2 + d)*log(c*x^n) + (a*e*x^2 + a*d)*sqrt(e*x^2 + d))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**(3/2)*(a+b*ln(c*x**n))/x,x)

[Out]

Integral((a + b*log(c*x**n))*(d + e*x**2)**(3/2)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*log(c*x^n))/x,x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^(3/2)*(b*log(c*x^n) + a)/x, x)