3.299 \(\int (f+g x^3) \log ^3(c (d+e x^2)^p) \, dx\)

Optimal. Leaf size=517 \[ 6 d f p \text{Unintegrable}\left (\frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )-\frac{24 i \sqrt{d} f p^3 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{\sqrt{e}}+\frac{3 g p^2 \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 e^2}-\frac{3 d g p^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac{3 g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{8 e^2}+\frac{3 d g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac{g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\frac{d g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{24 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{3 g p^3 \left (d+e x^2\right )^2}{16 e^2}-\frac{24 i \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{48 \sqrt{d} f p^3 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{3 d g p^3 x^2}{e}-48 f p^3 x \]

[Out]

-48*f*p^3*x + (3*d*g*p^3*x^2)/e - (3*g*p^3*(d + e*x^2)^2)/(16*e^2) + (48*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sqrt
[d]])/Sqrt[e] - ((24*I)*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] - (48*Sqrt[d]*f*p^3*ArcTan[(Sqrt[
e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] + 24*f*p^2*x*Log[c*(d + e*x^2)^p] - (3*d*g*p^
2*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e^2 + (3*g*p^2*(d + e*x^2)^2*Log[c*(d + e*x^2)^p])/(8*e^2) - (24*Sqrt[d]*f
*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] - 6*f*p*x*Log[c*(d + e*x^2)^p]^2 + (3*d*g*p*(d
+ e*x^2)*Log[c*(d + e*x^2)^p]^2)/(2*e^2) - (3*g*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^2)/(8*e^2) + f*x*Log[c*(d
+ e*x^2)^p]^3 - (d*g*(d + e*x^2)*Log[c*(d + e*x^2)^p]^3)/(2*e^2) + (g*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^3)/(
4*e^2) - ((24*I)*Sqrt[d]*f*p^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] + 6*d*f*p*Unintegr
able[Log[c*(d + e*x^2)^p]^2/(d + e*x^2), x]

________________________________________________________________________________________

Rubi [A]  time = 0.767616, 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 = {} \[ \int \left (f+g x^3\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-48*f*p^3*x + (3*d*g*p^3*x^2)/e - (3*g*p^3*(d + e*x^2)^2)/(16*e^2) + (48*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sqrt
[d]])/Sqrt[e] - ((24*I)*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] - (48*Sqrt[d]*f*p^3*ArcTan[(Sqrt[
e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] + 24*f*p^2*x*Log[c*(d + e*x^2)^p] - (3*d*g*p^
2*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e^2 + (3*g*p^2*(d + e*x^2)^2*Log[c*(d + e*x^2)^p])/(8*e^2) - (24*Sqrt[d]*f
*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] - 6*f*p*x*Log[c*(d + e*x^2)^p]^2 + (3*d*g*p*(d
+ e*x^2)*Log[c*(d + e*x^2)^p]^2)/(2*e^2) - (3*g*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^2)/(8*e^2) + f*x*Log[c*(d
+ e*x^2)^p]^3 - (d*g*(d + e*x^2)*Log[c*(d + e*x^2)^p]^3)/(2*e^2) + (g*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^3)/(
4*e^2) - ((24*I)*Sqrt[d]*f*p^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] + 6*d*f*p*Defer[In
t][Log[c*(d + e*x^2)^p]^2/(d + e*x^2), x]

Rubi steps

\begin{align*} \int \left (f+g x^3\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx &=\int \left (f \log ^3\left (c \left (d+e x^2\right )^p\right )+g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f \int \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx+g \int x^3 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} g \operatorname{Subst}\left (\int x \log ^3\left (c (d+e x)^p\right ) \, dx,x,x^2\right )-(6 e f p) \int \frac{x^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx\\ &=f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} g \operatorname{Subst}\left (\int \left (-\frac{d \log ^3\left (c (d+e x)^p\right )}{e}+\frac{(d+e x) \log ^3\left (c (d+e x)^p\right )}{e}\right ) \, dx,x,x^2\right )-(6 e f p) \int \left (\frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{d \log ^2\left (c \left (d+e x^2\right )^p\right )}{e \left (d+e x^2\right )}\right ) \, dx\\ &=f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{g \operatorname{Subst}\left (\int (d+e x) \log ^3\left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{2 e}-\frac{(d g) \operatorname{Subst}\left (\int \log ^3\left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{2 e}-(6 f p) \int \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx\\ &=-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{g \operatorname{Subst}\left (\int x \log ^3\left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}-\frac{(d g) \operatorname{Subst}\left (\int \log ^3\left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx+\left (24 e f p^2\right ) \int \frac{x^2 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx\\ &=-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{d g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac{g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{(3 g p) \operatorname{Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,d+e x^2\right )}{4 e^2}+\frac{(3 d g p) \operatorname{Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}+\left (24 e f p^2\right ) \int \left (\frac{\log \left (c \left (d+e x^2\right )^p\right )}{e}-\frac{d \log \left (c \left (d+e x^2\right )^p\right )}{e \left (d+e x^2\right )}\right ) \, dx\\ &=-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3 d g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac{3 g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{8 e^2}+f x \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{d g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac{g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx+\left (24 f p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx-\left (24 d f p^2\right ) \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx+\frac{\left (3 g p^2\right ) \operatorname{Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{4 e^2}-\frac{\left (3 d g p^2\right ) \operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}\\ &=\frac{3 d g p^3 x^2}{e}-\frac{3 g p^3 \left (d+e x^2\right )^2}{16 e^2}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{3 d g p^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{3 g p^2 \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 e^2}-\frac{24 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3 d g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac{3 g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{8 e^2}+f x \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{d g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac{g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\left (48 e f p^3\right ) \int \frac{x^2}{d+e x^2} \, dx+\left (48 d e f p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} \left (d+e x^2\right )} \, dx\\ &=-48 f p^3 x+\frac{3 d g p^3 x^2}{e}-\frac{3 g p^3 \left (d+e x^2\right )^2}{16 e^2}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{3 d g p^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{3 g p^2 \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 e^2}-\frac{24 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3 d g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac{3 g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{8 e^2}+f x \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{d g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac{g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx+\left (48 d f p^3\right ) \int \frac{1}{d+e x^2} \, dx+\left (48 \sqrt{d} \sqrt{e} f p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx\\ &=-48 f p^3 x+\frac{3 d g p^3 x^2}{e}-\frac{3 g p^3 \left (d+e x^2\right )^2}{16 e^2}+\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{24 i \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{3 d g p^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{3 g p^2 \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 e^2}-\frac{24 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3 d g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac{3 g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{8 e^2}+f x \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{d g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac{g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\left (48 f p^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx\\ &=-48 f p^3 x+\frac{3 d g p^3 x^2}{e}-\frac{3 g p^3 \left (d+e x^2\right )^2}{16 e^2}+\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{24 i \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}-\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{\sqrt{e}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{3 d g p^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{3 g p^2 \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 e^2}-\frac{24 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3 d g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac{3 g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{8 e^2}+f x \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{d g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac{g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx+\left (48 f p^3\right ) \int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{1+\frac{e x^2}{d}} \, dx\\ &=-48 f p^3 x+\frac{3 d g p^3 x^2}{e}-\frac{3 g p^3 \left (d+e x^2\right )^2}{16 e^2}+\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{24 i \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}-\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{\sqrt{e}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{3 d g p^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{3 g p^2 \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 e^2}-\frac{24 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3 d g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac{3 g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{8 e^2}+f x \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{d g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac{g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (48 i \sqrt{d} f p^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{\sqrt{e}}\\ &=-48 f p^3 x+\frac{3 d g p^3 x^2}{e}-\frac{3 g p^3 \left (d+e x^2\right )^2}{16 e^2}+\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{24 i \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}-\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{\sqrt{e}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{3 d g p^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{3 g p^2 \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 e^2}-\frac{24 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3 d g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac{3 g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{8 e^2}+f x \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{d g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac{g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\frac{24 i \sqrt{d} f p^3 \text{Li}_2\left (1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{\sqrt{e}}+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 2.38113, 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} \]

Warning: Unable to verify antiderivative.

[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)

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Maple [A]  time = 64.717, size = 0, normalized size = 0. \begin{align*} \int \left ( g{x}^{3}+f \right ) \left ( \ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x^3+f)*ln(c*(e*x^2+d)^p)^3,x)

[Out]

int((g*x^3+f)*ln(c*(e*x^2+d)^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((g*x^3+f)*log(c*(e*x^2+d)^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 ({\left (g x^{3} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^3+f)*log(c*(e*x^2+d)^p)^3,x, algorithm="fricas")

[Out]

integral((g*x^3 + f)*log((e*x^2 + d)^p*c)^3, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x**3+f)*ln(c*(e*x**2+d)**p)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^3+f)*log(c*(e*x^2+d)^p)^3,x, algorithm="giac")

[Out]

integrate((g*x^3 + f)*log((e*x^2 + d)^p*c)^3, x)