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

Optimal. Leaf size=682 \[ -\frac{2 d p (d g-3 e f) \text{Unintegrable}\left (\frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )}{e}+\frac{32 i d^{3/2} g p^3 \text{PolyLog}\left (2,-\frac{\sqrt{d}-i \sqrt{e} x}{\sqrt{d}+i \sqrt{e} x}\right )}{3 e^{3/2}}-\frac{24 i \sqrt{d} f p^3 \text{PolyLog}\left (2,-\frac{\sqrt{d}-i \sqrt{e} x}{\sqrt{d}+i \sqrt{e} x}\right )}{\sqrt{e}}+\frac{32 d^{3/2} g p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{24 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac{32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac{2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}+\frac{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{32 i d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{3 e^{3/2}}-\frac{208 d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{9 e^{3/2}}+\frac{64 d^{3/2} g p^3 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2}}-\frac{24 i \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{48 \sqrt{d} f p^3 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{208 d g p^3 x}{9 e}-48 f p^3 x-\frac{16}{27} g p^3 x^3 \]

[Out]

-48*f*p^3*x + (208*d*g*p^3*x)/(9*e) - (16*g*p^3*x^3)/27 + (48*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[
e] - (208*d^(3/2)*g*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*e^(3/2)) - ((24*I)*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sq
rt[d]]^2)/Sqrt[e] + (((32*I)/3)*d^(3/2)*g*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/e^(3/2) - (48*Sqrt[d]*f*p^3*ArcTa
n[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] + (64*d^(3/2)*g*p^3*ArcTan[(Sqrt[e]*x
)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(3*e^(3/2)) + 24*f*p^2*x*Log[c*(d + e*x^2)^p] - (32*d*g*p
^2*x*Log[c*(d + e*x^2)^p])/(3*e) + (8*g*p^2*x^3*Log[c*(d + e*x^2)^p])/9 - (24*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)
/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] + (32*d^(3/2)*g*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/
(3*e^(3/2)) - 6*f*p*x*Log[c*(d + e*x^2)^p]^2 + (2*d*g*p*x*Log[c*(d + e*x^2)^p]^2)/e - (2*g*p*x^3*Log[c*(d + e*
x^2)^p]^2)/3 + f*x*Log[c*(d + e*x^2)^p]^3 + (g*x^3*Log[c*(d + e*x^2)^p]^3)/3 - ((24*I)*Sqrt[d]*f*p^3*PolyLog[2
, -((Sqrt[d] - I*Sqrt[e]*x)/(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[e] + (((32*I)/3)*d^(3/2)*g*p^3*PolyLog[2, -((Sqrt[
d] - I*Sqrt[e]*x)/(Sqrt[d] + I*Sqrt[e]*x))])/e^(3/2) - (2*d*(-3*e*f + d*g)*p*Unintegrable[Log[c*(d + e*x^2)^p]
^2/(d + e*x^2), x])/e

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Rubi [A]  time = 1.38604, 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 = {} \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-48*f*p^3*x + (208*d*g*p^3*x)/(9*e) - (16*g*p^3*x^3)/27 + (48*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[
e] - (208*d^(3/2)*g*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*e^(3/2)) - ((24*I)*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sq
rt[d]]^2)/Sqrt[e] + (((32*I)/3)*d^(3/2)*g*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/e^(3/2) - (48*Sqrt[d]*f*p^3*ArcTa
n[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] + (64*d^(3/2)*g*p^3*ArcTan[(Sqrt[e]*x
)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(3*e^(3/2)) + 24*f*p^2*x*Log[c*(d + e*x^2)^p] - (32*d*g*p
^2*x*Log[c*(d + e*x^2)^p])/(3*e) + (8*g*p^2*x^3*Log[c*(d + e*x^2)^p])/9 - (24*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)
/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] + (32*d^(3/2)*g*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/
(3*e^(3/2)) - 6*f*p*x*Log[c*(d + e*x^2)^p]^2 + (2*d*g*p*x*Log[c*(d + e*x^2)^p]^2)/e - (2*g*p*x^3*Log[c*(d + e*
x^2)^p]^2)/3 + f*x*Log[c*(d + e*x^2)^p]^3 + (g*x^3*Log[c*(d + e*x^2)^p]^3)/3 - ((24*I)*Sqrt[d]*f*p^3*PolyLog[2
, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] + (((32*I)/3)*d^(3/2)*g*p^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sq
rt[d] + I*Sqrt[e]*x)])/e^(3/2) + 6*d*f*p*Defer[Int][Log[c*(d + e*x^2)^p]^2/(d + e*x^2), x] - (2*d^2*g*p*Defer[
Int][Log[c*(d + e*x^2)^p]^2/(d + e*x^2), x])/e

Rubi steps

\begin{align*} \int \left (f+g x^2\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^2 \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^2 \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}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\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-(2 e g p) \int \frac{x^4 \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}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\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-(2 e g p) \int \left (-\frac{d \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{x^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}+\frac{d^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )-(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-(2 g p) \int x^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+\frac{(2 d g p) \int \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx}{e}-\frac{\left (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}\\ &=-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \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{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(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 (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}+\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-\left (8 d g p^2\right ) \int \frac{x^2 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx+\frac{1}{3} \left (8 e g p^2\right ) \int \frac{x^4 \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 )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \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{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(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 (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}+\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-\left (8 d g 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+\frac{1}{3} \left (8 e g p^2\right ) \int \left (-\frac{d \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac{d^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^2 \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{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \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{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(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 (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}+\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{1}{3} \left (8 g p^2\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx-\frac{\left (8 d g p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{3 e}-\frac{\left (8 d g p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{e}+\frac{\left (8 d^2 g p^2\right ) \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{3 e}+\frac{\left (8 d^2 g p^2\right ) \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}\\ &=24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac{8}{9} g p^2 x^3 \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}}+\frac{32 d^{3/2} g p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \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{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(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 (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}-\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+\frac{1}{3} \left (16 d g p^3\right ) \int \frac{x^2}{d+e x^2} \, dx+\left (16 d g p^3\right ) \int \frac{x^2}{d+e x^2} \, dx-\frac{1}{3} \left (16 d^2 g 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-\left (16 d^2 g 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-\frac{1}{9} \left (16 e g p^3\right ) \int \frac{x^4}{d+e x^2} \, dx\\ &=-48 f p^3 x+\frac{64 d g p^3 x}{3 e}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac{8}{9} g p^2 x^3 \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}}+\frac{32 d^{3/2} g p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \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{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(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 (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}+\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-\frac{\left (16 d^2 g p^3\right ) \int \frac{1}{d+e x^2} \, dx}{3 e}-\frac{\left (16 d^2 g p^3\right ) \int \frac{1}{d+e x^2} \, dx}{e}-\frac{\left (16 d^{3/2} g p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx}{3 \sqrt{e}}-\frac{\left (16 d^{3/2} g p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx}{\sqrt{e}}-\frac{1}{9} \left (16 e g p^3\right ) \int \left (-\frac{d}{e^2}+\frac{x^2}{e}+\frac{d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=-48 f p^3 x+\frac{208 d g p^3 x}{9 e}-\frac{16}{27} g p^3 x^3+\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{64 d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2}}-\frac{24 i \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+\frac{32 i d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac{8}{9} g p^2 x^3 \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}}+\frac{32 d^{3/2} g p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \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{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(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 (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}-\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+\frac{\left (16 d g p^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx}{3 e}+\frac{\left (16 d g p^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx}{e}-\frac{\left (16 d^2 g p^3\right ) \int \frac{1}{d+e x^2} \, dx}{9 e}\\ &=-48 f p^3 x+\frac{208 d g p^3 x}{9 e}-\frac{16}{27} g p^3 x^3+\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{208 d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{9 e^{3/2}}-\frac{24 i \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+\frac{32 i d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{3 e^{3/2}}-\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}}+\frac{64 d^{3/2} g 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 )}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac{8}{9} g p^2 x^3 \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}}+\frac{32 d^{3/2} g p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \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{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(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 (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}+\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-\frac{\left (16 d g 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}{3 e}-\frac{\left (16 d g 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}{e}\\ &=-48 f p^3 x+\frac{208 d g p^3 x}{9 e}-\frac{16}{27} g p^3 x^3+\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{208 d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{9 e^{3/2}}-\frac{24 i \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+\frac{32 i d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{3 e^{3/2}}-\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}}+\frac{64 d^{3/2} g 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 )}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac{8}{9} g p^2 x^3 \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}}+\frac{32 d^{3/2} g p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \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{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(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 (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}-\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}}+\frac{\left (16 i d^{3/2} g 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 )}{3 e^{3/2}}+\frac{\left (16 i d^{3/2} g 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 )}{e^{3/2}}\\ &=-48 f p^3 x+\frac{208 d g p^3 x}{9 e}-\frac{16}{27} g p^3 x^3+\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{208 d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{9 e^{3/2}}-\frac{24 i \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+\frac{32 i d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{3 e^{3/2}}-\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}}+\frac{64 d^{3/2} g 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 )}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac{8}{9} g p^2 x^3 \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}}+\frac{32 d^{3/2} g p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \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{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )-\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}}+\frac{32 i d^{3/2} g p^3 \text{Li}_2\left (1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{3 e^{3/2}}+(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 (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}\\ \end{align*}

Mathematica [A]  time = 4.48061, size = 1460, normalized size = 2.14 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[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)

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Maple [A]  time = 19.954, size = 0, normalized size = 0. \begin{align*} \int \left ( g{x}^{2}+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^2+f)*ln(c*(e*x^2+d)^p)^3,x)

[Out]

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

[Out]

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

[Out]

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