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

Optimal. Leaf size=683 \[ -\frac {2 d p (d g-3 e f) \text {Int}\left (\frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )}{e}+\frac {32 d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac {24 \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}+f x \log ^3\left (c \left (d+e x^2\right )^p\right )-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac {8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac {2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {32 i d^{3/2} g p^3 \text {Li}_2\left (-\frac {\sqrt {d}-i \sqrt {e} x}{i \sqrt {e} x+\sqrt {d}}\right )}{3 e^{3/2}}+\frac {32 i d^{3/2} g p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}-\frac {208 d^{3/2} g p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}+\frac {64 d^{3/2} g p^3 \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}-\frac {24 i \sqrt {d} f p^3 \text {Li}_2\left (-\frac {\sqrt {d}-i \sqrt {e} x}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}-\frac {24 i \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {48 \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {48 \sqrt {d} f p^3 \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {208 d g p^3 x}{9 e}-48 f p^3 x-\frac {16}{27} g p^3 x^3 \]

[Out]

-48*f*p^3*x+208/9*d*g*p^3*x/e-16/27*g*p^3*x^3-208/9*d^(3/2)*g*p^3*arctan(x*e^(1/2)/d^(1/2))/e^(3/2)-24*I*f*p^3
*polylog(2,(-d^(1/2)+I*x*e^(1/2))/(d^(1/2)+I*x*e^(1/2)))*d^(1/2)/e^(1/2)+24*f*p^2*x*ln(c*(e*x^2+d)^p)-32/3*d*g
*p^2*x*ln(c*(e*x^2+d)^p)/e+8/9*g*p^2*x^3*ln(c*(e*x^2+d)^p)+32/3*d^(3/2)*g*p^2*arctan(x*e^(1/2)/d^(1/2))*ln(c*(
e*x^2+d)^p)/e^(3/2)-6*f*p*x*ln(c*(e*x^2+d)^p)^2+2*d*g*p*x*ln(c*(e*x^2+d)^p)^2/e-2/3*g*p*x^3*ln(c*(e*x^2+d)^p)^
2+f*x*ln(c*(e*x^2+d)^p)^3+1/3*g*x^3*ln(c*(e*x^2+d)^p)^3+64/3*d^(3/2)*g*p^3*arctan(x*e^(1/2)/d^(1/2))*ln(2*d^(1
/2)/(d^(1/2)+I*x*e^(1/2)))/e^(3/2)-24*I*f*p^3*arctan(x*e^(1/2)/d^(1/2))^2*d^(1/2)/e^(1/2)+48*f*p^3*arctan(x*e^
(1/2)/d^(1/2))*d^(1/2)/e^(1/2)+32/3*I*d^(3/2)*g*p^3*polylog(2,(-d^(1/2)+I*x*e^(1/2))/(d^(1/2)+I*x*e^(1/2)))/e^
(3/2)-24*f*p^2*arctan(x*e^(1/2)/d^(1/2))*ln(c*(e*x^2+d)^p)*d^(1/2)/e^(1/2)-48*f*p^3*arctan(x*e^(1/2)/d^(1/2))*
ln(2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)))*d^(1/2)/e^(1/2)+32/3*I*d^(3/2)*g*p^3*arctan(x*e^(1/2)/d^(1/2))^2/e^(3/2)-2
*d*(d*g-3*e*f)*p*Unintegrable(ln(c*(e*x^2+d)^p)^2/(e*x^2+d),x)/e

________________________________________________________________________________________

Rubi [A]  time = 1.39, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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 = 4.57, 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)

________________________________________________________________________________________

fricas [A]  time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3}, x\right ) \]

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)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3}\,{d x} \]

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)

________________________________________________________________________________________

maple [A]  time = 76.68, size = 0, normalized size = 0.00 \[ \int \left (g \,x^{2}+f \right ) \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )^{3}\, dx \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, {\left (g p^{3} x^{3} + 3 \, f p^{3} x\right )} \log \left (e x^{2} + d\right )^{3} + \int \frac {e g x^{4} \log \relax (c)^{3} + {\left (e f + d g\right )} x^{2} \log \relax (c)^{3} + d f \log \relax (c)^{3} - {\left ({\left (2 \, e g p^{3} - 3 \, e g p^{2} \log \relax (c)\right )} x^{4} - 3 \, d f p^{2} \log \relax (c) + 3 \, {\left (2 \, e f p^{3} - {\left (e f p^{2} + d g p^{2}\right )} \log \relax (c)\right )} x^{2}\right )} \log \left (e x^{2} + d\right )^{2} + 3 \, {\left (e g p x^{4} \log \relax (c)^{2} + d f p \log \relax (c)^{2} + {\left (e f p + d g p\right )} x^{2} \log \relax (c)^{2}\right )} \log \left (e x^{2} + d\right )}{e x^{2} + d}\,{d x} \]

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]

1/3*(g*p^3*x^3 + 3*f*p^3*x)*log(e*x^2 + d)^3 + integrate((e*g*x^4*log(c)^3 + (e*f + d*g)*x^2*log(c)^3 + d*f*lo
g(c)^3 - ((2*e*g*p^3 - 3*e*g*p^2*log(c))*x^4 - 3*d*f*p^2*log(c) + 3*(2*e*f*p^3 - (e*f*p^2 + d*g*p^2)*log(c))*x
^2)*log(e*x^2 + d)^2 + 3*(e*g*p*x^4*log(c)^2 + d*f*p*log(c)^2 + (e*f*p + d*g*p)*x^2*log(c)^2)*log(e*x^2 + d))/
(e*x^2 + d), x)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^3\,\left (g\,x^2+f\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (f + g x^{2}\right ) \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{3}\, dx \]

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]

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

________________________________________________________________________________________