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

Optimal. Leaf size=518 \[ 6 d f p \text {Int}\left (\frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )+\frac {3 g p^2 \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 e^2}-\frac {3 d g p^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\frac {d g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac {3 g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{8 e^2}+\frac {3 d g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac {24 \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}+f x \log ^3\left (c \left (d+e x^2\right )^p\right )-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {3 g p^3 \left (d+e x^2\right )^2}{16 e^2}-\frac {24 i \sqrt {d} f p^3 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}-\frac {24 i \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {48 \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {48 \sqrt {d} f p^3 \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {3 d g p^3 x^2}{e}-48 f p^3 x \]

[Out]

-48*f*p^3*x+3*d*g*p^3*x^2/e-3/16*g*p^3*(e*x^2+d)^2/e^2+24*f*p^2*x*ln(c*(e*x^2+d)^p)-3*d*g*p^2*(e*x^2+d)*ln(c*(
e*x^2+d)^p)/e^2+3/8*g*p^2*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)/e^2-6*f*p*x*ln(c*(e*x^2+d)^p)^2+3/2*d*g*p*(e*x^2+d)*ln
(c*(e*x^2+d)^p)^2/e^2-3/8*g*p*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)^2/e^2+f*x*ln(c*(e*x^2+d)^p)^3-1/2*d*g*(e*x^2+d)*ln
(c*(e*x^2+d)^p)^3/e^2+1/4*g*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)^3/e^2+48*f*p^3*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)/e^(
1/2)-24*I*f*p^3*polylog(2,1-2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)))*d^(1/2)/e^(1/2)-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)-24*I*f*p^3*arctan(x*e^(1/2)/d^(1/2))^2*d^(1/2)/e^(1/2)+6*d*f*p*Unintegrable(ln(c*(e*x^2+d)^p)^2/(
e*x^2+d),x)

________________________________________________________________________________________

Rubi [A]  time = 0.77, 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 = {} \[ \int \left (f+g x^3\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]

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*}

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Mathematica [A]  time = 4.54, size = 1146, normalized size = 2.21 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(g*p^3*(d + e*x^2)*(45*d - 3*e*x^2 + (-42*d + 6*e*x^2)*Log[d + e*x^2] + 6*(3*d - e*x^2)*Log[d + e*x^2]^2 - 4*(
d - e*x^2)*Log[d + e*x^2]^3))/(16*e^2) - (3*g*p^2*(e*x^2*(-6*d + e*x^2) + (6*d^2 + 4*d*e*x^2 - 2*e^2*x^4)*Log[
d + e*x^2] - 2*(d^2 - e^2*x^4)*Log[d + e*x^2]^2)*(p*Log[d + e*x^2] - Log[c*(d + e*x^2)^p]))/(8*e^2) + (3*d*g*p
*x^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(4*e) + (6*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-(p*L
og[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/Sqrt[e] - (3*d^2*g*p*Log[d + e*x^2]*(-(p*Log[d + e*x^2]) + Log[c*(d
+ e*x^2)^p])^2)/(4*e^2) + (3*p*x*(4*f + g*x^3)*Log[d + e*x^2]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/
4 - (g*x^4*(3*p + 2*p*Log[d + e*x^2] - 2*Log[c*(d + e*x^2)^p])*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)
/8 + 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])*((4*I)*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + 4*Sqrt[d]*ArcTa
n[(Sqrt[e]*x)/Sqrt[d]]*(-2 + 2*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)] + Log[d + e*x^2]) + Sqrt[e]*x*(8 - 4*L
og[d + e*x^2] + Log[d + e*x^2]^2) + (4*I)*Sqrt[d]*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqrt[d] + Sqrt[e]*x
)]))/Sqrt[e] + (f*p^3*(-48*Sqrt[-d^2]*Sqrt[(e*x^2)/(d + e*x^2)]*Sqrt[d + e*x^2]*ArcSin[Sqrt[d]/Sqrt[d + e*x^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) - 6*Sqrt[-d^2]*Sqrt[(e*x^
2)/(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)] + Log[d + e
*x^2]*(4*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^2)] + Sqrt[d + e*x^2]*ArcSin[Sqrt[d
]/Sqrt[d + e*x^2]]*Log[d + e*x^2])) + 24*d*Sqrt[e*x^2]*ArcTanh[Sqrt[e*x^2]/Sqrt[-d]]*(Log[d + e*x^2] - Log[1 +
(e*x^2)/d]) + 6*(-d)^(3/2)*Sqrt[-((e*x^2)/d)]*(Log[1 + (e*x^2)/d]^2 - 4*Log[1 + (e*x^2)/d]*Log[(1 + Sqrt[-((e
*x^2)/d)])/2] + 2*Log[(1 + Sqrt[-((e*x^2)/d)])/2]^2 - 4*PolyLog[2, 1/2 - Sqrt[-((e*x^2)/d)]/2])))/(Sqrt[-d]*e*
x)

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fricas [A]  time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (g x^{3} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^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)

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maple [A]  time = 175.46, size = 0, normalized size = 0.00 \[ \int \left (g \,x^{3}+f \right ) \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x^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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*d^2-4*sqrt(e)>0)', see `assu
me?` for more details)Is 4*d^2-4*sqrt(e) positive or negative?

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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