\(\int x \log ^2(c (d+e x^3)^p) \, dx\) [133]

Optimal result

Integrand size = 16, antiderivative size = 1294 \[ \int x \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {9 p^2 x^2}{4}+\frac {3 \sqrt {3} d^{2/3} p^2 \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{2 e^{2/3}}+\frac {3 d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac {d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac {d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} p^2 \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac {d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} p^2 \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac {3 d^{2/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 e^{2/3}}-\frac {3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac {d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac {\sqrt [3]{-1} d^{2/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )+\frac {d^{2/3} p^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} p^2 \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}+\frac {d^{2/3} p^2 \operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (3-i \sqrt {3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} p^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} p^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} p^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}} \]

[Out]

d^(2/3)*p^2*ln(d^(1/3)+e^(1/3)*x)*ln((-(-1)^(2/3)*d^(1/3)-e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3))/e^(2/3)-1/2*(-1)^
(1/3)*d^(2/3)*p^2*ln(d^(1/3)-(-1)^(1/3)*e^(1/3)*x)^2/e^(2/3)+1/2*(-1)^(2/3)*d^(2/3)*p^2*ln(d^(1/3)+(-1)^(2/3)*
e^(1/3)*x)^2/e^(2/3)+3/2*d^(2/3)*p^2*arctan(1/3*(d^(1/3)-2*e^(1/3)*x)/d^(1/3)*3^(1/2))*3^(1/2)/e^(2/3)+1/2*x^2
*ln(c*(e*x^3+d)^p)^2-3/2*p*x^2*ln(c*(e*x^3+d)^p)+d^(2/3)*p^2*polylog(2,2*(d^(1/3)+e^(1/3)*x)/d^(1/3)/(3-I*3^(1
/2)))/e^(2/3)+d^(2/3)*p^2*polylog(2,(d^(1/3)+e^(1/3)*x)/(1+(-1)^(1/3))/d^(1/3))/e^(2/3)+3/2*d^(2/3)*p^2*ln(d^(
1/3)+e^(1/3)*x)/e^(2/3)+1/2*d^(2/3)*p^2*ln(d^(1/3)+e^(1/3)*x)^2/e^(2/3)-3/4*d^(2/3)*p^2*ln(d^(2/3)-d^(1/3)*e^(
1/3)*x+e^(2/3)*x^2)/e^(2/3)+9/4*p^2*x^2+d^(2/3)*p^2*ln(d^(1/3)+e^(1/3)*x)*ln((-1)^(1/3)*(d^(1/3)+(-1)^(2/3)*e^
(1/3)*x)/(1+(-1)^(1/3))/d^(1/3))/e^(2/3)-d^(2/3)*p*ln(d^(1/3)+e^(1/3)*x)*ln(c*(e*x^3+d)^p)/e^(2/3)-(-1)^(1/3)*
d^(2/3)*p^2*ln((-1)^(1/3)*(d^(1/3)+e^(1/3)*x)/(1+(-1)^(1/3))/d^(1/3))*ln(d^(1/3)-(-1)^(1/3)*e^(1/3)*x)/e^(2/3)
+(-1)^(2/3)*d^(2/3)*p^2*ln((-1)^(1/3)*(d^(1/3)-(-1)^(1/3)*e^(1/3)*x)/(1+(-1)^(1/3))/d^(1/3))*ln(d^(1/3)+(-1)^(
2/3)*e^(1/3)*x)/e^(2/3)+(-1)^(2/3)*d^(2/3)*p^2*ln(-(-1)^(2/3)*(d^(1/3)+e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3))*ln(d
^(1/3)+(-1)^(2/3)*e^(1/3)*x)/e^(2/3)-(-1)^(2/3)*d^(2/3)*p^2*ln((-1)^(1/3)*(d^(1/3)-(-1)^(1/3)*e^(1/3)*x)/(1+(-
1)^(1/3))/d^(1/3))*ln((d^(1/3)+(-1)^(2/3)*e^(1/3)*x)/(1+(-1)^(1/3))/d^(1/3))/e^(2/3)-(-1)^(2/3)*d^(2/3)*p^2*ln
(-(-1)^(2/3)*(d^(1/3)+e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3))*ln((d^(1/3)+(-1)^(2/3)*e^(1/3)*x)/(1-(-1)^(2/3))/d^(1
/3))/e^(2/3)+(-1)^(1/3)*d^(2/3)*p^2*ln(-(-1)^(1/3)*((-1)^(2/3)*d^(1/3)+e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3))*ln(-
(-1)^(2/3)*(d^(1/3)+(-1)^(2/3)*e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3))/e^(2/3)-(-1)^(1/3)*d^(2/3)*p^2*ln(d^(1/3)-(-
1)^(1/3)*e^(1/3)*x)*ln(-(-1)^(2/3)*(d^(1/3)+(-1)^(2/3)*e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3))/e^(2/3)+(-1)^(1/3)*d
^(2/3)*p*ln(d^(1/3)-(-1)^(1/3)*e^(1/3)*x)*ln(c*(e*x^3+d)^p)/e^(2/3)-(-1)^(2/3)*d^(2/3)*p*ln(d^(1/3)+(-1)^(2/3)
*e^(1/3)*x)*ln(c*(e*x^3+d)^p)/e^(2/3)-(-1)^(2/3)*d^(2/3)*p^2*polylog(2,-(-1)^(2/3)*(d^(1/3)+e^(1/3)*x)/(1-(-1)
^(2/3))/d^(1/3))/e^(2/3)-(-1)^(1/3)*d^(2/3)*p^2*polylog(2,(d^(1/3)-(-1)^(1/3)*e^(1/3)*x)/(1+(-1)^(1/3))/d^(1/3
))/e^(2/3)-(-1)^(2/3)*d^(2/3)*p^2*polylog(2,(-1)^(1/3)*(d^(1/3)-(-1)^(1/3)*e^(1/3)*x)/(1+(-1)^(1/3))/d^(1/3))/
e^(2/3)+(-1)^(1/3)*d^(2/3)*p^2*polylog(2,-(-1)^(2/3)*(d^(1/3)+(-1)^(2/3)*e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3))/e^
(2/3)

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 1300, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.188, Rules used = {2507, 2526, 2505, 327, 298, 31, 648, 631, 210, 642, 2512, 266, 2463, 2437, 2338, 2441, 2440, 2438, 12} \[ \int x \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {9 x^2 p^2}{4}+\frac {d^{2/3} \log ^2\left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{2 e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) p^2}{2 e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} \log ^2\left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{2 e^{2/3}}+\frac {3 \sqrt {3} d^{2/3} \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) p^2}{2 e^{2/3}}+\frac {3 d^{2/3} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{2 e^{2/3}}+\frac {d^{2/3} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) \log \left (-\frac {\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) p^2}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) p^2}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{e^{2/3}}+\frac {d^{2/3} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) \log \left (\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) p^2}{e^{2/3}}-\frac {3 d^{2/3} \log \left (e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}\right ) p^2}{4 e^{2/3}}+\frac {d^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{e^{2/3}}+\frac {d^{2/3} \operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (3-i \sqrt {3}\right ) \sqrt [3]{d}}\right ) p^2}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{-1} \left (\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) p^2}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) p^2}{e^{2/3}}-\frac {3}{2} x^2 \log \left (c \left (e x^3+d\right )^p\right ) p-\frac {d^{2/3} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{e^{2/3}}+\frac {\sqrt [3]{-1} d^{2/3} \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} \log \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{e^{2/3}}+\frac {1}{2} x^2 \log ^2\left (c \left (e x^3+d\right )^p\right ) \]

[In]

Int[x*Log[c*(d + e*x^3)^p]^2,x]

[Out]

(9*p^2*x^2)/4 + (3*Sqrt[3]*d^(2/3)*p^2*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(2*e^(2/3)) + (3*d^(
2/3)*p^2*Log[d^(1/3) + e^(1/3)*x])/(2*e^(2/3)) + (d^(2/3)*p^2*Log[d^(1/3) + e^(1/3)*x]^2)/(2*e^(2/3)) + (d^(2/
3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[-(((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/e^(2/3) -
((-1)^(1/3)*d^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) - (-1)
^(1/3)*e^(1/3)*x])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]^2)/(2*e^(2/3)) + ((-1
)^(2/3)*d^(2/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[d^(1/3) + (-1)^(
2/3)*e^(1/3)*x])/e^(2/3) + ((-1)^(2/3)*d^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1
)^(1/3))*d^(1/3))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/e^(2/3) + ((-1)^(2/3)*d^(2/3)*p^2*Log[d^(1/3) + (-1)^(
2/3)*e^(1/3)*x]^2)/(2*e^(2/3)) - ((-1)^(2/3)*d^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1
+ (-1)^(1/3))*d^(1/3))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) + (d^(2/3)*
p^2*Log[d^(1/3) + e^(1/3)*x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/e^
(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^
(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(2/3) - (3*d^(2/3)*p^2*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2
])/(4*e^(2/3)) - (3*p*x^2*Log[c*(d + e*x^3)^p])/2 - (d^(2/3)*p*Log[d^(1/3) + e^(1/3)*x]*Log[c*(d + e*x^3)^p])/
e^(2/3) + ((-1)^(1/3)*d^(2/3)*p*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(2/3) - ((-1)^(2/3
)*d^(2/3)*p*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(2/3) + (x^2*Log[c*(d + e*x^3)^p]^2)/2
+ (d^(2/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) + (d^(2/3)*p^2*PolyLog[2
, (2*(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*PolyLog[2, -(((-1)^(
1/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*PolyLog
[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) - ((-1)^(2/3)*d^(2/3)*p^2*PolyLog[2,
((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) + ((-1)^(2/3)*d^(2/3)*p^2*
PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/e^(2/3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

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

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2438

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

Rule 2440

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

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2505

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)^(m +
1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Dist[b*e*n*(p/(f*(m + 1))), Int[x^(n - 1)*((f*x)^(m + 1)/
(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 2507

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_)*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)
^(m + 1)*((a + b*Log[c*(d + e*x^n)^p])^q/(f*(m + 1))), x] - Dist[b*e*n*p*(q/(f^n*(m + 1))), Int[(f*x)^(m + n)*
((a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[q, 1]
&& IntegerQ[n] && NeQ[m, -1]

Rule 2512

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[f +
g*x]*((a + b*Log[c*(d + e*x^n)^p])/g), x] - Dist[b*e*n*(p/g), Int[x^(n - 1)*(Log[f + g*x]/(d + e*x^n)), x], x]
/; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && RationalQ[n]

Rule 2526

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-(3 e p) \int \frac {x^4 \log \left (c \left (d+e x^3\right )^p\right )}{d+e x^3} \, dx \\ & = \frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-(3 e p) \int \left (\frac {x \log \left (c \left (d+e x^3\right )^p\right )}{e}-\frac {d x \log \left (c \left (d+e x^3\right )^p\right )}{e \left (d+e x^3\right )}\right ) \, dx \\ & = \frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-(3 p) \int x \log \left (c \left (d+e x^3\right )^p\right ) \, dx+(3 d p) \int \frac {x \log \left (c \left (d+e x^3\right )^p\right )}{d+e x^3} \, dx \\ & = -\frac {3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )+\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )+(3 d p) \int \left (-\frac {\log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac {(-1)^{2/3} \log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}+\frac {\sqrt [3]{-1} \log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}\right ) \, dx+\frac {1}{2} \left (9 e p^2\right ) \int \frac {x^4}{d+e x^3} \, dx \\ & = \frac {9 p^2 x^2}{4}-\frac {3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )+\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac {\left (d^{2/3} p\right ) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}+\frac {\left (\sqrt [3]{-1} d^{2/3} p\right ) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}-\frac {\left ((-1)^{2/3} d^{2/3} p\right ) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}-\frac {1}{2} \left (9 d p^2\right ) \int \frac {x}{d+e x^3} \, dx \\ & = \frac {9 p^2 x^2}{4}-\frac {3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac {d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac {\sqrt [3]{-1} d^{2/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )+\frac {\left (3 d^{2/3} p^2\right ) \int \frac {1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 \sqrt [3]{e}}-\frac {\left (3 d^{2/3} p^2\right ) \int \frac {\sqrt [3]{d}+\sqrt [3]{e} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{2 \sqrt [3]{e}}+\left (3 d^{2/3} \sqrt [3]{e} p^2\right ) \int \frac {x^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{d+e x^3} \, dx-\left (3 \sqrt [3]{-1} d^{2/3} \sqrt [3]{e} p^2\right ) \int \frac {x^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{d+e x^3} \, dx+\left (3 (-1)^{2/3} d^{2/3} \sqrt [3]{e} p^2\right ) \int \frac {x^2 \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{d+e x^3} \, dx \\ & = \frac {9 p^2 x^2}{4}+\frac {3 d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}-\frac {3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac {d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac {\sqrt [3]{-1} d^{2/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac {\left (3 d^{2/3} p^2\right ) \int \frac {-\sqrt [3]{d} \sqrt [3]{e}+2 e^{2/3} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{4 e^{2/3}}-\frac {\left (9 d p^2\right ) \int \frac {1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{4 \sqrt [3]{e}}+\left (3 d^{2/3} \sqrt [3]{e} p^2\right ) \int \left (\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx-\left (3 \sqrt [3]{-1} d^{2/3} \sqrt [3]{e} p^2\right ) \int \left (\frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx+\left (3 (-1)^{2/3} d^{2/3} \sqrt [3]{e} p^2\right ) \int \left (\frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx \\ & = \frac {9 p^2 x^2}{4}+\frac {3 d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}-\frac {3 d^{2/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 e^{2/3}}-\frac {3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac {d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac {\sqrt [3]{-1} d^{2/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac {\left (9 d^{2/3} p^2\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{2 e^{2/3}}+\frac {\left (d^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}+\frac {\left (d^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}+\frac {\left (d^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}-\frac {\left (\sqrt [3]{-1} d^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}-\frac {\left (\sqrt [3]{-1} d^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}-\frac {\left (\sqrt [3]{-1} d^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}+\frac {\left ((-1)^{2/3} d^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}+\frac {\left ((-1)^{2/3} d^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}+\frac {\left ((-1)^{2/3} d^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.52 (sec) , antiderivative size = 1041, normalized size of antiderivative = 0.80 \[ \int x \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-3 e p \left (-\frac {3 p x^2}{4 e}+\frac {3 p x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {e x^3}{d}\right )}{4 e}-\frac {d^{2/3} p \log ^2\left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{6 e^{5/3}}-\frac {d^{2/3} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{3 e^{5/3}}-\frac {d^{2/3} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{3 e^{5/3}}+\frac {x^2 \log \left (c \left (d+e x^3\right )^p\right )}{2 e}+\frac {d^{2/3} \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 e^{5/3}}-\frac {\sqrt [3]{-1} d^{2/3} \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 e^{5/3}}+\frac {(-1)^{2/3} d^{2/3} \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 e^{5/3}}-\frac {d^{2/3} p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{3 e^{5/3}}-\frac {d^{2/3} p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{3 e^{5/3}}+\frac {\sqrt [3]{-1} d^{2/3} p \left (\frac {2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac {\log ^2\left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac {2 \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}+\frac {2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}+\frac {2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}\right )}{6 e}-\frac {(-1)^{2/3} d^{2/3} p \left (\frac {2 \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac {2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac {\log ^2\left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac {2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}+\frac {2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}\right )}{6 e}\right ) \]

[In]

Integrate[x*Log[c*(d + e*x^3)^p]^2,x]

[Out]

(x^2*Log[c*(d + e*x^3)^p]^2)/2 - 3*e*p*((-3*p*x^2)/(4*e) + (3*p*x^2*Hypergeometric2F1[2/3, 1, 5/3, -((e*x^3)/d
)])/(4*e) - (d^(2/3)*p*Log[-d^(1/3) - e^(1/3)*x]^2)/(6*e^(5/3)) - (d^(2/3)*p*Log[-d^(1/3) - e^(1/3)*x]*Log[-((
(-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/(3*e^(5/3)) - (d^(2/3)*p*Log[-d^(1/3) - e^(1/3)*
x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/(3*e^(5/3)) + (x^2*Log[c*(d
+ e*x^3)^p])/(2*e) + (d^(2/3)*Log[-d^(1/3) - e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(3*e^(5/3)) - ((-1)^(1/3)*d^(2/3
)*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(3*e^(5/3)) + ((-1)^(2/3)*d^(2/3)*Log[-d^(1/3) -
(-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(3*e^(5/3)) - (d^(2/3)*p*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-
1)^(1/3))*d^(1/3))])/(3*e^(5/3)) - (d^(2/3)*p*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/(3
*e^(5/3)) + ((-1)^(1/3)*d^(2/3)*p*((2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[-
d^(1/3) + (-1)^(1/3)*e^(1/3)*x])/e^(2/3) + Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]^2/e^(2/3) + (2*Log[-d^(1/3) +
(-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(2/3
) + (2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) + (2*PolyLog[2, (d^(1/
3) - (-1)^(1/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/e^(2/3)))/(6*e) - ((-1)^(2/3)*d^(2/3)*p*((2*Log[-(((-1
)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x])/e^(2/3) + (2
*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[-d^(1/3) - (-1)^(2/3)*e^(1/
3)*x])/e^(2/3) + Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]^2/e^(2/3) + (2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*
x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) + (2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(
1/3))])/e^(2/3)))/(6*e))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.83 (sec) , antiderivative size = 1957, normalized size of antiderivative = 1.51

method	result	size
risch	\(\text {Expression too large to display}\)	\(1957\)

[In]

int(x*ln(c*(e*x^3+d)^p)^2,x,method=_RETURNVERBOSE)

[Out]

1/2*ln((e*x^3+d)^p)^2*x^2-3/2*p*x^2*ln((e*x^3+d)^p)+p^2/e*d/(d/e)^(1/3)*ln(x+(d/e)^(1/3))*ln(e*x^3+d)-p/e*d/(d
/e)^(1/3)*ln(x+(d/e)^(1/3))*ln((e*x^3+d)^p)-1/2*p^2/e*d/(d/e)^(1/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*ln(e*x^3
+d)+1/2*p/e*d/(d/e)^(1/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*ln((e*x^3+d)^p)-p^2/e*d*3^(1/2)/(d/e)^(1/3)*arctan
(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*ln(e*x^3+d)+p/e*d*3^(1/2)/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1)
)*ln((e*x^3+d)^p)+9/4*p^2*x^2+3/2*p^2/e*d/(d/e)^(1/3)*ln(x+(d/e)^(1/3))-3/4*p^2/e*d/(d/e)^(1/3)*ln(x^2-(d/e)^(
1/3)*x+(d/e)^(2/3))-3/2*p^2/e*d*3^(1/2)/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))-3*p^2*e*Sum(-1/3*(
ln(x-_alpha)*ln(e*x^3+d)-3*e*(1/6/_alpha^2/e*ln(x-_alpha)^2+1/3*_alpha*ln(x-_alpha)*(2*RootOf(_Z^2+3*_Z*_alpha
+3*_alpha^2,index=1)*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)*ln((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=
1)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1))+2*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)*RootOf
(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)*ln((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)-x+_alpha)/RootOf(_Z^2+3*
_Z*_alpha+3*_alpha^2,index=2))+3*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)*ln((RootOf(_Z^2+3*_Z*_alpha+3*_al
pha^2,index=1)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1))*_alpha+6*RootOf(_Z^2+3*_Z*_alpha+3*_alph
a^2,index=1)*ln((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,inde
x=2))*_alpha+6*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)*ln((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)-x+_
alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1))*_alpha+3*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)*ln((R
ootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2))*_alpha+9*ln(
(RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1))*_alpha^2+9
*ln((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2))*_alpha
^2)/(3*_alpha+2*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1))/d/(3*_alpha+2*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,
index=2))+1/3*_alpha*(2*dilog((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3
*_alpha^2,index=2))*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)+6*
dilog((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2))*Root
Of(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)*_alpha+3*dilog((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)-x+_alpha)/
RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2))*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)*_alpha+9*dilog((RootO
f(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2))*_alpha^2+2*RootO
f(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)*dilog((RootOf(_Z^2+3*_Z*_al
pha+3*_alpha^2,index=1)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1))+3*RootOf(_Z^2+3*_Z*_alpha+3*_al
pha^2,index=1)*dilog((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2
,index=1))*_alpha+6*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)*dilog((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,inde
x=1)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1))*_alpha+9*dilog((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2
,index=1)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1))*_alpha^2)/(3*_alpha+2*RootOf(_Z^2+3*_Z*_alpha
+3*_alpha^2,index=1))/d/(3*_alpha+2*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2))))*d/_alpha/e^2,_alpha=RootOf(
_Z^3*e+d))+(I*Pi*csgn(I*(e*x^3+d)^p)*csgn(I*c*(e*x^3+d)^p)^2-I*Pi*csgn(I*(e*x^3+d)^p)*csgn(I*c*(e*x^3+d)^p)*cs
gn(I*c)-I*Pi*csgn(I*c*(e*x^3+d)^p)^3+I*Pi*csgn(I*c*(e*x^3+d)^p)^2*csgn(I*c)+2*ln(c))*(1/2*ln((e*x^3+d)^p)*x^2-
3/2*p*e*(1/2*x^2/e-(-1/3/e/(d/e)^(1/3)*ln(x+(d/e)^(1/3))+1/6/e/(d/e)^(1/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))+1
/3*3^(1/2)/e/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1)))*d/e))+1/8*(I*Pi*csgn(I*(e*x^3+d)^p)*csgn(I*c
*(e*x^3+d)^p)^2-I*Pi*csgn(I*(e*x^3+d)^p)*csgn(I*c*(e*x^3+d)^p)*csgn(I*c)-I*Pi*csgn(I*c*(e*x^3+d)^p)^3+I*Pi*csg
n(I*c*(e*x^3+d)^p)^2*csgn(I*c)+2*ln(c))^2*x^2

Fricas [F]

\[ \int x \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\int { x \log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2} \,d x } \]

[In]

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

[Out]

integral(x*log((e*x^3 + d)^p*c)^2, x)

Sympy [F]

\[ \int x \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\int x \log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int x \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x*log(c*(e*x^3+d)^p)^2,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(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int x \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\int { x \log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\int x\,{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2 \,d x \]

[In]

int(x*log(c*(d + e*x^3)^p)^2,x)

[Out]

int(x*log(c*(d + e*x^3)^p)^2, x)