\(\int \frac {\log ^2(c (d+e x^3)^p)}{x^5} \, dx\) [137]

Optimal result

Integrand size = 18, antiderivative size = 1328 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^5} \, dx=-\frac {3 \sqrt {3} e^{4/3} p^2 \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{2 d^{4/3}}-\frac {3 e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {e^{4/3} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{4 d^{4/3}}-\frac {e^{4/3} 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 )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} 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 )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} p^2 \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{4 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} 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 )}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} 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 )}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} p^2 \log ^2\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{4 d^{4/3}}-\frac {e^{4/3} 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 )}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} 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 )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} 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 )}{2 d^{4/3}}+\frac {3 e^{4/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 d^{4/3}}-\frac {3 e p \log \left (c \left (d+e x^3\right )^p\right )}{2 d x}+\frac {e^{4/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\sqrt [3]{-1} e^{4/3} 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 )}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} 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 )}{2 d^{4/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}-\frac {e^{4/3} p^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} 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 )}{2 d^{4/3}}-\frac {e^{4/3} 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 )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} 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 )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} 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 )}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} 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 )}{2 d^{4/3}} \]

[Out]

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

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 1334, normalized size of antiderivative = 1.00, number of steps used = 48, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2507, 2526, 2505, 298, 31, 648, 631, 210, 642, 2512, 266, 2463, 2437, 2338, 2441, 2440, 2438, 12} \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^5} \, dx=-\frac {e^{4/3} \log ^2\left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{4 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) p^2}{4 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} \log ^2\left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{4 d^{4/3}}-\frac {3 \sqrt {3} e^{4/3} \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) p^2}{2 d^{4/3}}-\frac {3 e^{4/3} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{2 d^{4/3}}-\frac {e^{4/3} \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}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} \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}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} \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}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} \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}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} \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}{2 d^{4/3}}-\frac {e^{4/3} \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}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} \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}{2 d^{4/3}}+\frac {3 e^{4/3} \log \left (e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}\right ) p^2}{4 d^{4/3}}-\frac {e^{4/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{2 d^{4/3}}-\frac {e^{4/3} \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}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} \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}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} \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}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} \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}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} \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}{2 d^{4/3}}+\frac {e^{4/3} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{2 d^{4/3}}-\frac {\sqrt [3]{-1} e^{4/3} \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}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} \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}{2 d^{4/3}}-\frac {3 e \log \left (c \left (e x^3+d\right )^p\right ) p}{2 d x}-\frac {\log ^2\left (c \left (e x^3+d\right )^p\right )}{4 x^4} \]

[In]

Int[Log[c*(d + e*x^3)^p]^2/x^5,x]

[Out]

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

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 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 {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}+\frac {1}{2} (3 e p) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{x^2 \left (d+e x^3\right )} \, dx \\ & = -\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}+\frac {1}{2} (3 e p) \int \left (\frac {\log \left (c \left (d+e x^3\right )^p\right )}{d x^2}-\frac {e x \log \left (c \left (d+e x^3\right )^p\right )}{d \left (d+e x^3\right )}\right ) \, dx \\ & = -\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}+\frac {(3 e p) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{x^2} \, dx}{2 d}-\frac {\left (3 e^2 p\right ) \int \frac {x \log \left (c \left (d+e x^3\right )^p\right )}{d+e x^3} \, dx}{2 d} \\ & = -\frac {3 e p \log \left (c \left (d+e x^3\right )^p\right )}{2 d x}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}-\frac {\left (3 e^2 p\right ) \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}{2 d}+\frac {\left (9 e^2 p^2\right ) \int \frac {x}{d+e x^3} \, dx}{2 d} \\ & = -\frac {3 e p \log \left (c \left (d+e x^3\right )^p\right )}{2 d x}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}+\frac {\left (e^{5/3} p\right ) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}-\frac {\left (\sqrt [3]{-1} e^{5/3} 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}{2 d^{4/3}}+\frac {\left ((-1)^{2/3} e^{5/3} 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}{2 d^{4/3}}-\frac {\left (3 e^{5/3} p^2\right ) \int \frac {1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}+\frac {\left (3 e^{5/3} 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 d^{4/3}} \\ & = -\frac {3 e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {3 e p \log \left (c \left (d+e x^3\right )^p\right )}{2 d x}+\frac {e^{4/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\sqrt [3]{-1} e^{4/3} 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 )}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} 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 )}{2 d^{4/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}+\frac {\left (3 e^{4/3} 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 d^{4/3}}+\frac {\left (9 e^{5/3} p^2\right ) \int \frac {1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{4 d}-\frac {\left (3 e^{7/3} p^2\right ) \int \frac {x^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{d+e x^3} \, dx}{2 d^{4/3}}+\frac {\left (3 \sqrt [3]{-1} e^{7/3} 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}{2 d^{4/3}}-\frac {\left (3 (-1)^{2/3} e^{7/3} 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}{2 d^{4/3}} \\ & = -\frac {3 e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 d^{4/3}}+\frac {3 e^{4/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 d^{4/3}}-\frac {3 e p \log \left (c \left (d+e x^3\right )^p\right )}{2 d x}+\frac {e^{4/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\sqrt [3]{-1} e^{4/3} 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 )}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} 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 )}{2 d^{4/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}+\frac {\left (9 e^{4/3} 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 d^{4/3}}-\frac {\left (3 e^{7/3} 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}{2 d^{4/3}}+\frac {\left (3 \sqrt [3]{-1} e^{7/3} 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}{2 d^{4/3}}-\frac {\left (3 (-1)^{2/3} e^{7/3} 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}{2 d^{4/3}} \\ & = -\frac {3 \sqrt {3} e^{4/3} p^2 \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{2 d^{4/3}}-\frac {3 e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 d^{4/3}}+\frac {3 e^{4/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 d^{4/3}}-\frac {3 e p \log \left (c \left (d+e x^3\right )^p\right )}{2 d x}+\frac {e^{4/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\sqrt [3]{-1} e^{4/3} 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 )}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} 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 )}{2 d^{4/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}-\frac {\left (e^{5/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}-\frac {\left (e^{5/3} 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}{2 d^{4/3}}-\frac {\left (e^{5/3} 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}{2 d^{4/3}}+\frac {\left (\sqrt [3]{-1} e^{5/3} 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}{2 d^{4/3}}+\frac {\left (\sqrt [3]{-1} e^{5/3} 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}{2 d^{4/3}}+\frac {\left (\sqrt [3]{-1} e^{5/3} 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}{2 d^{4/3}}-\frac {\left ((-1)^{2/3} e^{5/3} 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}{2 d^{4/3}}-\frac {\left ((-1)^{2/3} e^{5/3} 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}{2 d^{4/3}}-\frac {\left ((-1)^{2/3} e^{5/3} 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}{2 d^{4/3}} \\ & = \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 = 1.01 (sec) , antiderivative size = 912, normalized size of antiderivative = 0.69 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^5} \, dx=\frac {-\log ^2\left (c \left (d+e x^3\right )^p\right )+\frac {e p x^3 \left (9 e p x^3 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {e x^3}{d}\right )-d^{2/3} \sqrt [3]{e} p x \log ^2\left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )-2 d^{2/3} \sqrt [3]{e} p x \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (\frac {\sqrt [3]{-1} \sqrt [3]{d}-\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )-2 d^{2/3} \sqrt [3]{e} p x \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (\frac {i+\sqrt {3}-\frac {2 i \sqrt [3]{e} x}{\sqrt [3]{d}}}{3 i+\sqrt {3}}\right )-6 d \log \left (c \left (d+e x^3\right )^p\right )+2 d^{2/3} \sqrt [3]{e} x \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )-2 \sqrt [3]{-1} d^{2/3} \sqrt [3]{e} x \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )+2 (-1)^{2/3} d^{2/3} \sqrt [3]{e} x \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )-2 d^{2/3} \sqrt [3]{e} p x \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+\sqrt [3]{-1} d^{2/3} \sqrt [3]{e} p x \left (\log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right ) \left (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 )+2 \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 )\right )+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 )+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 )\right )-(-1)^{2/3} d^{2/3} \sqrt [3]{e} p x \left (\log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \left (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 )+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 )\right )+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 )+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 )\right )-2 d^{2/3} \sqrt [3]{e} p x \operatorname {PolyLog}\left (2,\frac {2 i \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 i+\sqrt {3}}\right )\right )}{d^2}}{4 x^4} \]

[In]

Integrate[Log[c*(d + e*x^3)^p]^2/x^5,x]

[Out]

(-Log[c*(d + e*x^3)^p]^2 + (e*p*x^3*(9*e*p*x^3*Hypergeometric2F1[2/3, 1, 5/3, -((e*x^3)/d)] - d^(2/3)*e^(1/3)*
p*x*Log[-d^(1/3) - e^(1/3)*x]^2 - 2*d^(2/3)*e^(1/3)*p*x*Log[-d^(1/3) - e^(1/3)*x]*Log[((-1)^(1/3)*d^(1/3) - e^
(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))] - 2*d^(2/3)*e^(1/3)*p*x*Log[-d^(1/3) - e^(1/3)*x]*Log[(I + Sqrt[3] - ((2*
I)*e^(1/3)*x)/d^(1/3))/(3*I + Sqrt[3])] - 6*d*Log[c*(d + e*x^3)^p] + 2*d^(2/3)*e^(1/3)*x*Log[-d^(1/3) - e^(1/3
)*x]*Log[c*(d + e*x^3)^p] - 2*(-1)^(1/3)*d^(2/3)*e^(1/3)*x*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x
^3)^p] + 2*(-1)^(2/3)*d^(2/3)*e^(1/3)*x*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p] - 2*d^(2/3)*
e^(1/3)*p*x*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))] + (-1)^(1/3)*d^(2/3)*e^(1/3)*p*x*(Log
[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*(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] + 2*Log[((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((-1 + (-1)^(2/3))*d^(1
/3))]) + 2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))] + 2*PolyLog[2, (-d^(1/3) +
(-1)^(1/3)*e^(1/3)*x)/((-1 + (-1)^(2/3))*d^(1/3))]) - (-1)^(2/3)*d^(2/3)*e^(1/3)*p*x*(Log[-d^(1/3) - (-1)^(2/3
)*e^(1/3)*x]*(2*Log[((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((-1 + (-1)^(2/3))*d^(1/3))] + 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]) + 2*PolyLog[2,
(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))] + 2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/
((1 - (-1)^(2/3))*d^(1/3))]) - 2*d^(2/3)*e^(1/3)*p*x*PolyLog[2, ((2*I)*(1 + (e^(1/3)*x)/d^(1/3)))/(3*I + Sqrt[
3])]))/d^2)/(4*x^4)

Maple [C] (warning: unable to verify)

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

Time = 1.17 (sec) , antiderivative size = 1954, normalized size of antiderivative = 1.47

method	result	size
risch	\(\text {Expression too large to display}\)	\(1954\)

[In]

int(ln(c*(e*x^3+d)^p)^2/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*ln((e*x^3+d)^p)^2/x^4-3/2*p*e*ln((e*x^3+d)^p)/d/x-1/2*p^2*e/d/(d/e)^(1/3)*ln(x+(d/e)^(1/3))*ln(e*x^3+d)+1
/2*p*e/d/(d/e)^(1/3)*ln(x+(d/e)^(1/3))*ln((e*x^3+d)^p)+1/4*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)-1/4*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)+1/2*p^2*e/d*3^(1/2)/(d/
e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*ln(e*x^3+d)-1/2*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)-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))+3/2*p^2*e*S
um(-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)/RootO
f(_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*_al
pha+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*_alph
a+3*_alpha^2,index=1)*ln((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alp
ha^2,index=2))*_alpha+6*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)*ln((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,ind
ex=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((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2))*_al
pha+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,in
dex=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))*RootOf(_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*dil
og((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+2*RootOf(_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*_alpha+3*_alpha^2,index=1)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1))+3*RootOf(_Z^2+3*_Z*_al
pha+3*_alpha^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*_alp
ha^2,index=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,_alpha=Ro
otOf(_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)*csgn(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))*(-1/4*ln((e*x^3+d)^p
)/x^4+3/4*p*e*(-1/d/x-(-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)))*e/d))-1/16*(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
*csgn(I*c*(e*x^3+d)^p)^2*csgn(I*c)+2*ln(c))^2/x^4

Fricas [F]

\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^5} \, dx=\int { \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x^{5}} \,d x } \]

[In]

integrate(log(c*(e*x^3+d)^p)^2/x^5,x, algorithm="fricas")

[Out]

integral(log((e*x^3 + d)^p*c)^2/x^5, x)

Sympy [F]

\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^5} \, dx=\int \frac {\log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}}{x^{5}}\, dx \]

[In]

integrate(ln(c*(e*x**3+d)**p)**2/x**5,x)

[Out]

Integral(log(c*(d + e*x**3)**p)**2/x**5, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^5} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(log(c*(e*x^3+d)^p)^2/x^5,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]

\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^5} \, dx=\int { \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x^{5}} \,d x } \]

[In]

integrate(log(c*(e*x^3+d)^p)^2/x^5,x, algorithm="giac")

[Out]

integrate(log((e*x^3 + d)^p*c)^2/x^5, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^5} \, dx=\int \frac {{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2}{x^5} \,d x \]

[In]

int(log(c*(d + e*x^3)^p)^2/x^5,x)

[Out]

int(log(c*(d + e*x^3)^p)^2/x^5, x)