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

3.2.33.1 Optimal result

Integrand size = 16, antiderivative size = 1294 \[ \int x \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx =\text {Too large to display} \]

output

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)+1/2*x^2*ln(c*(e*x^3+d)^p)^2+9/4*p^2*x^2-3/2*p*x^2*ln(c*(e*x^3+ 
d)^p)-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)+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)+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-(...
 

3.2.33.2 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 ) \]

input

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

output

(x^2*Log[c*(d + e*x^3)^p]^2)/2 - 3*e*p*((-3*p*x^2)/(4*e) + (3*p*x^2*Hyperg 
eometric2F1[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)...
 

3.2.33.3 Rubi [A] (verified)

Time = 1.97 (sec) , antiderivative size = 1307, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2907, 2926, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx\)

\(\Big \downarrow \) 2907

\(\displaystyle \frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-3 e p \int \frac {x^4 \log \left (c \left (e x^3+d\right )^p\right )}{e x^3+d}dx\)

\(\Big \downarrow \) 2926

\(\displaystyle \frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-3 e p \int \left (\frac {x \log \left (c \left (e x^3+d\right )^p\right )}{e}-\frac {d x \log \left (c \left (e x^3+d\right )^p\right )}{e \left (e x^3+d\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} x^2 \log ^2\left (c \left (e x^3+d\right )^p\right )-3 e p \left (-\frac {3 p x^2}{4 e}+\frac {\log \left (c \left (e x^3+d\right )^p\right ) x^2}{2 e}-\frac {d^{2/3} p \log ^2\left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{6 e^{5/3}}+\frac {\sqrt [3]{-1} d^{2/3} p \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{6 e^{5/3}}-\frac {(-1)^{2/3} d^{2/3} p \log ^2\left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}{6 e^{5/3}}-\frac {\sqrt {3} d^{2/3} p \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{2 e^{5/3}}-\frac {d^{2/3} p \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{2 e^{5/3}}-\frac {d^{2/3} p \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 )}{3 e^{5/3}}+\frac {\sqrt [3]{-1} d^{2/3} p \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 )}{3 e^{5/3}}-\frac {(-1)^{2/3} d^{2/3} p \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 )}{3 e^{5/3}}-\frac {(-1)^{2/3} d^{2/3} p \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 )}{3 e^{5/3}}+\frac {(-1)^{2/3} d^{2/3} p \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 )}{3 e^{5/3}}-\frac {d^{2/3} p \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 )}{3 e^{5/3}}+\frac {\sqrt [3]{-1} d^{2/3} p \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 )}{3 e^{5/3}}+\frac {d^{2/3} p \log \left (e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}\right )}{4 e^{5/3}}+\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 )}{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 (e x^3+d\right )^p\right )}{3 e^{5/3}}+\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 )}{3 e^{5/3}}-\frac {d^{2/3} p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{3 e^{5/3}}-\frac {d^{2/3} p \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 )}{3 e^{5/3}}+\frac {\sqrt [3]{-1} d^{2/3} p \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 )}{3 e^{5/3}}+\frac {\sqrt [3]{-1} d^{2/3} p \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 )}{3 e^{5/3}}+\frac {(-1)^{2/3} d^{2/3} p \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 )}{3 e^{5/3}}-\frac {(-1)^{2/3} d^{2/3} p \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 )}{3 e^{5/3}}\right )\)

input

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

output

(x^2*Log[c*(d + e*x^3)^p]^2)/2 - 3*e*p*((-3*p*x^2)/(4*e) - (Sqrt[3]*d^(2/3 
)*p*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(2*e^(5/3)) - (d^(2 
/3)*p*Log[d^(1/3) + e^(1/3)*x])/(2*e^(5/3)) - (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)) + ((-1 
)^(1/3)*d^(2/3)*p*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])/(3*e^(5/3)) + ((-1)^(1/3)* 
d^(2/3)*p*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]^2)/(6*e^(5/3)) - ((-1)^(2/3) 
*d^(2/3)*p*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])/(3*e^(5/3)) - ((-1)^(2/3)*d^(2 
/3)*p*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])/(3*e^(5/3)) - ((-1)^(2/3)*d 
^(2/3)*p*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]^2)/(6*e^(5/3)) + ((-1)^(2/3)* 
d^(2/3)*p*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))])/(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)) + 
 ((-1)^(1/3)*d^(2/3)*p*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)))])/(3*e^(5 
/3)) + (d^(2/3)*p*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(4*e^...
 

3.2.33.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2907

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] - Simp[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 2926

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]
 

3.2.33.4 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	\(\text {Expression too large to display}\)	\(1957\)

input

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

output

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-_alp 
ha)*(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+_al 
pha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1))+2*RootOf(_Z^2+3*_Z*_alph 
a+3*_alpha^2,index=1)*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)*ln((Root 
Of(_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((Roo 
tOf(_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=1) 
*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+6*RootOf(_Z^2+3*_Z*_alpha+3*_alpha...
 

3.2.33.5 Fricas [F]

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

input

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

output

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

3.2.33.6 Sympy [F]

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

input

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

output

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

3.2.33.7 Maxima [F(-2)]

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

input

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

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

3.2.33.8 Giac [F]

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

input

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

output

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

3.2.33.9 Mupad [F(-1)]

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

input

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

output

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