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

Optimal result

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

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

Mathematica [C] (verified)

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

Time = 0.88 (sec) , antiderivative size = 1041, normalized size of antiderivative = 0.80 \[ \int x \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx =\text {Too large to display} \] 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)...
 

Rubi [A] (verified)

Time = 3.36 (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^...
 

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]
 

Maple [C] (warning: unable to verify)

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

Time = 1.35 (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)+1/e*p^2*d/(d/e)^(1/3)* 
ln(x+(d/e)^(1/3))*ln(e*x^3+d)-1/e*p*d/(d/e)^(1/3)*ln(x+(d/e)^(1/3))*ln((e* 
x^3+d)^p)-1/2/e*p^2*d/(d/e)^(1/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*ln(e*x 
^3+d)+1/2/e*p*d/(d/e)^(1/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*ln((e*x^3+d) 
^p)-1/e*p^2*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/e*p*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/e*p^2*d/(d/e)^(1/3)*ln(x+(d/e)^(1/3) 
)-3/4/e*p^2*d/(d/e)^(1/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))-3/2/e*p^2*d*3^ 
(1/2)/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))-3*e*p^2*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*l 
n(x-_alpha)*(9*_alpha^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))+9*_alpha^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))+6*_alpha*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))+3*_alpha*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,ind 
ex=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*_alpha*RootOf(_Z^2+3*_Z*_alpha+3*_alp 
ha^2,index=1)*ln((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)-x+_alpha)/Ro 
otOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1))+6*_alpha*RootOf(_Z^2+3*_Z*_a...
 

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)
 

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)
 

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
 

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)
 

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)
 

Reduce [F]

\[ \int x \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {6 \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}-2 e^{\frac {1}{3}} x}{d^{\frac {1}{3}} \sqrt {3}}\right ) d \,p^{2}+12 e^{\frac {2}{3}} d^{\frac {4}{3}} \left (\int \frac {\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right ) x}{e \,x^{3}+d}d x \right ) p +2 e^{\frac {2}{3}} d^{\frac {1}{3}} {\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right )}^{2} x^{2}-6 e^{\frac {2}{3}} d^{\frac {1}{3}} \mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right ) p \,x^{2}+9 e^{\frac {2}{3}} d^{\frac {1}{3}} p^{2} x^{2}+9 \,\mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) d \,p^{2}-3 \,\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right ) d p}{4 e^{\frac {2}{3}} d^{\frac {1}{3}}} \] Input:

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

Output:

(6*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**(1/3)*sqrt(3)))*d*p**2 + 12* 
e**(2/3)*d**(1/3)*int((log((d + e*x**3)**p*c)*x)/(d + e*x**3),x)*d*p + 2*e 
**(2/3)*d**(1/3)*log((d + e*x**3)**p*c)**2*x**2 - 6*e**(2/3)*d**(1/3)*log( 
(d + e*x**3)**p*c)*p*x**2 + 9*e**(2/3)*d**(1/3)*p**2*x**2 + 9*log(d**(1/3) 
 + e**(1/3)*x)*d*p**2 - 3*log((d + e*x**3)**p*c)*d*p)/(4*e**(2/3)*d**(1/3) 
)