Integrand size = 18, antiderivative size = 1328 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^5} \, dx =\text {Too large to display} \]
-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*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+1/2*e^(4/3)*p*ln(d^(1/3)+e^ (1/3)*x)*ln(c*(e*x^3+d)^p)/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*(-1)^(2/3)*e^(4/3)*p^2*poly log(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)-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*(-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*(-1)^(1/3)*e^(4/3)*p^2*l n((-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...
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} \]
(-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*Lo g[-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...
Time = 1.76 (sec) , antiderivative size = 1292, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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.
| \(\displaystyle \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^5} \, dx\) |
| \(\Big \downarrow \) 2907 |
| \(\displaystyle \frac {3}{2} e p \int \frac {\log \left (c \left (e x^3+d\right )^p\right )}{x^2 \left (e x^3+d\right )}dx-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}\) |
| \(\Big \downarrow \) 2926 |
| \(\displaystyle \frac {3}{2} e p \int \left (\frac {\log \left (c \left (e x^3+d\right )^p\right )}{d x^2}-\frac {e x \log \left (c \left (e x^3+d\right )^p\right )}{d \left (e x^3+d\right )}\right )dx-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{2} e p \left (-\frac {\sqrt [3]{e} p \log ^2\left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{6 d^{4/3}}-\frac {\sqrt [3]{e} p \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{d^{4/3}}-\frac {\sqrt [3]{e} p \log \left (-\frac {\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{3 d^{4/3}}-\frac {\sqrt [3]{e} p \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 ) \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{3 d^{4/3}}+\frac {\sqrt [3]{e} \log \left (c \left (e x^3+d\right )^p\right ) \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{3 d^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{e} p \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{6 d^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{e} p \log ^2\left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}{6 d^{4/3}}-\frac {\sqrt {3} \sqrt [3]{e} p \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{d^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{e} 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 d^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{e} 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 d^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{e} 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 d^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{e} 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 d^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{e} 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 d^{4/3}}+\frac {\sqrt [3]{e} p \log \left (e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}\right )}{2 d^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{e} \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 d^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{e} \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 d^{4/3}}-\frac {\log \left (c \left (e x^3+d\right )^p\right )}{d x}-\frac {\sqrt [3]{e} p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{3 d^{4/3}}-\frac {\sqrt [3]{e} 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 d^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{e} 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 d^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{e} 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 d^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{e} 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 d^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{e} 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 d^{4/3}}\right )-\frac {\log ^2\left (c \left (e x^3+d\right )^p\right )}{4 x^4}\) |
-1/4*Log[c*(d + e*x^3)^p]^2/x^4 + (3*e*p*(-((Sqrt[3]*e^(1/3)*p*ArcTan[(d^( 1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/d^(4/3)) - (e^(1/3)*p*Log[d^(1/3) + e^(1/3)*x])/d^(4/3) - (e^(1/3)*p*Log[d^(1/3) + e^(1/3)*x]^2)/(6*d^(4/3)) - (e^(1/3)*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*d^(4/3)) + ((-1)^(1/3)*e^(1/3)*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*d^(4/3)) + ((-1)^(1/3)*e^(1/3)*p*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]^2)/(6*d^(4/3)) - ((-1)^(2/3)*e^(1/3)*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*d^(4/3)) - ((-1)^(2/3)*e^(1/3)*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*d^(4/3)) - ((-1)^(2/3)*e^(1/3)*p*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]^2)/(6*d^(4/3)) + ((-1)^(2/3)*e^(1/3)*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*d^(4/3)) - (e ^(1/3)*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*d^(4/3)) + ((-1)^(1/3)*e^(1/3)*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*d^(4/3)) + (e^(1/3)*p*Log[ d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(2*d^(4/3)) - Log[c*(d + e*...
3.2.37.3.1 Defintions of rubi rules used
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]
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]
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
-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*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*_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...
\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^5} \, dx=\int { \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x^{5}} \,d x } \]
\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^5} \, dx=\int \frac {\log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}}{x^{5}}\, dx \]
Exception generated. \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^5} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^5} \, dx=\int { \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^5} \, dx=\int \frac {{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2}{x^5} \,d x \]