Optimal. Leaf size=105 \[ \frac{9 e^{\frac{3 a}{b n}} \left (c x^n\right )^{3/n} \text{Ei}\left (-\frac{3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b^3 n^3 x^3}+\frac{3}{2 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}-\frac{1}{2 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.106838, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2306, 2310, 2178} \[ \frac{9 e^{\frac{3 a}{b n}} \left (c x^n\right )^{3/n} \text{Ei}\left (-\frac{3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b^3 n^3 x^3}+\frac{3}{2 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}-\frac{1}{2 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2306
Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (a+b \log \left (c x^n\right )\right )^3} \, dx &=-\frac{1}{2 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}-\frac{3 \int \frac{1}{x^4 \left (a+b \log \left (c x^n\right )\right )^2} \, dx}{2 b n}\\ &=-\frac{1}{2 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}+\frac{3}{2 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}+\frac{9 \int \frac{1}{x^4 \left (a+b \log \left (c x^n\right )\right )} \, dx}{2 b^2 n^2}\\ &=-\frac{1}{2 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}+\frac{3}{2 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}+\frac{\left (9 \left (c x^n\right )^{3/n}\right ) \operatorname{Subst}\left (\int \frac{e^{-\frac{3 x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 b^2 n^3 x^3}\\ &=\frac{9 e^{\frac{3 a}{b n}} \left (c x^n\right )^{3/n} \text{Ei}\left (-\frac{3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b^3 n^3 x^3}-\frac{1}{2 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}+\frac{3}{2 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.114823, size = 89, normalized size = 0.85 \[ \frac{9 e^{\frac{3 a}{b n}} \left (c x^n\right )^{3/n} \text{Ei}\left (-\frac{3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac{b n \left (3 a+3 b \log \left (c x^n\right )-b n\right )}{\left (a+b \log \left (c x^n\right )\right )^2}}{2 b^3 n^3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.741, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b{\left (n - 3 \, \log \left (c\right )\right )} - 3 \, b \log \left (x^{n}\right ) - 3 \, a}{2 \,{\left (b^{4} n^{2} x^{3} \log \left (x^{n}\right )^{2} + 2 \,{\left (b^{4} n^{2} \log \left (c\right ) + a b^{3} n^{2}\right )} x^{3} \log \left (x^{n}\right ) +{\left (b^{4} n^{2} \log \left (c\right )^{2} + 2 \, a b^{3} n^{2} \log \left (c\right ) + a^{2} b^{2} n^{2}\right )} x^{3}\right )}} + 9 \, \int \frac{1}{2 \,{\left (b^{3} n^{2} x^{4} \log \left (x^{n}\right ) +{\left (b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} x^{4}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.738719, size = 524, normalized size = 4.99 \begin{align*} \frac{3 \, b^{2} n^{2} \log \left (x\right ) - b^{2} n^{2} + 3 \, b^{2} n \log \left (c\right ) + 3 \, a b n + 9 \,{\left (b^{2} n^{2} x^{3} \log \left (x\right )^{2} + b^{2} x^{3} \log \left (c\right )^{2} + 2 \, a b x^{3} \log \left (c\right ) + a^{2} x^{3} + 2 \,{\left (b^{2} n x^{3} \log \left (c\right ) + a b n x^{3}\right )} \log \left (x\right )\right )} e^{\left (\frac{3 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \logintegral \left (\frac{e^{\left (-\frac{3 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{x^{3}}\right )}{2 \,{\left (b^{5} n^{5} x^{3} \log \left (x\right )^{2} + b^{5} n^{3} x^{3} \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} x^{3} \log \left (c\right ) + a^{2} b^{3} n^{3} x^{3} + 2 \,{\left (b^{5} n^{4} x^{3} \log \left (c\right ) + a b^{4} n^{4} x^{3}\right )} \log \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} \left (a + b \log{\left (c x^{n} \right )}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]