Optimal. Leaf size=148 \[ -\frac{b d^3 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^4}-\frac{d^3 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac{d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac{a d^2 x}{e^3}+\frac{b d^2 x \log \left (c x^n\right )}{e^3}-\frac{b d^2 n x}{e^3}+\frac{b d n x^2}{4 e^2}-\frac{b n x^3}{9 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.177357, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {43, 2351, 2295, 2304, 2317, 2391} \[ -\frac{b d^3 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^4}-\frac{d^3 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac{d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac{a d^2 x}{e^3}+\frac{b d^2 x \log \left (c x^n\right )}{e^3}-\frac{b d^2 n x}{e^3}+\frac{b d n x^2}{4 e^2}-\frac{b n x^3}{9 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2351
Rule 2295
Rule 2304
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx &=\int \left (\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac{d x \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{d^2 \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^3}-\frac{d^3 \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^3}-\frac{d \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}+\frac{\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}\\ &=\frac{a d^2 x}{e^3}+\frac{b d n x^2}{4 e^2}-\frac{b n x^3}{9 e}-\frac{d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^4}+\frac{\left (b d^2\right ) \int \log \left (c x^n\right ) \, dx}{e^3}+\frac{\left (b d^3 n\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^4}\\ &=\frac{a d^2 x}{e^3}-\frac{b d^2 n x}{e^3}+\frac{b d n x^2}{4 e^2}-\frac{b n x^3}{9 e}+\frac{b d^2 x \log \left (c x^n\right )}{e^3}-\frac{d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^4}-\frac{b d^3 n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^4}\\ \end{align*}
Mathematica [A] time = 0.0761221, size = 142, normalized size = 0.96 \[ \frac{-36 b d^3 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+36 a d^2 e x-36 a d^3 \log \left (\frac{e x}{d}+1\right )-18 a d e^2 x^2+12 a e^3 x^3+6 b \log \left (c x^n\right ) \left (e x \left (6 d^2-3 d e x+2 e^2 x^2\right )-6 d^3 \log \left (\frac{e x}{d}+1\right )\right )-36 b d^2 e n x+9 b d e^2 n x^2-4 b e^3 n x^3}{36 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.217, size = 693, normalized size = 4.7 \begin{align*} \text{result too large to display} \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{1}{6} \, a{\left (\frac{6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac{2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} + b \int \frac{x^{3} \log \left (c\right ) + x^{3} \log \left (x^{n}\right )}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{3} \log \left (c x^{n}\right ) + a x^{3}}{e x + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 41.4876, size = 248, normalized size = 1.68 \begin{align*} - \frac{a d^{3} \left (\begin{cases} \frac{x}{d} & \text{for}\: e = 0 \\\frac{\log{\left (d + e x \right )}}{e} & \text{otherwise} \end{cases}\right )}{e^{3}} + \frac{a d^{2} x}{e^{3}} - \frac{a d x^{2}}{2 e^{2}} + \frac{a x^{3}}{3 e} + \frac{b d^{3} n \left (\begin{cases} \frac{x}{d} & \text{for}\: e = 0 \\\frac{\begin{cases} \log{\left (d \right )} \log{\left (x \right )} - \operatorname{Li}_{2}\left (\frac{e x e^{i \pi }}{d}\right ) & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (d \right )} \log{\left (\frac{1}{x} \right )} - \operatorname{Li}_{2}\left (\frac{e x e^{i \pi }}{d}\right ) & \text{for}\: \frac{1}{\left |{x}\right |} < 1 \\-{G_{2, 2}^{2, 0}\left (\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle |{x} \right )} \log{\left (d \right )} +{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x} \right )} \log{\left (d \right )} - \operatorname{Li}_{2}\left (\frac{e x e^{i \pi }}{d}\right ) & \text{otherwise} \end{cases}}{e} & \text{otherwise} \end{cases}\right )}{e^{3}} - \frac{b d^{3} \left (\begin{cases} \frac{x}{d} & \text{for}\: e = 0 \\\frac{\log{\left (d + e x \right )}}{e} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )}}{e^{3}} - \frac{b d^{2} n x}{e^{3}} + \frac{b d^{2} x \log{\left (c x^{n} \right )}}{e^{3}} + \frac{b d n x^{2}}{4 e^{2}} - \frac{b d x^{2} \log{\left (c x^{n} \right )}}{2 e^{2}} - \frac{b n x^{3}}{9 e} + \frac{b x^{3} \log{\left (c x^{n} \right )}}{3 e} \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{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]