Optimal. Leaf size=121 \[ -\frac{b d^2 \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{e^3}+\frac{d^2 (a+b \log (c x))^2}{2 b e^3}-\frac{d^2 \log \left (\frac{d x}{e}+1\right ) (a+b \log (c x))}{e^3}+\frac{d (a+b \log (c x))}{e^2 x}-\frac{a+b \log (c x)}{2 e x^2}+\frac{b d}{e^2 x}-\frac{b}{4 e x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.155915, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {263, 44, 2351, 2304, 2301, 2317, 2391} \[ -\frac{b d^2 \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{e^3}+\frac{d^2 (a+b \log (c x))^2}{2 b e^3}-\frac{d^2 \log \left (\frac{d x}{e}+1\right ) (a+b \log (c x))}{e^3}+\frac{d (a+b \log (c x))}{e^2 x}-\frac{a+b \log (c x)}{2 e x^2}+\frac{b d}{e^2 x}-\frac{b}{4 e x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 263
Rule 44
Rule 2351
Rule 2304
Rule 2301
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log (c x)}{\left (d+\frac{e}{x}\right ) x^4} \, dx &=\int \left (\frac{a+b \log (c x)}{e x^3}-\frac{d (a+b \log (c x))}{e^2 x^2}+\frac{d^2 (a+b \log (c x))}{e^3 x}-\frac{d^3 (a+b \log (c x))}{e^3 (e+d x)}\right ) \, dx\\ &=\frac{d^2 \int \frac{a+b \log (c x)}{x} \, dx}{e^3}-\frac{d^3 \int \frac{a+b \log (c x)}{e+d x} \, dx}{e^3}-\frac{d \int \frac{a+b \log (c x)}{x^2} \, dx}{e^2}+\frac{\int \frac{a+b \log (c x)}{x^3} \, dx}{e}\\ &=-\frac{b}{4 e x^2}+\frac{b d}{e^2 x}-\frac{a+b \log (c x)}{2 e x^2}+\frac{d (a+b \log (c x))}{e^2 x}+\frac{d^2 (a+b \log (c x))^2}{2 b e^3}-\frac{d^2 (a+b \log (c x)) \log \left (1+\frac{d x}{e}\right )}{e^3}+\frac{\left (b d^2\right ) \int \frac{\log \left (1+\frac{d x}{e}\right )}{x} \, dx}{e^3}\\ &=-\frac{b}{4 e x^2}+\frac{b d}{e^2 x}-\frac{a+b \log (c x)}{2 e x^2}+\frac{d (a+b \log (c x))}{e^2 x}+\frac{d^2 (a+b \log (c x))^2}{2 b e^3}-\frac{d^2 (a+b \log (c x)) \log \left (1+\frac{d x}{e}\right )}{e^3}-\frac{b d^2 \text{Li}_2\left (-\frac{d x}{e}\right )}{e^3}\\ \end{align*}
Mathematica [A] time = 0.151095, size = 110, normalized size = 0.91 \[ -\frac{4 b d^2 \text{PolyLog}\left (2,-\frac{d x}{e}\right )+4 d^2 \log \left (\frac{d x}{e}+1\right ) (a+b \log (c x))-\frac{2 d^2 (a+b \log (c x))^2}{b}-\frac{4 d e (a+b \log (c x))}{x}+\frac{2 e^2 (a+b \log (c x))}{x^2}-\frac{4 b d e}{x}+\frac{b e^2}{x^2}}{4 e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.055, size = 163, normalized size = 1.4 \begin{align*} -{\frac{a{d}^{2}\ln \left ( cdx+ce \right ) }{{e}^{3}}}-{\frac{a}{2\,e{x}^{2}}}+{\frac{a{d}^{2}\ln \left ( cx \right ) }{{e}^{3}}}+{\frac{ad}{{e}^{2}x}}-{\frac{b{d}^{2}}{{e}^{3}}{\it dilog} \left ({\frac{cdx+ce}{ce}} \right ) }-{\frac{b{d}^{2}\ln \left ( cx \right ) }{{e}^{3}}\ln \left ({\frac{cdx+ce}{ce}} \right ) }+{\frac{b{d}^{2} \left ( \ln \left ( cx \right ) \right ) ^{2}}{2\,{e}^{3}}}+{\frac{bd\ln \left ( cx \right ) }{{e}^{2}x}}+{\frac{bd}{{e}^{2}x}}-{\frac{b\ln \left ( cx \right ) }{2\,e{x}^{2}}}-{\frac{b}{4\,e{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.38922, size = 204, normalized size = 1.69 \begin{align*} -\frac{{\left (\log \left (\frac{d x}{e} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{d x}{e}\right )\right )} b d^{2}}{e^{3}} - \frac{{\left (b d^{2} \log \left (c\right ) + a d^{2}\right )} \log \left (d x + e\right )}{e^{3}} + \frac{2 \, b d^{2} x^{2} \log \left (x\right )^{2} - 2 \, a e^{2} -{\left (2 \, e^{2} \log \left (c\right ) + e^{2}\right )} b + 4 \,{\left (a d e +{\left (d e \log \left (c\right ) + d e\right )} b\right )} x + 2 \,{\left (2 \, b d e x - b e^{2} + 2 \,{\left (b d^{2} \log \left (c\right ) + a d^{2}\right )} x^{2}\right )} \log \left (x\right )}{4 \, e^{3} x^{2}} \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 \log \left (c x\right ) + a}{d x^{4} + e x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 68.6833, size = 233, normalized size = 1.93 \begin{align*} - \frac{a d^{3} \left (\begin{cases} \frac{x}{e} & \text{for}\: d = 0 \\\frac{\log{\left (d x + e \right )}}{d} & \text{otherwise} \end{cases}\right )}{e^{3}} + \frac{a d^{2} \log{\left (x \right )}}{e^{3}} + \frac{a d}{e^{2} x} - \frac{a}{2 e x^{2}} + \frac{b d^{3} \left (\begin{cases} \frac{x}{e} & \text{for}\: d = 0 \\\frac{\begin{cases} \log{\left (e \right )} \log{\left (x \right )} - \operatorname{Li}_{2}\left (\frac{d x e^{i \pi }}{e}\right ) & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (e \right )} \log{\left (\frac{1}{x} \right )} - \operatorname{Li}_{2}\left (\frac{d x e^{i \pi }}{e}\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 (e \right )} +{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x} \right )} \log{\left (e \right )} - \operatorname{Li}_{2}\left (\frac{d x e^{i \pi }}{e}\right ) & \text{otherwise} \end{cases}}{d} & \text{otherwise} \end{cases}\right )}{e^{3}} - \frac{b d^{3} \left (\begin{cases} \frac{x}{e} & \text{for}\: d = 0 \\\frac{\log{\left (d x + e \right )}}{d} & \text{otherwise} \end{cases}\right ) \log{\left (c x \right )}}{e^{3}} - \frac{b d^{2} \log{\left (x \right )}^{2}}{2 e^{3}} + \frac{b d^{2} \log{\left (x \right )} \log{\left (c x \right )}}{e^{3}} + \frac{b d \log{\left (c x \right )}}{e^{2} x} + \frac{b d}{e^{2} x} - \frac{b \log{\left (c x \right )}}{2 e x^{2}} - \frac{b}{4 e x^{2}} \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{b \log \left (c x\right ) + a}{{\left (d + \frac{e}{x}\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]