Optimal. Leaf size=170 \[ \frac {e^4 \log \left (\frac {d x}{e}+1\right ) (a+b \log (c x))}{d^5}+\frac {e^2 x^2 (a+b \log (c x))}{2 d^3}-\frac {e x^3 (a+b \log (c x))}{3 d^2}+\frac {x^4 (a+b \log (c x))}{4 d}-\frac {a e^3 x}{d^4}-\frac {b e^3 x \log (c x)}{d^4}+\frac {b e^4 \text {Li}_2\left (-\frac {d x}{e}\right )}{d^5}+\frac {b e^3 x}{d^4}-\frac {b e^2 x^2}{4 d^3}+\frac {b e x^3}{9 d^2}-\frac {b x^4}{16 d} \]
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Rubi [A] time = 0.18, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {263, 43, 2351, 2295, 2304, 2317, 2391} \[ \frac {b e^4 \text {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^5}+\frac {e^2 x^2 (a+b \log (c x))}{2 d^3}+\frac {e^4 \log \left (\frac {d x}{e}+1\right ) (a+b \log (c x))}{d^5}-\frac {e x^3 (a+b \log (c x))}{3 d^2}+\frac {x^4 (a+b \log (c x))}{4 d}-\frac {a e^3 x}{d^4}-\frac {b e^3 x \log (c x)}{d^4}-\frac {b e^2 x^2}{4 d^3}+\frac {b e^3 x}{d^4}+\frac {b e x^3}{9 d^2}-\frac {b x^4}{16 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 2295
Rule 2304
Rule 2317
Rule 2351
Rule 2391
Rubi steps
\begin {align*} \int \frac {x^3 (a+b \log (c x))}{d+\frac {e}{x}} \, dx &=\int \left (-\frac {e^3 (a+b \log (c x))}{d^4}+\frac {e^2 x (a+b \log (c x))}{d^3}-\frac {e x^2 (a+b \log (c x))}{d^2}+\frac {x^3 (a+b \log (c x))}{d}+\frac {e^4 (a+b \log (c x))}{d^4 (e+d x)}\right ) \, dx\\ &=\frac {\int x^3 (a+b \log (c x)) \, dx}{d}-\frac {e \int x^2 (a+b \log (c x)) \, dx}{d^2}+\frac {e^2 \int x (a+b \log (c x)) \, dx}{d^3}-\frac {e^3 \int (a+b \log (c x)) \, dx}{d^4}+\frac {e^4 \int \frac {a+b \log (c x)}{e+d x} \, dx}{d^4}\\ &=-\frac {a e^3 x}{d^4}-\frac {b e^2 x^2}{4 d^3}+\frac {b e x^3}{9 d^2}-\frac {b x^4}{16 d}+\frac {e^2 x^2 (a+b \log (c x))}{2 d^3}-\frac {e x^3 (a+b \log (c x))}{3 d^2}+\frac {x^4 (a+b \log (c x))}{4 d}+\frac {e^4 (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{d^5}-\frac {\left (b e^3\right ) \int \log (c x) \, dx}{d^4}-\frac {\left (b e^4\right ) \int \frac {\log \left (1+\frac {d x}{e}\right )}{x} \, dx}{d^5}\\ &=-\frac {a e^3 x}{d^4}+\frac {b e^3 x}{d^4}-\frac {b e^2 x^2}{4 d^3}+\frac {b e x^3}{9 d^2}-\frac {b x^4}{16 d}-\frac {b e^3 x \log (c x)}{d^4}+\frac {e^2 x^2 (a+b \log (c x))}{2 d^3}-\frac {e x^3 (a+b \log (c x))}{3 d^2}+\frac {x^4 (a+b \log (c x))}{4 d}+\frac {e^4 (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{d^5}+\frac {b e^4 \text {Li}_2\left (-\frac {d x}{e}\right )}{d^5}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 156, normalized size = 0.92 \[ \frac {36 d^4 x^4 (a+b \log (c x))-48 d^3 e x^3 (a+b \log (c x))+72 d^2 e^2 x^2 (a+b \log (c x))+144 e^4 \log \left (\frac {d x}{e}+1\right ) (a+b \log (c x))-144 a d e^3 x-144 b d e^3 x \log (c x)-9 b d^4 x^4+16 b d^3 e x^3-36 b d^2 e^2 x^2+144 b e^4 \text {Li}_2\left (-\frac {d x}{e}\right )+144 b d e^3 x}{144 d^5} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{4} \log \left (c x\right ) + a x^{4}}{d x + e}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x\right ) + a\right )} x^{3}}{d + \frac {e}{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 209, normalized size = 1.23 \[ \frac {b \,x^{4} \ln \left (c x \right )}{4 d}+\frac {a \,x^{4}}{4 d}-\frac {b \,x^{4}}{16 d}-\frac {b e \,x^{3} \ln \left (c x \right )}{3 d^{2}}-\frac {a e \,x^{3}}{3 d^{2}}+\frac {b e \,x^{3}}{9 d^{2}}+\frac {b \,e^{2} x^{2} \ln \left (c x \right )}{2 d^{3}}+\frac {a \,e^{2} x^{2}}{2 d^{3}}-\frac {b \,e^{2} x^{2}}{4 d^{3}}-\frac {b \,e^{3} x \ln \left (c x \right )}{d^{4}}+\frac {b \,e^{4} \ln \left (c x \right ) \ln \left (\frac {c d x +c e}{c e}\right )}{d^{5}}-\frac {a \,e^{3} x}{d^{4}}+\frac {a \,e^{4} \ln \left (c d x +c e \right )}{d^{5}}+\frac {b \,e^{3} x}{d^{4}}+\frac {b \,e^{4} \dilog \left (\frac {c d x +c e}{c e}\right )}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 210, normalized size = 1.24 \[ \frac {{\left (\log \left (\frac {d x}{e} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {d x}{e}\right )\right )} b e^{4}}{d^{5}} + \frac {9 \, {\left (4 \, a d^{3} + {\left (4 \, d^{3} \log \relax (c) - d^{3}\right )} b\right )} x^{4} - 16 \, {\left (3 \, a d^{2} e + {\left (3 \, d^{2} e \log \relax (c) - d^{2} e\right )} b\right )} x^{3} + 36 \, {\left (2 \, a d e^{2} + {\left (2 \, d e^{2} \log \relax (c) - d e^{2}\right )} b\right )} x^{2} - 144 \, {\left (a e^{3} + {\left (e^{3} \log \relax (c) - e^{3}\right )} b\right )} x + 12 \, {\left (3 \, b d^{3} x^{4} - 4 \, b d^{2} e x^{3} + 6 \, b d e^{2} x^{2} - 12 \, b e^{3} x\right )} \log \relax (x)}{144 \, d^{4}} + \frac {{\left (b e^{4} \log \relax (c) + a e^{4}\right )} \log \left (d x + e\right )}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\left (a+b\,\ln \left (c\,x\right )\right )}{d+\frac {e}{x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 161.98, size = 280, normalized size = 1.65 \[ \frac {a x^{4}}{4 d} - \frac {a e x^{3}}{3 d^{2}} + \frac {a e^{2} x^{2}}{2 d^{3}} + \frac {a e^{4} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{4}} - \frac {a e^{3} x}{d^{4}} + \frac {b x^{4} \log {\left (c x \right )}}{4 d} - \frac {b x^{4}}{16 d} - \frac {b e x^{3} \log {\left (c x \right )}}{3 d^{2}} + \frac {b e x^{3}}{9 d^{2}} + \frac {b e^{2} x^{2} \log {\left (c x \right )}}{2 d^{3}} - \frac {b e^{2} x^{2}}{4 d^{3}} - \frac {b e^{4} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\begin {cases} \log {\relax (e )} \log {\relax (x )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (e )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (e )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (e )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{d^{4}} + \frac {b e^{4} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x \right )}}{d^{4}} - \frac {b e^{3} x \log {\left (c x \right )}}{d^{4}} + \frac {b e^{3} x}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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