Optimal. Leaf size=98 \[ \frac {e^2 \log \left (\frac {d x}{e}+1\right ) (a+b \log (c x))}{d^3}+\frac {x^2 (a+b \log (c x))}{2 d}-\frac {a e x}{d^2}-\frac {b e x \log (c x)}{d^2}+\frac {b e^2 \text {Li}_2\left (-\frac {d x}{e}\right )}{d^3}+\frac {b e x}{d^2}-\frac {b x^2}{4 d} \]
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Rubi [A] time = 0.11, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {263, 43, 2351, 2295, 2304, 2317, 2391} \[ \frac {b e^2 \text {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^3}+\frac {e^2 \log \left (\frac {d x}{e}+1\right ) (a+b \log (c x))}{d^3}+\frac {x^2 (a+b \log (c x))}{2 d}-\frac {a e x}{d^2}-\frac {b e x \log (c x)}{d^2}+\frac {b e x}{d^2}-\frac {b x^2}{4 d} \]
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 (a+b \log (c x))}{d+\frac {e}{x}} \, dx &=\int \left (-\frac {e (a+b \log (c x))}{d^2}+\frac {x (a+b \log (c x))}{d}+\frac {e^2 (a+b \log (c x))}{d^2 (e+d x)}\right ) \, dx\\ &=\frac {\int x (a+b \log (c x)) \, dx}{d}-\frac {e \int (a+b \log (c x)) \, dx}{d^2}+\frac {e^2 \int \frac {a+b \log (c x)}{e+d x} \, dx}{d^2}\\ &=-\frac {a e x}{d^2}-\frac {b x^2}{4 d}+\frac {x^2 (a+b \log (c x))}{2 d}+\frac {e^2 (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{d^3}-\frac {(b e) \int \log (c x) \, dx}{d^2}-\frac {\left (b e^2\right ) \int \frac {\log \left (1+\frac {d x}{e}\right )}{x} \, dx}{d^3}\\ &=-\frac {a e x}{d^2}+\frac {b e x}{d^2}-\frac {b x^2}{4 d}-\frac {b e x \log (c x)}{d^2}+\frac {x^2 (a+b \log (c x))}{2 d}+\frac {e^2 (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{d^3}+\frac {b e^2 \text {Li}_2\left (-\frac {d x}{e}\right )}{d^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 99, normalized size = 1.01 \[ \frac {e^2 \log \left (\frac {d x+e}{e}\right ) (a+b \log (c x))}{d^3}+\frac {x^2 (a+b \log (c x))}{2 d}-\frac {a e x}{d^2}-\frac {b e x \log (c x)}{d^2}+\frac {b e^2 \text {Li}_2\left (-\frac {d x}{e}\right )}{d^3}+\frac {b e x}{d^2}-\frac {b x^2}{4 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \log \left (c x\right ) + a x^{2}}{d x + e}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x\right ) + a\right )} x}{d + \frac {e}{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 129, normalized size = 1.32 \[ \frac {b \,x^{2} \ln \left (c x \right )}{2 d}+\frac {a \,x^{2}}{2 d}-\frac {b \,x^{2}}{4 d}-\frac {b e x \ln \left (c x \right )}{d^{2}}+\frac {b \,e^{2} \ln \left (c x \right ) \ln \left (\frac {c d x +c e}{c e}\right )}{d^{3}}-\frac {a e x}{d^{2}}+\frac {a \,e^{2} \ln \left (c d x +c e \right )}{d^{3}}+\frac {b e x}{d^{2}}+\frac {b \,e^{2} \dilog \left (\frac {c d x +c e}{c e}\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 112, normalized size = 1.14 \[ \frac {{\left (\log \left (\frac {d x}{e} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {d x}{e}\right )\right )} b e^{2}}{d^{3}} + \frac {{\left ({\left (2 \, d \log \relax (c) - d\right )} b + 2 \, a d\right )} x^{2} - 4 \, {\left ({\left (e \log \relax (c) - e\right )} b + a e\right )} x + 2 \, {\left (b d x^{2} - 2 \, b e x\right )} \log \relax (x)}{4 \, d^{2}} + \frac {{\left (b e^{2} \log \relax (c) + a e^{2}\right )} \log \left (d x + e\right )}{d^{3}} \]
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 \[ \int \frac {x\,\left (a+b\,\ln \left (c\,x\right )\right )}{d+\frac {e}{x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 112.58, size = 189, normalized size = 1.93 \[ \frac {a x^{2}}{2 d} + \frac {a e^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{2}} - \frac {a e x}{d^{2}} + \frac {b x^{2} \log {\left (c x \right )}}{2 d} - \frac {b x^{2}}{4 d} - \frac {b e^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\begin {cases} \log {\relax (e )} \log {\relax (x )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (e )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (e )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (e )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{d^{2}} + \frac {b e^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x \right )}}{d^{2}} - \frac {b e x \log {\left (c x \right )}}{d^{2}} + \frac {b e x}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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