Optimal. Leaf size=84 \[ -\frac {b}{e x}-\frac {a+b \log (c x)}{e x}-\frac {d (a+b \log (c x))^2}{2 b e^2}+\frac {d (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{e^2}+\frac {b d \text {Li}_2\left (-\frac {d x}{e}\right )}{e^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {269, 46, 2393,
2341, 2338, 2354, 2438} \begin {gather*} \frac {b d \text {PolyLog}\left (2,-\frac {d x}{e}\right )}{e^2}-\frac {d (a+b \log (c x))^2}{2 b e^2}+\frac {d \log \left (\frac {d x}{e}+1\right ) (a+b \log (c x))}{e^2}-\frac {a+b \log (c x)}{e x}-\frac {b}{e x} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 269
Rule 2338
Rule 2341
Rule 2354
Rule 2393
Rule 2438
Rubi steps
\begin {align*} \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x^3} \, dx &=\int \left (\frac {a+b \log (c x)}{e x^2}-\frac {d (a+b \log (c x))}{e^2 x}+\frac {d^2 (a+b \log (c x))}{e^2 (e+d x)}\right ) \, dx\\ &=-\frac {d \int \frac {a+b \log (c x)}{x} \, dx}{e^2}+\frac {d^2 \int \frac {a+b \log (c x)}{e+d x} \, dx}{e^2}+\frac {\int \frac {a+b \log (c x)}{x^2} \, dx}{e}\\ &=-\frac {b}{e x}-\frac {a+b \log (c x)}{e x}-\frac {d (a+b \log (c x))^2}{2 b e^2}+\frac {d (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{e^2}-\frac {(b d) \int \frac {\log \left (1+\frac {d x}{e}\right )}{x} \, dx}{e^2}\\ &=-\frac {b}{e x}-\frac {a+b \log (c x)}{e x}-\frac {d (a+b \log (c x))^2}{2 b e^2}+\frac {d (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{e^2}+\frac {b d \text {Li}_2\left (-\frac {d x}{e}\right )}{e^2}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 77, normalized size = 0.92 \begin {gather*} -\frac {\frac {2 b e}{x}+\frac {2 e (a+b \log (c x))}{x}+\frac {d (a+b \log (c x))^2}{b}-2 d (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )-2 b d \text {Li}_2\left (-\frac {d x}{e}\right )}{2 e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 148, normalized size = 1.76
method | result | size |
risch | \(-\frac {a}{e x}-\frac {a d \ln \left (x \right )}{e^{2}}+\frac {a d \ln \left (d x +e \right )}{e^{2}}-\frac {b \ln \left (c x \right )^{2} d}{2 e^{2}}-\frac {b \ln \left (c x \right )}{e x}-\frac {b}{e x}+\frac {b d \dilog \left (\frac {c d x +c e}{e c}\right )}{e^{2}}+\frac {b d \ln \left (c x \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{e^{2}}\) | \(115\) |
derivativedivides | \(c^{2} \left (-\frac {a}{e \,c^{2} x}-\frac {a d \ln \left (c x \right )}{e^{2} c^{2}}+\frac {a d \ln \left (c d x +c e \right )}{e^{2} c^{2}}-\frac {b \ln \left (c x \right )^{2} d}{2 e^{2} c^{2}}+\frac {b d \dilog \left (\frac {c d x +c e}{e c}\right )}{e^{2} c^{2}}+\frac {b d \ln \left (c x \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{e^{2} c^{2}}-\frac {b \ln \left (c x \right )}{e \,c^{2} x}-\frac {b}{e \,c^{2} x}\right )\) | \(148\) |
default | \(c^{2} \left (-\frac {a}{e \,c^{2} x}-\frac {a d \ln \left (c x \right )}{e^{2} c^{2}}+\frac {a d \ln \left (c d x +c e \right )}{e^{2} c^{2}}-\frac {b \ln \left (c x \right )^{2} d}{2 e^{2} c^{2}}+\frac {b d \dilog \left (\frac {c d x +c e}{e c}\right )}{e^{2} c^{2}}+\frac {b d \ln \left (c x \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{e^{2} c^{2}}-\frac {b \ln \left (c x \right )}{e \,c^{2} x}-\frac {b}{e \,c^{2} x}\right )\) | \(148\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 92, normalized size = 1.10 \begin {gather*} {\left (\log \left (d x e^{\left (-1\right )} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-d x e^{\left (-1\right )}\right )\right )} b d e^{\left (-2\right )} + {\left (b d \log \left (c\right ) + a d\right )} e^{\left (-2\right )} \log \left (d x + e\right ) - \frac {{\left (b d x \log \left (x\right )^{2} + 2 \, {\left (b {\left (\log \left (c\right ) + 1\right )} + a\right )} e + 2 \, {\left ({\left (b d \log \left (c\right ) + a d\right )} x + b e\right )} \log \left (x\right )\right )} e^{\left (-2\right )}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x\right )}{x^3\,\left (d+\frac {e}{x}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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