Optimal. Leaf size=58 \[ \frac {x^2}{2 a}-\frac {x \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a d}-\frac {\text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a d^2} \]
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Rubi [A]
time = 0.07, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2215, 2221,
2317, 2438} \begin {gather*} -\frac {\text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{a d^2}-\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{a d}+\frac {x^2}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2215
Rule 2221
Rule 2317
Rule 2438
Rubi steps
\begin {align*} \int \frac {x}{a+b e^{c+d x}} \, dx &=\frac {x^2}{2 a}-\frac {b \int \frac {e^{c+d x} x}{a+b e^{c+d x}} \, dx}{a}\\ &=\frac {x^2}{2 a}-\frac {x \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a d}+\frac {\int \log \left (1+\frac {b e^{c+d x}}{a}\right ) \, dx}{a d}\\ &=\frac {x^2}{2 a}-\frac {x \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a d}+\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a}\right )}{x} \, dx,x,e^{c+d x}\right )}{a d^2}\\ &=\frac {x^2}{2 a}-\frac {x \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a d}-\frac {\text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a d^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 53, normalized size = 0.91 \begin {gather*} -\frac {x \log \left (1+\frac {a e^{-c-d x}}{b}\right )}{a d}+\frac {\text {Li}_2\left (-\frac {a e^{-c-d x}}{b}\right )}{a d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 90, normalized size = 1.55
method | result | size |
derivativedivides | \(\frac {\frac {\left (d x +c \right )^{2}}{2 a}-\frac {\left (d x +c \right ) \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}-\frac {\polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}-\frac {c \ln \left ({\mathrm e}^{d x +c}\right )}{a}+\frac {c \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{a}}{d^{2}}\) | \(90\) |
default | \(\frac {\frac {\left (d x +c \right )^{2}}{2 a}-\frac {\left (d x +c \right ) \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}-\frac {\polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}-\frac {c \ln \left ({\mathrm e}^{d x +c}\right )}{a}+\frac {c \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{a}}{d^{2}}\) | \(90\) |
risch | \(\frac {x^{2}}{2 a}+\frac {c x}{d a}+\frac {c^{2}}{2 d^{2} a}-\frac {x \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a d}-\frac {\ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right ) c}{d^{2} a}-\frac {\polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a \,d^{2}}-\frac {c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} a}+\frac {c \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{d^{2} a}\) | \(133\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 48, normalized size = 0.83 \begin {gather*} \frac {x^{2}}{2 \, a} - \frac {d x \log \left (\frac {b e^{\left (d x + c\right )}}{a} + 1\right ) + {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )}}{a}\right )}{a d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 72, normalized size = 1.24 \begin {gather*} \frac {d^{2} x^{2} + 2 \, c \log \left (b e^{\left (d x + c\right )} + a\right ) - 2 \, {\left (d x + c\right )} \log \left (\frac {b e^{\left (d x + c\right )} + a}{a}\right ) - 2 \, {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )} + a}{a} + 1\right )}{2 \, a d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{a + b e^{c} e^{d x}}\, dx \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 = -0.02 \begin {gather*} \int \frac {x}{a+b\,{\mathrm {e}}^{c+d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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