Optimal. Leaf size=54 \[ \frac {x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}+\frac {\text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)} \]
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Rubi [A]
time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2221, 2317,
2438} \begin {gather*} \frac {\text {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}+\frac {x \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rubi steps
\begin {align*} \int \frac {F^{c+d x} x}{a+b F^{c+d x}} \, dx &=\frac {x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}-\frac {\int \log \left (1+\frac {b F^{c+d x}}{a}\right ) \, dx}{b d \log (F)}\\ &=\frac {x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}-\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a}\right )}{x} \, dx,x,F^{c+d x}\right )}{b d^2 \log ^2(F)}\\ &=\frac {x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}+\frac {\text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 54, normalized size = 1.00 \begin {gather*} \frac {x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}+\frac {\text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(153\) vs.
\(2(54)=108\).
time = 0.03, size = 154, normalized size = 2.85
method | result | size |
risch | \(-\frac {c x}{d b}-\frac {c^{2}}{2 d^{2} b}+\frac {\ln \left (1+\frac {b \,F^{d x} F^{c}}{a}\right ) x}{d \ln \left (F \right ) b}+\frac {\ln \left (1+\frac {b \,F^{d x} F^{c}}{a}\right ) c}{d^{2} \ln \left (F \right ) b}+\frac {\polylog \left (2, -\frac {b \,F^{d x} F^{c}}{a}\right )}{d^{2} \ln \left (F \right )^{2} b}-\frac {c \ln \left (a +F^{c} F^{d x} b \right )}{d^{2} \ln \left (F \right ) b}+\frac {c \ln \left (F^{d x} F^{c}\right )}{d^{2} \ln \left (F \right ) b}\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 48, normalized size = 0.89 \begin {gather*} \frac {d x \log \left (\frac {F^{d x} F^{c} b}{a} + 1\right ) \log \left (F\right ) + {\rm Li}_2\left (-\frac {F^{d x} F^{c} b}{a}\right )}{b d^{2} \log \left (F\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 75, normalized size = 1.39 \begin {gather*} -\frac {c \log \left (F^{d x + c} b + a\right ) \log \left (F\right ) - {\left (d x + c\right )} \log \left (F\right ) \log \left (\frac {F^{d x + c} b + a}{a}\right ) - {\rm Li}_2\left (-\frac {F^{d x + c} b + a}{a} + 1\right )}{b d^{2} \log \left (F\right )^{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 {F^{c + d x} x}{F^{c} F^{d x} b + a}\, 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 {F^{c+d\,x}\,x}{a+F^{c+d\,x}\,b} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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