Optimal. Leaf size=106 \[ \frac {1}{2 a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac {x}{2 a^2 b d \log (F)}-\frac {x}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}-\frac {\log \left (a+b F^{c+d x}\right )}{2 a^2 b d^2 \log ^2(F)} \]
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Rubi [A]
time = 0.07, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2222, 2320, 46}
\begin {gather*} -\frac {\log \left (a+b F^{c+d x}\right )}{2 a^2 b d^2 \log ^2(F)}+\frac {x}{2 a^2 b d \log (F)}+\frac {1}{2 a b d^2 \log ^2(F) \left (a+b F^{c+d x}\right )}-\frac {x}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 2222
Rule 2320
Rubi steps
\begin {align*} \int \frac {F^{c+d x} x}{\left (a+b F^{c+d x}\right )^3} \, dx &=-\frac {x}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}+\frac {\int \frac {1}{\left (a+b F^{c+d x}\right )^2} \, dx}{2 b d \log (F)}\\ &=-\frac {x}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}+\frac {\text {Subst}\left (\int \frac {1}{x (a+b x)^2} \, dx,x,F^{c+d x}\right )}{2 b d^2 \log ^2(F)}\\ &=-\frac {x}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}+\frac {\text {Subst}\left (\int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx,x,F^{c+d x}\right )}{2 b d^2 \log ^2(F)}\\ &=\frac {1}{2 a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac {x}{2 a^2 b d \log (F)}-\frac {x}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}-\frac {\log \left (a+b F^{c+d x}\right )}{2 a^2 b d^2 \log ^2(F)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 98, normalized size = 0.92 \begin {gather*} \frac {b d F^{c+d x} \left (2 a+b F^{c+d x}\right ) x \log (F)-\left (a+b F^{c+d x}\right ) \left (-a+\left (a+b F^{c+d x}\right ) \log \left (a+b F^{c+d x}\right )\right )}{2 a^2 b d^2 \left (a+b F^{c+d x}\right )^2 \log ^2(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 111, normalized size = 1.05
method | result | size |
risch | \(\frac {x}{2 a^{2} b d \ln \left (F \right )}+\frac {c}{2 \ln \left (F \right ) b \,d^{2} a^{2}}-\frac {\ln \left (F \right ) a d x -b \,F^{d x +c}-a}{2 \ln \left (F \right )^{2} d^{2} b \left (a +b \,F^{d x +c}\right )^{2} a}-\frac {\ln \left (F^{d x +c}+\frac {a}{b}\right )}{2 \ln \left (F \right )^{2} b \,d^{2} a^{2}}\) | \(111\) |
norman | \(\frac {\frac {{\mathrm e}^{\left (d x +c \right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{2} a \,d^{2}}+\frac {x \,{\mathrm e}^{\left (d x +c \right ) \ln \left (F \right )}}{\ln \left (F \right ) a d}+\frac {b x \,{\mathrm e}^{\left (2 d x +2 c \right ) \ln \left (F \right )}}{2 \ln \left (F \right ) a^{2} d}+\frac {1}{2 \ln \left (F \right )^{2} b \,d^{2}}}{\left (a +b \,{\mathrm e}^{\left (d x +c \right ) \ln \left (F \right )}\right )^{2}}-\frac {\ln \left (a +b \,{\mathrm e}^{\left (d x +c \right ) \ln \left (F \right )}\right )}{2 \ln \left (F \right )^{2} b \,d^{2} a^{2}}\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 150, normalized size = 1.42 \begin {gather*} \frac {F^{2 \, d x} F^{2 \, c} b^{2} d x \log \left (F\right ) + {\left (2 \, F^{c} a b d x \log \left (F\right ) + F^{c} a b\right )} F^{d x} + a^{2}}{2 \, {\left (2 \, F^{d x} F^{c} a^{3} b^{2} d^{2} \log \left (F\right )^{2} + F^{2 \, d x} F^{2 \, c} a^{2} b^{3} d^{2} \log \left (F\right )^{2} + a^{4} b d^{2} \log \left (F\right )^{2}\right )}} - \frac {\log \left (\frac {F^{d x} F^{c} b + a}{F^{c} b}\right )}{2 \, a^{2} 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 = 0.44, size = 148, normalized size = 1.40 \begin {gather*} \frac {F^{2 \, d x + 2 \, c} b^{2} d x \log \left (F\right ) + {\left (2 \, a b d x \log \left (F\right ) + a b\right )} F^{d x + c} + a^{2} - {\left (2 \, F^{d x + c} a b + F^{2 \, d x + 2 \, c} b^{2} + a^{2}\right )} \log \left (F^{d x + c} b + a\right )}{2 \, {\left (2 \, F^{d x + c} a^{3} b^{2} d^{2} \log \left (F\right )^{2} + F^{2 \, d x + 2 \, c} a^{2} b^{3} d^{2} \log \left (F\right )^{2} + a^{4} b d^{2} \log \left (F\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 122, normalized size = 1.15 \begin {gather*} \frac {F^{c + d x} b - a d x \log {\left (F \right )} + a}{4 F^{c + d x} a^{2} b^{2} d^{2} \log {\left (F \right )}^{2} + 2 F^{2 c + 2 d x} a b^{3} d^{2} \log {\left (F \right )}^{2} + 2 a^{3} b d^{2} \log {\left (F \right )}^{2}} + \frac {x}{2 a^{2} b d \log {\left (F \right )}} - \frac {\log {\left (F^{c + d x} + \frac {a}{b} \right )}}{2 a^{2} b d^{2} \log {\left (F \right )}^{2}} \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 [B]
time = 3.75, size = 155, normalized size = 1.46 \begin {gather*} -\frac {\frac {F^c\,F^{d\,x}}{2\,a\,d^2\,{\ln \left (F\right )}^2}-\frac {F^c\,F^{d\,x}\,x}{a\,d\,\ln \left (F\right )}+\frac {F^{2\,c}\,F^{2\,d\,x}\,b}{2\,a^2\,d^2\,{\ln \left (F\right )}^2}-\frac {F^{2\,c}\,F^{2\,d\,x}\,b\,x}{2\,a^2\,d\,\ln \left (F\right )}}{a^2+F^{2\,c}\,F^{2\,d\,x}\,b^2+2\,F^c\,F^{d\,x}\,a\,b}-\frac {\ln \left (a+F^c\,F^{d\,x}\,b\right )}{2\,a^2\,b\,d^2\,{\ln \left (F\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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