Optimal. Leaf size=107 \[ \frac {x^2}{a b d \log (F)}-\frac {x^2}{b d \left (a+b F^{c+d x}\right ) \log (F)}-\frac {2 x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{a b d^2 \log ^2(F)}-\frac {2 \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{a b d^3 \log ^3(F)} \]
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Rubi [A]
time = 0.15, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2222, 2215,
2221, 2317, 2438} \begin {gather*} -\frac {2 \text {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )}{a b d^3 \log ^3(F)}-\frac {2 x \log \left (\frac {b F^{c+d x}}{a}+1\right )}{a b d^2 \log ^2(F)}-\frac {x^2}{b d \log (F) \left (a+b F^{c+d x}\right )}+\frac {x^2}{a b d \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2215
Rule 2221
Rule 2222
Rule 2317
Rule 2438
Rubi steps
\begin {align*} \int \frac {F^{c+d x} x^2}{\left (a+b F^{c+d x}\right )^2} \, dx &=-\frac {x^2}{b d \left (a+b F^{c+d x}\right ) \log (F)}+\frac {2 \int \frac {x}{a+b F^{c+d x}} \, dx}{b d \log (F)}\\ &=\frac {x^2}{a b d \log (F)}-\frac {x^2}{b d \left (a+b F^{c+d x}\right ) \log (F)}-\frac {2 \int \frac {F^{c+d x} x}{a+b F^{c+d x}} \, dx}{a d \log (F)}\\ &=\frac {x^2}{a b d \log (F)}-\frac {x^2}{b d \left (a+b F^{c+d x}\right ) \log (F)}-\frac {2 x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{a b d^2 \log ^2(F)}+\frac {2 \int \log \left (1+\frac {b F^{c+d x}}{a}\right ) \, dx}{a b d^2 \log ^2(F)}\\ &=\frac {x^2}{a b d \log (F)}-\frac {x^2}{b d \left (a+b F^{c+d x}\right ) \log (F)}-\frac {2 x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{a b d^2 \log ^2(F)}+\frac {2 \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a}\right )}{x} \, dx,x,F^{c+d x}\right )}{a b d^3 \log ^3(F)}\\ &=\frac {x^2}{a b d \log (F)}-\frac {x^2}{b d \left (a+b F^{c+d x}\right ) \log (F)}-\frac {2 x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{a b d^2 \log ^2(F)}-\frac {2 \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{a b d^3 \log ^3(F)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 103, normalized size = 0.96 \begin {gather*} \frac {d x \log (F) \left (b d F^{c+d x} x \log (F)-2 \left (a+b F^{c+d x}\right ) \log \left (1+\frac {b F^{c+d x}}{a}\right )\right )-2 \left (a+b F^{c+d x}\right ) \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{a b d^3 \left (a+b F^{c+d x}\right ) \log ^3(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. \(230\) vs.
\(2(107)=214\).
time = 0.02, size = 231, normalized size = 2.16
method | result | size |
risch | \(-\frac {x^{2}}{b d \left (a +b \,F^{d x +c}\right ) \ln \left (F \right )}+\frac {x^{2}}{a b d \ln \left (F \right )}+\frac {2 c x}{b \,d^{2} \ln \left (F \right ) a}+\frac {c^{2}}{b \,d^{3} \ln \left (F \right ) a}-\frac {2 \ln \left (1+\frac {b \,F^{d x} F^{c}}{a}\right ) x}{b \,d^{2} \ln \left (F \right )^{2} a}-\frac {2 \ln \left (1+\frac {b \,F^{d x} F^{c}}{a}\right ) c}{b \,d^{3} \ln \left (F \right )^{2} a}-\frac {2 \polylog \left (2, -\frac {b \,F^{d x} F^{c}}{a}\right )}{b \,d^{3} \ln \left (F \right )^{3} a}+\frac {2 c \ln \left (a +F^{c} F^{d x} b \right )}{b \,d^{3} \ln \left (F \right )^{2} a}-\frac {2 c \ln \left (F^{d x} F^{c}\right )}{b \,d^{3} \ln \left (F \right )^{2} a}\) | \(231\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 99, normalized size = 0.93 \begin {gather*} -\frac {x^{2}}{F^{d x} F^{c} b^{2} d \log \left (F\right ) + a b d \log \left (F\right )} + \frac {x^{2}}{a b d \log \left (F\right )} - \frac {2 \, {\left (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 )\right )}}{a b d^{3} \log \left (F\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 186, normalized size = 1.74 \begin {gather*} -\frac {a c^{2} \log \left (F\right )^{2} - {\left (b d^{2} x^{2} - b c^{2}\right )} F^{d x + c} \log \left (F\right )^{2} + 2 \, {\left (F^{d x + c} b + a\right )} {\rm Li}_2\left (-\frac {F^{d x + c} b + a}{a} + 1\right ) - 2 \, {\left (F^{d x + c} b c \log \left (F\right ) + a c \log \left (F\right )\right )} \log \left (F^{d x + c} b + a\right ) + 2 \, {\left ({\left (b d x + b c\right )} F^{d x + c} \log \left (F\right ) + {\left (a d x + a c\right )} \log \left (F\right )\right )} \log \left (\frac {F^{d x + c} b + a}{a}\right )}{F^{d x + c} a b^{2} d^{3} \log \left (F\right )^{3} + a^{2} b d^{3} \log \left (F\right )^{3}} \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*} - \frac {x^{2}}{F^{c + d x} b^{2} d \log {\left (F \right )} + a b d \log {\left (F \right )}} + \frac {2 \int \frac {x}{a + b e^{c \log {\left (F \right )}} e^{d x \log {\left (F \right )}}}\, dx}{b d \log {\left (F \right )}} \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 {F^{c+d\,x}\,x^2}{{\left (a+F^{c+d\,x}\,b\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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