Optimal. Leaf size=85 \[ \frac {x^2 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}+\frac {2 x \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}-\frac {2 \text {Li}_3\left (-\frac {b F^{c+d x}}{a}\right )}{b d^3 \log ^3(F)} \]
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Rubi [A]
time = 0.08, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2221, 2611,
2320, 6724} \begin {gather*} -\frac {2 \text {PolyLog}\left (3,-\frac {b F^{c+d x}}{a}\right )}{b d^3 \log ^3(F)}+\frac {2 x \text {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}+\frac {x^2 \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 2320
Rule 2611
Rule 6724
Rubi steps
\begin {align*} \int \frac {F^{c+d x} x^2}{a+b F^{c+d x}} \, dx &=\frac {x^2 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}-\frac {2 \int x \log \left (1+\frac {b F^{c+d x}}{a}\right ) \, dx}{b d \log (F)}\\ &=\frac {x^2 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}+\frac {2 x \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}-\frac {2 \int \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right ) \, dx}{b d^2 \log ^2(F)}\\ &=\frac {x^2 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}+\frac {2 x \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}-\frac {2 \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{a}\right )}{x} \, dx,x,F^{c+d x}\right )}{b d^3 \log ^3(F)}\\ &=\frac {x^2 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}+\frac {2 x \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}-\frac {2 \text {Li}_3\left (-\frac {b F^{c+d x}}{a}\right )}{b d^3 \log ^3(F)}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 85, normalized size = 1.00 \begin {gather*} \frac {x^2 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}+\frac {2 x \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}-\frac {2 \text {Li}_3\left (-\frac {b F^{c+d x}}{a}\right )}{b d^3 \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. \(193\) vs.
\(2(85)=170\).
time = 0.04, size = 194, normalized size = 2.28
method | result | size |
risch | \(\frac {c^{2} x}{d^{2} b}+\frac {2 c^{3}}{3 d^{3} b}+\frac {\ln \left (1+\frac {b \,F^{d x} F^{c}}{a}\right ) x^{2}}{d \ln \left (F \right ) b}-\frac {\ln \left (1+\frac {b \,F^{d x} F^{c}}{a}\right ) c^{2}}{d^{3} \ln \left (F \right ) b}+\frac {2 \polylog \left (2, -\frac {b \,F^{d x} F^{c}}{a}\right ) x}{d^{2} \ln \left (F \right )^{2} b}-\frac {2 \polylog \left (3, -\frac {b \,F^{d x} F^{c}}{a}\right )}{d^{3} \ln \left (F \right )^{3} b}+\frac {c^{2} \ln \left (a +F^{c} F^{d x} b \right )}{d^{3} \ln \left (F \right ) b}-\frac {c^{2} \ln \left (F^{d x} F^{c}\right )}{d^{3} \ln \left (F \right ) b}\) | \(194\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.60, size = 78, normalized size = 0.92 \begin {gather*} \frac {d^{2} x^{2} \log \left (\frac {F^{d x} F^{c} b}{a} + 1\right ) \log \left (F\right )^{2} + 2 \, d x {\rm Li}_2\left (-\frac {F^{d x} F^{c} b}{a}\right ) \log \left (F\right ) - 2 \, {\rm Li}_{3}(-\frac {F^{d x} F^{c} b}{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 = 0.43, size = 108, normalized size = 1.27 \begin {gather*} \frac {c^{2} \log \left (F^{d x + c} b + a\right ) \log \left (F\right )^{2} + 2 \, d x {\rm Li}_2\left (-\frac {F^{d x + c} b + a}{a} + 1\right ) \log \left (F\right ) + {\left (d^{2} x^{2} - c^{2}\right )} \log \left (F\right )^{2} \log \left (\frac {F^{d x + c} b + a}{a}\right ) - 2 \, {\rm polylog}\left (3, -\frac {F^{d x + c} b}{a}\right )}{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*} \int \frac {F^{c + d x} x^{2}}{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 = -0.01 \begin {gather*} \int \frac {F^{c+d\,x}\,x^2}{a+F^{c+d\,x}\,b} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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