Optimal. Leaf size=115 \[ \frac {6 \text {Li}_4\left (-\frac {b F^{c+d x}}{a}\right )}{b d^4 \log ^4(F)}-\frac {6 x \text {Li}_3\left (-\frac {b F^{c+d x}}{a}\right )}{b d^3 \log ^3(F)}+\frac {3 x^2 \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}+\frac {x^3 \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)} \]
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Rubi [A] time = 0.13, antiderivative size = 115, 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 = {2190, 2531, 6609, 2282, 6589} \[ \frac {3 x^2 \text {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}-\frac {6 x \text {PolyLog}\left (3,-\frac {b F^{c+d x}}{a}\right )}{b d^3 \log ^3(F)}+\frac {6 \text {PolyLog}\left (4,-\frac {b F^{c+d x}}{a}\right )}{b d^4 \log ^4(F)}+\frac {x^3 \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {F^{c+d x} x^3}{a+b F^{c+d x}} \, dx &=\frac {x^3 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}-\frac {3 \int x^2 \log \left (1+\frac {b F^{c+d x}}{a}\right ) \, dx}{b d \log (F)}\\ &=\frac {x^3 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}+\frac {3 x^2 \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}-\frac {6 \int x \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right ) \, dx}{b d^2 \log ^2(F)}\\ &=\frac {x^3 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}+\frac {3 x^2 \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}-\frac {6 x \text {Li}_3\left (-\frac {b F^{c+d x}}{a}\right )}{b d^3 \log ^3(F)}+\frac {6 \int \text {Li}_3\left (-\frac {b F^{c+d x}}{a}\right ) \, dx}{b d^3 \log ^3(F)}\\ &=\frac {x^3 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}+\frac {3 x^2 \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}-\frac {6 x \text {Li}_3\left (-\frac {b F^{c+d x}}{a}\right )}{b d^3 \log ^3(F)}+\frac {6 \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{a}\right )}{x} \, dx,x,F^{c+d x}\right )}{b d^4 \log ^4(F)}\\ &=\frac {x^3 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}+\frac {3 x^2 \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}-\frac {6 x \text {Li}_3\left (-\frac {b F^{c+d x}}{a}\right )}{b d^3 \log ^3(F)}+\frac {6 \text {Li}_4\left (-\frac {b F^{c+d x}}{a}\right )}{b d^4 \log ^4(F)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 115, normalized size = 1.00 \[ \frac {6 \text {Li}_4\left (-\frac {b F^{c+d x}}{a}\right )}{b d^4 \log ^4(F)}-\frac {6 x \text {Li}_3\left (-\frac {b F^{c+d x}}{a}\right )}{b d^3 \log ^3(F)}+\frac {3 x^2 \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}+\frac {x^3 \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.44, size = 134, normalized size = 1.17 \[ \frac {3 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {F^{d x + c} b + a}{a} + 1\right ) \log \relax (F)^{2} - c^{3} \log \left (F^{d x + c} b + a\right ) \log \relax (F)^{3} + {\left (d^{3} x^{3} + c^{3}\right )} \log \relax (F)^{3} \log \left (\frac {F^{d x + c} b + a}{a}\right ) - 6 \, d x \log \relax (F) {\rm polylog}\left (3, -\frac {F^{d x + c} b}{a}\right ) + 6 \, {\rm polylog}\left (4, -\frac {F^{d x + c} b}{a}\right )}{b d^{4} \log \relax (F)^{4}} \]
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 \[ \int \frac {F^{d x + c} x^{3}}{F^{d x + c} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 225, normalized size = 1.96 \[ \frac {x^{3} \ln \left (\frac {b \,F^{c} F^{d x}}{a}+1\right )}{b d \ln \relax (F )}-\frac {c^{3} x}{b \,d^{3}}-\frac {3 c^{4}}{4 b \,d^{4}}+\frac {c^{3} \ln \left (F^{c} F^{d x}\right )}{b \,d^{4} \ln \relax (F )}+\frac {c^{3} \ln \left (\frac {b \,F^{c} F^{d x}}{a}+1\right )}{b \,d^{4} \ln \relax (F )}-\frac {c^{3} \ln \left (b \,F^{c} F^{d x}+a \right )}{b \,d^{4} \ln \relax (F )}+\frac {3 x^{2} \polylog \left (2, -\frac {b \,F^{c} F^{d x}}{a}\right )}{b \,d^{2} \ln \relax (F )^{2}}-\frac {6 x \polylog \left (3, -\frac {b \,F^{c} F^{d x}}{a}\right )}{b \,d^{3} \ln \relax (F )^{3}}+\frac {6 \polylog \left (4, -\frac {b \,F^{c} F^{d x}}{a}\right )}{b \,d^{4} \ln \relax (F )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 133, normalized size = 1.16 \[ \frac {x^{4}}{4 \, b} - \frac {\log \left (F^{d x}\right )^{4}}{4 \, b d^{4} \log \relax (F)^{4}} + \frac {\log \left (\frac {F^{d x} F^{c} b}{a} + 1\right ) \log \left (F^{d x}\right )^{3} + 3 \, {\rm Li}_2\left (-\frac {F^{d x} F^{c} b}{a}\right ) \log \left (F^{d x}\right )^{2} - 6 \, \log \left (F^{d x}\right ) {\rm Li}_{3}(-\frac {F^{d x} F^{c} b}{a}) + 6 \, {\rm Li}_{4}(-\frac {F^{d x} F^{c} b}{a})}{b d^{4} \log \relax (F)^{4}} \]
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 \[ \int \frac {F^{c+d\,x}\,x^3}{a+F^{c+d\,x}\,b} \,d x \]
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 \[ \int \frac {F^{c + d x} x^{3}}{F^{c} F^{d x} b + a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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