Optimal. Leaf size=261 \[ -\frac {3 x^2}{2 a^2 b d^2 \log ^2(F)}+\frac {3 x^2}{2 a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac {x^3}{2 a^2 b d \log (F)}-\frac {x^3}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}+\frac {3 x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}-\frac {3 x^2 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{2 a^2 b d^2 \log ^2(F)}+\frac {3 \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}-\frac {3 x \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}+\frac {3 \text {Li}_3\left (-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)} \]
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Rubi [A]
time = 0.34, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2222, 2216,
2215, 2221, 2611, 2320, 6724, 2317, 2438} \begin {gather*} \frac {3 \text {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}+\frac {3 \text {PolyLog}\left (3,-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}-\frac {3 x \text {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}+\frac {3 x \log \left (\frac {b F^{c+d x}}{a}+1\right )}{a^2 b d^3 \log ^3(F)}-\frac {3 x^2 \log \left (\frac {b F^{c+d x}}{a}+1\right )}{2 a^2 b d^2 \log ^2(F)}-\frac {3 x^2}{2 a^2 b d^2 \log ^2(F)}+\frac {x^3}{2 a^2 b d \log (F)}+\frac {3 x^2}{2 a b d^2 \log ^2(F) \left (a+b F^{c+d x}\right )}-\frac {x^3}{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 2215
Rule 2216
Rule 2221
Rule 2222
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 6724
Rubi steps
\begin {align*} \int \frac {F^{c+d x} x^3}{\left (a+b F^{c+d x}\right )^3} \, dx &=-\frac {x^3}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}+\frac {3 \int \frac {x^2}{\left (a+b F^{c+d x}\right )^2} \, dx}{2 b d \log (F)}\\ &=-\frac {x^3}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}-\frac {3 \int \frac {F^{c+d x} x^2}{\left (a+b F^{c+d x}\right )^2} \, dx}{2 a d \log (F)}+\frac {3 \int \frac {x^2}{a+b F^{c+d x}} \, dx}{2 a b d \log (F)}\\ &=\frac {3 x^2}{2 a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac {x^3}{2 a^2 b d \log (F)}-\frac {x^3}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}-\frac {3 \int \frac {x}{a+b F^{c+d x}} \, dx}{a b d^2 \log ^2(F)}-\frac {3 \int \frac {F^{c+d x} x^2}{a+b F^{c+d x}} \, dx}{2 a^2 d \log (F)}\\ &=-\frac {3 x^2}{2 a^2 b d^2 \log ^2(F)}+\frac {3 x^2}{2 a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac {x^3}{2 a^2 b d \log (F)}-\frac {x^3}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}-\frac {3 x^2 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{2 a^2 b d^2 \log ^2(F)}+\frac {3 \int \frac {F^{c+d x} x}{a+b F^{c+d x}} \, dx}{a^2 d^2 \log ^2(F)}+\frac {3 \int x \log \left (1+\frac {b F^{c+d x}}{a}\right ) \, dx}{a^2 b d^2 \log ^2(F)}\\ &=-\frac {3 x^2}{2 a^2 b d^2 \log ^2(F)}+\frac {3 x^2}{2 a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac {x^3}{2 a^2 b d \log (F)}-\frac {x^3}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}+\frac {3 x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}-\frac {3 x^2 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{2 a^2 b d^2 \log ^2(F)}-\frac {3 x \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}-\frac {3 \int \log \left (1+\frac {b F^{c+d x}}{a}\right ) \, dx}{a^2 b d^3 \log ^3(F)}+\frac {3 \int \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right ) \, dx}{a^2 b d^3 \log ^3(F)}\\ &=-\frac {3 x^2}{2 a^2 b d^2 \log ^2(F)}+\frac {3 x^2}{2 a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac {x^3}{2 a^2 b d \log (F)}-\frac {x^3}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}+\frac {3 x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}-\frac {3 x^2 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{2 a^2 b d^2 \log ^2(F)}-\frac {3 x \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}-\frac {3 \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a}\right )}{x} \, dx,x,F^{c+d x}\right )}{a^2 b d^4 \log ^4(F)}+\frac {3 \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{a}\right )}{x} \, dx,x,F^{c+d x}\right )}{a^2 b d^4 \log ^4(F)}\\ &=-\frac {3 x^2}{2 a^2 b d^2 \log ^2(F)}+\frac {3 x^2}{2 a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac {x^3}{2 a^2 b d \log (F)}-\frac {x^3}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}+\frac {3 x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}-\frac {3 x^2 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{2 a^2 b d^2 \log ^2(F)}+\frac {3 \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}-\frac {3 x \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}+\frac {3 \text {Li}_3\left (-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 220, normalized size = 0.84 \begin {gather*} \frac {d x \log (F) \left (b d^2 F^{c+d x} \left (2 a+b F^{c+d x}\right ) x^2 \log ^2(F)+6 \left (a+b F^{c+d x}\right )^2 \log \left (1+\frac {b F^{c+d x}}{a}\right )-3 d \left (a+b F^{c+d x}\right ) x \log (F) \left (b F^{c+d x}+\left (a+b F^{c+d x}\right ) \log \left (1+\frac {b F^{c+d x}}{a}\right )\right )\right )-6 \left (a+b F^{c+d x}\right )^2 (-1+d x \log (F)) \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )+6 \left (a+b F^{c+d x}\right )^2 \text {Li}_3\left (-\frac {b F^{c+d x}}{a}\right )}{2 a^2 b d^4 \left (a+b F^{c+d x}\right )^2 \log ^4(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 501, normalized size = 1.92
method | result | size |
risch | \(-\frac {x^{2} \left (\ln \left (F \right ) a d x -3 b \,F^{d x +c}-3 a \right )}{2 \ln \left (F \right )^{2} d^{2} b \left (a +b \,F^{d x +c}\right )^{2} a}+\frac {x^{3}}{2 a^{2} b d \ln \left (F \right )}-\frac {3 c^{2} x}{2 b \,a^{2} d^{3} \ln \left (F \right )}-\frac {c^{3}}{b \,a^{2} d^{4} \ln \left (F \right )}-\frac {3 \ln \left (1+\frac {b \,F^{d x} F^{c}}{a}\right ) x^{2}}{2 b \,a^{2} d^{2} \ln \left (F \right )^{2}}+\frac {3 \ln \left (1+\frac {b \,F^{d x} F^{c}}{a}\right ) c^{2}}{2 b \,a^{2} d^{4} \ln \left (F \right )^{2}}-\frac {3 \polylog \left (2, -\frac {b \,F^{d x} F^{c}}{a}\right ) x}{b \,a^{2} d^{3} \ln \left (F \right )^{3}}+\frac {3 \polylog \left (3, -\frac {b \,F^{d x} F^{c}}{a}\right )}{b \,a^{2} d^{4} \ln \left (F \right )^{4}}-\frac {3 c^{2} \ln \left (a +F^{c} F^{d x} b \right )}{2 b \,a^{2} d^{4} \ln \left (F \right )^{2}}+\frac {3 c^{2} \ln \left (F^{d x} F^{c}\right )}{2 b \,a^{2} d^{4} \ln \left (F \right )^{2}}-\frac {3 x^{2}}{2 a^{2} b \,d^{2} \ln \left (F \right )^{2}}-\frac {3 c x}{b \,a^{2} d^{3} \ln \left (F \right )^{2}}-\frac {3 c^{2}}{2 b \,a^{2} d^{4} \ln \left (F \right )^{2}}+\frac {3 \ln \left (1+\frac {b \,F^{d x} F^{c}}{a}\right ) x}{b \,a^{2} d^{3} \ln \left (F \right )^{3}}+\frac {3 \ln \left (1+\frac {b \,F^{d x} F^{c}}{a}\right ) c}{b \,a^{2} d^{4} \ln \left (F \right )^{3}}+\frac {3 \polylog \left (2, -\frac {b \,F^{d x} F^{c}}{a}\right )}{b \,a^{2} d^{4} \ln \left (F \right )^{4}}-\frac {3 c \ln \left (a +F^{c} F^{d x} b \right )}{b \,a^{2} d^{4} \ln \left (F \right )^{3}}+\frac {3 c \ln \left (F^{d x} F^{c}\right )}{b \,a^{2} d^{4} \ln \left (F \right )^{3}}\) | \(501\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 266, normalized size = 1.02 \begin {gather*} -\frac {a d x^{3} \log \left (F\right ) - 3 \, F^{d x} F^{c} b x^{2} - 3 \, a x^{2}}{2 \, {\left (2 \, F^{d x} F^{c} a^{2} b^{2} d^{2} \log \left (F\right )^{2} + F^{2 \, d x} F^{2 \, c} a b^{3} d^{2} \log \left (F\right )^{2} + a^{3} b d^{2} \log \left (F\right )^{2}\right )}} + \frac {d^{3} x^{3} \log \left (F\right )^{3} - 3 \, d^{2} x^{2} \log \left (F\right )^{2}}{2 \, a^{2} b d^{4} \log \left (F\right )^{4}} - \frac {3 \, {\left (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})\right )}}{2 \, a^{2} b d^{4} \log \left (F\right )^{4}} + \frac {3 \, {\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^{2} b d^{4} \log \left (F\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 577 vs.
\(2 (249) = 498\).
time = 0.36, size = 577, normalized size = 2.21 \begin {gather*} \frac {a^{2} c^{3} \log \left (F\right )^{3} + 3 \, a^{2} c^{2} \log \left (F\right )^{2} + {\left ({\left (b^{2} d^{3} x^{3} + b^{2} c^{3}\right )} \log \left (F\right )^{3} - 3 \, {\left (b^{2} d^{2} x^{2} - b^{2} c^{2}\right )} \log \left (F\right )^{2}\right )} F^{2 \, d x + 2 \, c} + {\left (2 \, {\left (a b d^{3} x^{3} + a b c^{3}\right )} \log \left (F\right )^{3} - 3 \, {\left (a b d^{2} x^{2} - 2 \, a b c^{2}\right )} \log \left (F\right )^{2}\right )} F^{d x + c} - 6 \, {\left (a^{2} d x \log \left (F\right ) + {\left (b^{2} d x \log \left (F\right ) - b^{2}\right )} F^{2 \, d x + 2 \, c} + 2 \, {\left (a b d x \log \left (F\right ) - a b\right )} F^{d x + c} - a^{2}\right )} {\rm Li}_2\left (-\frac {F^{d x + c} b + a}{a} + 1\right ) - 3 \, {\left (a^{2} c^{2} \log \left (F\right )^{2} + 2 \, a^{2} c \log \left (F\right ) + {\left (b^{2} c^{2} \log \left (F\right )^{2} + 2 \, b^{2} c \log \left (F\right )\right )} F^{2 \, d x + 2 \, c} + 2 \, {\left (a b c^{2} \log \left (F\right )^{2} + 2 \, a b c \log \left (F\right )\right )} F^{d x + c}\right )} \log \left (F^{d x + c} b + a\right ) - 3 \, {\left ({\left (a^{2} d^{2} x^{2} - a^{2} c^{2}\right )} \log \left (F\right )^{2} + {\left ({\left (b^{2} d^{2} x^{2} - b^{2} c^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (b^{2} d x + b^{2} c\right )} \log \left (F\right )\right )} F^{2 \, d x + 2 \, c} + 2 \, {\left ({\left (a b d^{2} x^{2} - a b c^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (a b d x + a b c\right )} \log \left (F\right )\right )} F^{d x + c} - 2 \, {\left (a^{2} d x + a^{2} c\right )} \log \left (F\right )\right )} \log \left (\frac {F^{d x + c} b + a}{a}\right ) + 6 \, {\left (2 \, F^{d x + c} a b + F^{2 \, d x + 2 \, c} b^{2} + a^{2}\right )} {\rm polylog}\left (3, -\frac {F^{d x + c} b}{a}\right )}{2 \, {\left (2 \, F^{d x + c} a^{3} b^{2} d^{4} \log \left (F\right )^{4} + F^{2 \, d x + 2 \, c} a^{2} b^{3} d^{4} \log \left (F\right )^{4} + a^{4} b d^{4} \log \left (F\right )^{4}\right )}} \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 {3 F^{c + d x} b x^{2} - a d x^{3} \log {\left (F \right )} + 3 a x^{2}}{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 {3 \left (\int \left (- \frac {2 x}{a + b e^{c \log {\left (F \right )}} e^{d x \log {\left (F \right )}}}\right )\, dx + \int \frac {d x^{2} \log {\left (F \right )}}{a + b e^{c \log {\left (F \right )}} e^{d x \log {\left (F \right )}}}\, dx\right )}{2 a 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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {F^{c+d\,x}\,x^3}{{\left (a+F^{c+d\,x}\,b\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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