Optimal. Leaf size=182 \[ -\frac {\text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}+\frac {\log \left (a+b F^{c+d x}\right )}{a^2 b d^3 \log ^3(F)}-\frac {x \log \left (\frac {b F^{c+d x}}{a}+1\right )}{a^2 b d^2 \log ^2(F)}-\frac {x}{a^2 b d^2 \log ^2(F)}+\frac {x^2}{2 a^2 b d \log (F)}+\frac {x}{a b d^2 \log ^2(F) \left (a+b F^{c+d x}\right )}-\frac {x^2}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]
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Rubi [A] time = 0.29, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2191, 2185, 2184, 2190, 2279, 2391, 2282, 36, 29, 31} \[ -\frac {\text {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}-\frac {x \log \left (\frac {b F^{c+d x}}{a}+1\right )}{a^2 b d^2 \log ^2(F)}+\frac {\log \left (a+b F^{c+d x}\right )}{a^2 b d^3 \log ^3(F)}-\frac {x}{a^2 b d^2 \log ^2(F)}+\frac {x^2}{2 a^2 b d \log (F)}+\frac {x}{a b d^2 \log ^2(F) \left (a+b F^{c+d x}\right )}-\frac {x^2}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2184
Rule 2185
Rule 2190
Rule 2191
Rule 2279
Rule 2282
Rule 2391
Rubi steps
\begin {align*} \int \frac {F^{c+d x} x^2}{\left (a+b F^{c+d x}\right )^3} \, dx &=-\frac {x^2}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}+\frac {\int \frac {x}{\left (a+b F^{c+d x}\right )^2} \, dx}{b d \log (F)}\\ &=-\frac {x^2}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}-\frac {\int \frac {F^{c+d x} x}{\left (a+b F^{c+d x}\right )^2} \, dx}{a d \log (F)}+\frac {\int \frac {x}{a+b F^{c+d x}} \, dx}{a b d \log (F)}\\ &=\frac {x}{a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac {x^2}{2 a^2 b d \log (F)}-\frac {x^2}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}-\frac {\int \frac {1}{a+b F^{c+d x}} \, dx}{a b d^2 \log ^2(F)}-\frac {\int \frac {F^{c+d x} x}{a+b F^{c+d x}} \, dx}{a^2 d \log (F)}\\ &=\frac {x}{a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac {x^2}{2 a^2 b d \log (F)}-\frac {x^2}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}-\frac {x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{a^2 b d^2 \log ^2(F)}-\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,F^{c+d x}\right )}{a b d^3 \log ^3(F)}+\frac {\int \log \left (1+\frac {b F^{c+d x}}{a}\right ) \, dx}{a^2 b d^2 \log ^2(F)}\\ &=\frac {x}{a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac {x^2}{2 a^2 b d \log (F)}-\frac {x^2}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}-\frac {x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{a^2 b d^2 \log ^2(F)}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x} \, dx,x,F^{c+d x}\right )}{a^2 d^3 \log ^3(F)}-\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,F^{c+d x}\right )}{a^2 b d^3 \log ^3(F)}+\frac {\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a}\right )}{x} \, dx,x,F^{c+d x}\right )}{a^2 b d^3 \log ^3(F)}\\ &=-\frac {x}{a^2 b d^2 \log ^2(F)}+\frac {x}{a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac {x^2}{2 a^2 b d \log (F)}-\frac {x^2}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}+\frac {\log \left (a+b F^{c+d x}\right )}{a^2 b d^3 \log ^3(F)}-\frac {x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{a^2 b d^2 \log ^2(F)}-\frac {\text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 177, normalized size = 0.97 \[ \frac {b d^2 x^2 \log ^2(F) F^{c+d x} \left (2 a+b F^{c+d x}\right )-2 \left (a+b F^{c+d x}\right )^2 \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )+2 \left (a+b F^{c+d x}\right )^2 \log \left (\frac {b F^{c+d x}}{a}+1\right )-2 d x \log (F) \left (a+b F^{c+d x}\right ) \left (\left (a+b F^{c+d x}\right ) \log \left (\frac {b F^{c+d x}}{a}+1\right )+b F^{c+d x}\right )}{2 a^2 b d^3 \log ^3(F) \left (a+b F^{c+d x}\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 379, normalized size = 2.08 \[ -\frac {a^{2} c^{2} \log \relax (F)^{2} + 2 \, a^{2} c \log \relax (F) - {\left ({\left (b^{2} d^{2} x^{2} - b^{2} c^{2}\right )} \log \relax (F)^{2} - 2 \, {\left (b^{2} d x + b^{2} c\right )} \log \relax (F)\right )} F^{2 \, d x + 2 \, c} - 2 \, {\left ({\left (a b d^{2} x^{2} - a b c^{2}\right )} \log \relax (F)^{2} - {\left (a b d x + 2 \, a b c\right )} \log \relax (F)\right )} F^{d x + c} + 2 \, {\left (2 \, F^{d x + c} a b + F^{2 \, d x + 2 \, c} b^{2} + a^{2}\right )} {\rm Li}_2\left (-\frac {F^{d x + c} b + a}{a} + 1\right ) - 2 \, {\left (a^{2} c \log \relax (F) + {\left (b^{2} c \log \relax (F) + b^{2}\right )} F^{2 \, d x + 2 \, c} + 2 \, {\left (a b c \log \relax (F) + a b\right )} F^{d x + c} + a^{2}\right )} \log \left (F^{d x + c} b + a\right ) + 2 \, {\left ({\left (b^{2} d x + b^{2} c\right )} F^{2 \, d x + 2 \, c} \log \relax (F) + 2 \, {\left (a b d x + a b c\right )} F^{d x + c} \log \relax (F) + {\left (a^{2} d x + a^{2} c\right )} \log \relax (F)\right )} \log \left (\frac {F^{d x + c} b + a}{a}\right )}{2 \, {\left (2 \, F^{d x + c} a^{3} b^{2} d^{3} \log \relax (F)^{3} + F^{2 \, d x + 2 \, c} a^{2} b^{3} d^{3} \log \relax (F)^{3} + a^{4} b d^{3} \log \relax (F)^{3}\right )}} \]
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 \[ \int \frac {F^{d x + c} x^{2}}{{\left (F^{d x + c} b + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 304, normalized size = 1.67 \[ \frac {x^{2}}{2 a^{2} b d \ln \relax (F )}+\frac {c x}{a^{2} b \,d^{2} \ln \relax (F )}-\frac {\left (a d x \ln \relax (F )-2 b \,F^{d x +c}-2 a \right ) x}{2 \left (b \,F^{d x +c}+a \right )^{2} a b \,d^{2} \ln \relax (F )^{2}}+\frac {c^{2}}{2 a^{2} b \,d^{3} \ln \relax (F )}-\frac {x \ln \left (\frac {b \,F^{c} F^{d x}}{a}+1\right )}{a^{2} b \,d^{2} \ln \relax (F )^{2}}-\frac {c \ln \left (F^{c} F^{d x}\right )}{a^{2} b \,d^{3} \ln \relax (F )^{2}}-\frac {c \ln \left (\frac {b \,F^{c} F^{d x}}{a}+1\right )}{a^{2} b \,d^{3} \ln \relax (F )^{2}}+\frac {c \ln \left (b \,F^{c} F^{d x}+a \right )}{a^{2} b \,d^{3} \ln \relax (F )^{2}}-\frac {\polylog \left (2, -\frac {b \,F^{c} F^{d x}}{a}\right )}{a^{2} b \,d^{3} \ln \relax (F )^{3}}-\frac {\ln \left (F^{c} F^{d x}\right )}{a^{2} b \,d^{3} \ln \relax (F )^{3}}+\frac {\ln \left (b \,F^{c} F^{d x}+a \right )}{a^{2} b \,d^{3} \ln \relax (F )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 214, normalized size = 1.18 \[ -\frac {a d x^{2} \log \relax (F) - 2 \, F^{d x} F^{c} b x - 2 \, a x}{2 \, {\left (2 \, F^{d x} F^{c} a^{2} b^{2} d^{2} \log \relax (F)^{2} + F^{2 \, d x} F^{2 \, c} a b^{3} d^{2} \log \relax (F)^{2} + a^{3} b d^{2} \log \relax (F)^{2}\right )}} + \frac {\log \left (F^{d x}\right )^{2}}{2 \, a^{2} b d^{3} \log \relax (F)^{3}} - \frac {\log \left (\frac {F^{d x} F^{c} b}{a} + 1\right ) \log \left (F^{d x}\right ) + {\rm Li}_2\left (-\frac {F^{d x} F^{c} b}{a}\right )}{a^{2} b d^{3} \log \relax (F)^{3}} + \frac {\log \left (F^{d x} F^{c} b + a\right )}{a^{2} b d^{3} \log \relax (F)^{3}} - \frac {\log \left (F^{d x}\right )}{a^{2} b d^{3} \log \relax (F)^{3}} \]
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 \[ \int \frac {F^{c+d\,x}\,x^2}{{\left (a+F^{c+d\,x}\,b\right )}^3} \,d x \]
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 \[ \frac {2 F^{c + d x} b x - a d x^{2} \log {\relax (F )} + 2 a x}{4 F^{c + d x} a^{2} b^{2} d^{2} \log {\relax (F )}^{2} + 2 F^{2 c + 2 d x} a b^{3} d^{2} \log {\relax (F )}^{2} + 2 a^{3} b d^{2} \log {\relax (F )}^{2}} + \frac {\int \frac {d x \log {\relax (F )}}{a + b e^{c \log {\relax (F )}} e^{d x \log {\relax (F )}}}\, dx + \int \left (- \frac {1}{a + b e^{c \log {\relax (F )}} e^{d x \log {\relax (F )}}}\right )\, dx}{a b d^{2} \log {\relax (F )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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