Optimal. Leaf size=63 \[ -\frac {\text {Int}\left (\frac {1}{x^3 \left (a+b F^{c+d x}\right )^2},x\right )}{b d \log (F)}-\frac {1}{2 b d x^2 \log (F) \left (a+b F^{c+d x}\right )^2} \]
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Rubi [A] time = 0.11, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx &=-\frac {1}{2 b d \left (a+b F^{c+d x}\right )^2 x^2 \log (F)}-\frac {\int \frac {1}{\left (a+b F^{c+d x}\right )^2 x^3} \, dx}{b d \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.78, size = 0, normalized size = 0.00 \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {F^{d x + c}}{3 \, F^{d x + c} a^{2} b x^{2} + 3 \, F^{2 \, d x + 2 \, c} a b^{2} x^{2} + F^{3 \, d x + 3 \, c} b^{3} x^{2} + a^{3} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{d x + c}}{{\left (F^{d x + c} b + a\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {F^{d x +c}}{\left (b \,F^{d x +c}+a \right )^{3} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a d x \log \relax (F) + 2 \, F^{d x} F^{c} b + 2 \, a}{2 \, {\left (2 \, F^{d x} F^{c} a^{2} b^{2} d^{2} x^{3} \log \relax (F)^{2} + F^{2 \, d x} F^{2 \, c} a b^{3} d^{2} x^{3} \log \relax (F)^{2} + a^{3} b d^{2} x^{3} \log \relax (F)^{2}\right )}} - \int \frac {d x \log \relax (F) + 3}{F^{d x} F^{c} a b^{2} d^{2} x^{4} \log \relax (F)^{2} + a^{2} b d^{2} x^{4} \log \relax (F)^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {F^{c+d\,x}}{x^2\,{\left (a+F^{c+d\,x}\,b\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {- 2 F^{c + d x} b - a d x \log {\relax (F )} - 2 a}{4 F^{c + d x} a^{2} b^{2} d^{2} x^{3} \log {\relax (F )}^{2} + 2 F^{2 c + 2 d x} a b^{3} d^{2} x^{3} \log {\relax (F )}^{2} + 2 a^{3} b d^{2} x^{3} \log {\relax (F )}^{2}} - \frac {\int \frac {d x \log {\relax (F )}}{a x^{4} + b x^{4} e^{c \log {\relax (F )}} e^{d x \log {\relax (F )}}}\, dx + \int \frac {3}{a x^{4} + b x^{4} e^{c \log {\relax (F )}} e^{d x \log {\relax (F )}}}\, dx}{a b d^{2} \log {\relax (F )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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