Optimal. Leaf size=62 \[ \frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^2} \]
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Rubi [A] time = 0.09, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2212, 2209} \[ \frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^2} \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2212
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^5} \, dx &=-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^2 \log (F)}-\frac {\int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^3} \, dx}{b \log (F)}\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^2 \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 0.76 \[ \frac {F^{a+\frac {b}{(c+d x)^2}} \left ((c+d x)^2-b \log (F)\right )}{2 b^2 d \log ^2(F) (c+d x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 100, normalized size = 1.61 \[ \frac {{\left (d^{2} x^{2} + 2 \, c d x + c^{2} - b \log \relax (F)\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \log \relax (F)^{2}} \]
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 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 185, normalized size = 2.98 \[ \frac {\frac {d^{3} x^{4} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{2 b^{2} \ln \relax (F )^{2}}+\frac {2 c \,d^{2} x^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{2} \ln \relax (F )^{2}}-\frac {\left (b \ln \relax (F )-6 c^{2}\right ) d \,x^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{2 b^{2} \ln \relax (F )^{2}}-\frac {\left (b \ln \relax (F )-2 c^{2}\right ) c x \,{\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{2} \ln \relax (F )^{2}}-\frac {\left (b \ln \relax (F )-c^{2}\right ) c^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{2 b^{2} d \ln \relax (F )^{2}}}{\left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 101, normalized size = 1.63 \[ \frac {{\left (F^{a} d^{2} x^{2} + 2 \, F^{a} c d x + F^{a} c^{2} - F^{a} b \log \relax (F)\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{2} d^{3} x^{2} \log \relax (F)^{2} + 2 \, b^{2} c d^{2} x \log \relax (F)^{2} + b^{2} c^{2} d \log \relax (F)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.71, size = 97, normalized size = 1.56 \[ \frac {F^a\,F^{\frac {b}{c^2+2\,c\,d\,x+d^2\,x^2}}\,\left (\frac {x^2}{2\,b^2\,d\,{\ln \relax (F)}^2}-\frac {b\,\ln \relax (F)-c^2}{2\,b^2\,d^3\,{\ln \relax (F)}^2}+\frac {c\,x}{b^2\,d^2\,{\ln \relax (F)}^2}\right )}{x^2+\frac {c^2}{d^2}+\frac {2\,c\,x}{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 82, normalized size = 1.32 \[ \frac {F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (- b \log {\relax (F )} + c^{2} + 2 c d x + d^{2} x^{2}\right )}{2 b^{2} c^{2} d \log {\relax (F )}^{2} + 4 b^{2} c d^{2} x \log {\relax (F )}^{2} + 2 b^{2} d^{3} x^{2} \log {\relax (F )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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