Optimal. Leaf size=91 \[ -\frac {F^{a+\frac {b}{(c+d x)^2}}}{b^3 d \log ^3(F)}+\frac {F^{a+\frac {b}{(c+d x)^2}}}{b^2 d \log ^2(F) (c+d x)^2}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^4} \]
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Rubi [A] time = 0.14, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2212, 2209} \[ \frac {F^{a+\frac {b}{(c+d x)^2}}}{b^2 d \log ^2(F) (c+d x)^2}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{b^3 d \log ^3(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^4} \]
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)^7} \, dx &=-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^4 \log (F)}-\frac {2 \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^5} \, dx}{b \log (F)}\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}}}{b^2 d (c+d x)^2 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^4 \log (F)}+\frac {2 \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^3} \, dx}{b^2 \log ^2(F)}\\ &=-\frac {F^{a+\frac {b}{(c+d x)^2}}}{b^3 d \log ^3(F)}+\frac {F^{a+\frac {b}{(c+d x)^2}}}{b^2 d (c+d x)^2 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^4 \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 64, normalized size = 0.70 \[ -\frac {F^{a+\frac {b}{(c+d x)^2}} \left (b^2 \log ^2(F)-2 b \log (F) (c+d x)^2+2 (c+d x)^4\right )}{2 b^3 d \log ^3(F) (c+d x)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 180, normalized size = 1.98 \[ -\frac {{\left (2 \, d^{4} x^{4} + 8 \, c d^{3} x^{3} + 12 \, c^{2} d^{2} x^{2} + 8 \, c^{3} d x + 2 \, c^{4} + b^{2} \log \relax (F)^{2} - 2 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \relax (F)\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{3} d^{5} x^{4} + 4 \, b^{3} c d^{4} x^{3} + 6 \, b^{3} c^{2} d^{3} x^{2} + 4 \, b^{3} c^{3} d^{2} x + b^{3} c^{4} d\right )} \log \relax (F)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.56, size = 1656, normalized size = 18.20 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 301, normalized size = 3.31 \[ \frac {-\frac {d^{5} x^{6} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{3} \ln \relax (F )^{3}}-\frac {6 c \,d^{4} x^{5} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{3} \ln \relax (F )^{3}}+\frac {\left (b \ln \relax (F )-15 c^{2}\right ) d^{3} x^{4} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{3} \ln \relax (F )^{3}}+\frac {4 \left (b \ln \relax (F )-5 c^{2}\right ) c \,d^{2} x^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{3} \ln \relax (F )^{3}}-\frac {\left (b^{2} \ln \relax (F )^{2}-12 b \,c^{2} \ln \relax (F )+30 c^{4}\right ) d \,x^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{2 b^{3} \ln \relax (F )^{3}}-\frac {\left (b^{2} \ln \relax (F )^{2}-4 b \,c^{2} \ln \relax (F )+6 c^{4}\right ) c x \,{\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{3} \ln \relax (F )^{3}}-\frac {\left (b^{2} \ln \relax (F )^{2}-2 b \,c^{2} \ln \relax (F )+2 c^{4}\right ) c^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{2 b^{3} d \ln \relax (F )^{3}}}{\left (d x +c \right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.01, size = 208, normalized size = 2.29 \[ -\frac {{\left (2 \, F^{a} d^{4} x^{4} + 8 \, F^{a} c d^{3} x^{3} + 2 \, F^{a} c^{4} - 2 \, F^{a} b c^{2} \log \relax (F) + F^{a} b^{2} \log \relax (F)^{2} + 2 \, {\left (6 \, F^{a} c^{2} d^{2} - F^{a} b d^{2} \log \relax (F)\right )} x^{2} + 4 \, {\left (2 \, F^{a} c^{3} d - F^{a} b c d \log \relax (F)\right )} x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{3} d^{5} x^{4} \log \relax (F)^{3} + 4 \, b^{3} c d^{4} x^{3} \log \relax (F)^{3} + 6 \, b^{3} c^{2} d^{3} x^{2} \log \relax (F)^{3} + 4 \, b^{3} c^{3} d^{2} x \log \relax (F)^{3} + b^{3} c^{4} d \log \relax (F)^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.96, size = 183, normalized size = 2.01 \[ -\frac {F^a\,F^{\frac {b}{c^2+2\,c\,d\,x+d^2\,x^2}}\,\left (\frac {x^4}{b^3\,d\,{\ln \relax (F)}^3}+\frac {b^2\,{\ln \relax (F)}^2-2\,b\,c^2\,\ln \relax (F)+2\,c^4}{2\,b^3\,d^5\,{\ln \relax (F)}^3}+\frac {4\,c\,x^3}{b^3\,d^2\,{\ln \relax (F)}^3}-\frac {x^2\,\left (b\,\ln \relax (F)-6\,c^2\right )}{b^3\,d^3\,{\ln \relax (F)}^3}-\frac {2\,c\,x\,\left (b\,\ln \relax (F)-2\,c^2\right )}{b^3\,d^4\,{\ln \relax (F)}^3}\right )}{x^4+\frac {c^4}{d^4}+\frac {4\,c\,x^3}{d}+\frac {4\,c^3\,x}{d^3}+\frac {6\,c^2\,x^2}{d^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.39, size = 189, normalized size = 2.08 \[ \frac {F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (- b^{2} \log {\relax (F )}^{2} + 2 b c^{2} \log {\relax (F )} + 4 b c d x \log {\relax (F )} + 2 b d^{2} x^{2} \log {\relax (F )} - 2 c^{4} - 8 c^{3} d x - 12 c^{2} d^{2} x^{2} - 8 c d^{3} x^{3} - 2 d^{4} x^{4}\right )}{2 b^{3} c^{4} d \log {\relax (F )}^{3} + 8 b^{3} c^{3} d^{2} x \log {\relax (F )}^{3} + 12 b^{3} c^{2} d^{3} x^{2} \log {\relax (F )}^{3} + 8 b^{3} c d^{4} x^{3} \log {\relax (F )}^{3} + 2 b^{3} d^{5} x^{4} \log {\relax (F )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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