Optimal. Leaf size=122 \[ \frac {6 F^{a+\frac {b}{c+d x}}}{b^4 d \log ^4(F)}-\frac {6 F^{a+\frac {b}{c+d x}}}{b^3 d \log ^3(F) (c+d x)}+\frac {3 F^{a+\frac {b}{c+d x}}}{b^2 d \log ^2(F) (c+d x)^2}-\frac {F^{a+\frac {b}{c+d x}}}{b d \log (F) (c+d x)^3} \]
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Rubi [A] time = 0.19, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2212, 2209} \[ \frac {3 F^{a+\frac {b}{c+d x}}}{b^2 d \log ^2(F) (c+d x)^2}-\frac {6 F^{a+\frac {b}{c+d x}}}{b^3 d \log ^3(F) (c+d x)}+\frac {6 F^{a+\frac {b}{c+d x}}}{b^4 d \log ^4(F)}-\frac {F^{a+\frac {b}{c+d x}}}{b d \log (F) (c+d x)^3} \]
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}}}{(c+d x)^5} \, dx &=-\frac {F^{a+\frac {b}{c+d x}}}{b d (c+d x)^3 \log (F)}-\frac {3 \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^4} \, dx}{b \log (F)}\\ &=\frac {3 F^{a+\frac {b}{c+d x}}}{b^2 d (c+d x)^2 \log ^2(F)}-\frac {F^{a+\frac {b}{c+d x}}}{b d (c+d x)^3 \log (F)}+\frac {6 \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^3} \, dx}{b^2 \log ^2(F)}\\ &=-\frac {6 F^{a+\frac {b}{c+d x}}}{b^3 d (c+d x) \log ^3(F)}+\frac {3 F^{a+\frac {b}{c+d x}}}{b^2 d (c+d x)^2 \log ^2(F)}-\frac {F^{a+\frac {b}{c+d x}}}{b d (c+d x)^3 \log (F)}-\frac {6 \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2} \, dx}{b^3 \log ^3(F)}\\ &=\frac {6 F^{a+\frac {b}{c+d x}}}{b^4 d \log ^4(F)}-\frac {6 F^{a+\frac {b}{c+d x}}}{b^3 d (c+d x) \log ^3(F)}+\frac {3 F^{a+\frac {b}{c+d x}}}{b^2 d (c+d x)^2 \log ^2(F)}-\frac {F^{a+\frac {b}{c+d x}}}{b d (c+d x)^3 \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 76, normalized size = 0.62 \[ \frac {F^{a+\frac {b}{c+d x}} \left (-b^3 \log ^3(F)+3 b^2 \log ^2(F) (c+d x)-6 b \log (F) (c+d x)^2+6 (c+d x)^3\right )}{b^4 d \log ^4(F) (c+d x)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 150, normalized size = 1.23 \[ \frac {{\left (6 \, d^{3} x^{3} - b^{3} \log \relax (F)^{3} + 18 \, c d^{2} x^{2} + 18 \, c^{2} d x + 6 \, c^{3} + 3 \, {\left (b^{2} d x + b^{2} c\right )} \log \relax (F)^{2} - 6 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \relax (F)\right )} F^{\frac {a d x + a c + b}{d x + c}}}{{\left (b^{4} d^{4} x^{3} + 3 \, b^{4} c d^{3} x^{2} + 3 \, b^{4} c^{2} d^{2} x + b^{4} c^{3} d\right )} \log \relax (F)^{4}} \]
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 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (d x + c\right )}^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 243, normalized size = 1.99 \[ \frac {\frac {6 d^{3} x^{4} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}-\frac {6 \left (b \ln \relax (F )-4 c \right ) d^{2} x^{3} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}+\frac {3 \left (b^{2} \ln \relax (F )^{2}-6 b c \ln \relax (F )+12 c^{2}\right ) d \,x^{2} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}-\frac {\left (b^{3} \ln \relax (F )^{3}-6 b^{2} c \ln \relax (F )^{2}+18 b \,c^{2} \ln \relax (F )-24 c^{3}\right ) x \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}-\frac {\left (b^{3} \ln \relax (F )^{3}-3 b^{2} c \ln \relax (F )^{2}+6 b \,c^{2} \ln \relax (F )-6 c^{3}\right ) c \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{4} d \ln \relax (F )^{4}}}{\left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (d x + c\right )}^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.80, size = 161, normalized size = 1.32 \[ \frac {F^{a+\frac {b}{c+d\,x}}\,\left (\frac {6\,x^3}{b^4\,d\,{\ln \relax (F)}^4}-\frac {b^3\,{\ln \relax (F)}^3-3\,b^2\,c\,{\ln \relax (F)}^2+6\,b\,c^2\,\ln \relax (F)-6\,c^3}{b^4\,d^4\,{\ln \relax (F)}^4}+\frac {x^2\,\left (18\,c-6\,b\,\ln \relax (F)\right )}{b^4\,d^2\,{\ln \relax (F)}^4}+\frac {3\,x\,\left (b^2\,{\ln \relax (F)}^2-4\,b\,c\,\ln \relax (F)+6\,c^2\right )}{b^4\,d^3\,{\ln \relax (F)}^4}\right )}{x^3+\frac {c^3}{d^3}+\frac {3\,c\,x^2}{d}+\frac {3\,c^2\,x}{d^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 177, normalized size = 1.45 \[ \frac {F^{a + \frac {b}{c + d x}} \left (- b^{3} \log {\relax (F )}^{3} + 3 b^{2} c \log {\relax (F )}^{2} + 3 b^{2} d x \log {\relax (F )}^{2} - 6 b c^{2} \log {\relax (F )} - 12 b c d x \log {\relax (F )} - 6 b d^{2} x^{2} \log {\relax (F )} + 6 c^{3} + 18 c^{2} d x + 18 c d^{2} x^{2} + 6 d^{3} x^{3}\right )}{b^{4} c^{3} d \log {\relax (F )}^{4} + 3 b^{4} c^{2} d^{2} x \log {\relax (F )}^{4} + 3 b^{4} c d^{3} x^{2} \log {\relax (F )}^{4} + b^{4} d^{4} x^{3} \log {\relax (F )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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