Optimal. Leaf size=108 \[ \frac {F^{a+\frac {b}{c+d x}} \left (-b^5 \log ^5(F)+5 b^4 \log ^4(F) (c+d x)-20 b^3 \log ^3(F) (c+d x)^2+60 b^2 \log ^2(F) (c+d x)^3-120 b \log (F) (c+d x)^4+120 (c+d x)^5\right )}{b^6 d \log ^6(F) (c+d x)^5} \]
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Rubi [C] time = 0.05, antiderivative size = 28, normalized size of antiderivative = 0.26, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2218} \[ \frac {F^a \text {Gamma}\left (6,-\frac {b \log (F)}{c+d x}\right )}{b^6 d \log ^6(F)} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^7} \, dx &=\frac {F^a \Gamma \left (6,-\frac {b \log (F)}{c+d x}\right )}{b^6 d \log ^6(F)}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 28, normalized size = 0.26 \[ \frac {F^a \Gamma \left (6,-\frac {b \log (F)}{c+d x}\right )}{b^6 d \log ^6(F)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 302, normalized size = 2.80 \[ \frac {{\left (120 \, d^{5} x^{5} - b^{5} \log \relax (F)^{5} + 600 \, c d^{4} x^{4} + 1200 \, c^{2} d^{3} x^{3} + 1200 \, c^{3} d^{2} x^{2} + 600 \, c^{4} d x + 120 \, c^{5} + 5 \, {\left (b^{4} d x + b^{4} c\right )} \log \relax (F)^{4} - 20 \, {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \log \relax (F)^{3} + 60 \, {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \relax (F)^{2} - 120 \, {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \log \relax (F)\right )} F^{\frac {a d x + a c + b}{d x + c}}}{{\left (b^{6} d^{6} x^{5} + 5 \, b^{6} c d^{5} x^{4} + 10 \, b^{6} c^{2} d^{4} x^{3} + 10 \, b^{6} c^{3} d^{3} x^{2} + 5 \, b^{6} c^{4} d^{2} x + b^{6} c^{5} d\right )} \log \relax (F)^{6}} \]
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 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (d x + c\right )}^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 427, normalized size = 3.95 \[ \frac {\frac {120 d^{5} x^{6} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{6} \ln \relax (F )^{6}}-\frac {120 \left (b \ln \relax (F )-6 c \right ) d^{4} x^{5} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{6} \ln \relax (F )^{6}}+\frac {60 \left (b^{2} \ln \relax (F )^{2}-10 b c \ln \relax (F )+30 c^{2}\right ) d^{3} x^{4} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{6} \ln \relax (F )^{6}}-\frac {20 \left (b^{3} \ln \relax (F )^{3}-12 b^{2} c \ln \relax (F )^{2}+60 b \,c^{2} \ln \relax (F )-120 c^{3}\right ) d^{2} x^{3} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{6} \ln \relax (F )^{6}}+\frac {5 \left (b^{4} \ln \relax (F )^{4}-12 b^{3} c \ln \relax (F )^{3}+72 b^{2} c^{2} \ln \relax (F )^{2}-240 b \,c^{3} \ln \relax (F )+360 c^{4}\right ) d \,x^{2} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{6} \ln \relax (F )^{6}}-\frac {\left (b^{5} \ln \relax (F )^{5}-10 b^{4} c \ln \relax (F )^{4}+60 b^{3} c^{2} \ln \relax (F )^{3}-240 b^{2} c^{3} \ln \relax (F )^{2}+600 b \,c^{4} \ln \relax (F )-720 c^{5}\right ) x \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{6} \ln \relax (F )^{6}}-\frac {\left (b^{5} \ln \relax (F )^{5}-5 b^{4} c \ln \relax (F )^{4}+20 b^{3} c^{2} \ln \relax (F )^{3}-60 b^{2} c^{3} \ln \relax (F )^{2}+120 b \,c^{4} \ln \relax (F )-120 c^{5}\right ) c \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{6} d \ln \relax (F )^{6}}}{\left (d x +c \right )^{6}} \]
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 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (d x + c\right )}^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.94, size = 315, normalized size = 2.92 \[ \frac {F^a\,F^{\frac {b}{c+d\,x}}\,\left (\frac {120\,x^5}{b^6\,d\,{\ln \relax (F)}^6}-\frac {b^5\,{\ln \relax (F)}^5-5\,b^4\,c\,{\ln \relax (F)}^4+20\,b^3\,c^2\,{\ln \relax (F)}^3-60\,b^2\,c^3\,{\ln \relax (F)}^2+120\,b\,c^4\,\ln \relax (F)-120\,c^5}{b^6\,d^6\,{\ln \relax (F)}^6}-\frac {20\,x^2\,\left (b^3\,{\ln \relax (F)}^3-9\,b^2\,c\,{\ln \relax (F)}^2+36\,b\,c^2\,\ln \relax (F)-60\,c^3\right )}{b^6\,d^4\,{\ln \relax (F)}^6}+\frac {60\,x^3\,\left (b^2\,{\ln \relax (F)}^2-8\,b\,c\,\ln \relax (F)+20\,c^2\right )}{b^6\,d^3\,{\ln \relax (F)}^6}+\frac {120\,x^4\,\left (5\,c-b\,\ln \relax (F)\right )}{b^6\,d^2\,{\ln \relax (F)}^6}+\frac {5\,x\,\left (b^4\,{\ln \relax (F)}^4-8\,b^3\,c\,{\ln \relax (F)}^3+36\,b^2\,c^2\,{\ln \relax (F)}^2-96\,b\,c^3\,\ln \relax (F)+120\,c^4\right )}{b^6\,d^5\,{\ln \relax (F)}^6}\right )}{x^5+\frac {c^5}{d^5}+\frac {5\,c\,x^4}{d}+\frac {5\,c^4\,x}{d^4}+\frac {10\,c^2\,x^3}{d^2}+\frac {10\,c^3\,x^2}{d^3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.40, size = 388, normalized size = 3.59 \[ \frac {F^{a + \frac {b}{c + d x}} \left (- b^{5} \log {\relax (F )}^{5} + 5 b^{4} c \log {\relax (F )}^{4} + 5 b^{4} d x \log {\relax (F )}^{4} - 20 b^{3} c^{2} \log {\relax (F )}^{3} - 40 b^{3} c d x \log {\relax (F )}^{3} - 20 b^{3} d^{2} x^{2} \log {\relax (F )}^{3} + 60 b^{2} c^{3} \log {\relax (F )}^{2} + 180 b^{2} c^{2} d x \log {\relax (F )}^{2} + 180 b^{2} c d^{2} x^{2} \log {\relax (F )}^{2} + 60 b^{2} d^{3} x^{3} \log {\relax (F )}^{2} - 120 b c^{4} \log {\relax (F )} - 480 b c^{3} d x \log {\relax (F )} - 720 b c^{2} d^{2} x^{2} \log {\relax (F )} - 480 b c d^{3} x^{3} \log {\relax (F )} - 120 b d^{4} x^{4} \log {\relax (F )} + 120 c^{5} + 600 c^{4} d x + 1200 c^{3} d^{2} x^{2} + 1200 c^{2} d^{3} x^{3} + 600 c d^{4} x^{4} + 120 d^{5} x^{5}\right )}{b^{6} c^{5} d \log {\relax (F )}^{6} + 5 b^{6} c^{4} d^{2} x \log {\relax (F )}^{6} + 10 b^{6} c^{3} d^{3} x^{2} \log {\relax (F )}^{6} + 10 b^{6} c^{2} d^{4} x^{3} \log {\relax (F )}^{6} + 5 b^{6} c d^{5} x^{4} \log {\relax (F )}^{6} + b^{6} d^{6} x^{5} \log {\relax (F )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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