Optimal. Leaf size=123 \[ \frac {2 F^{a+\frac {b}{(c+d x)^3}}}{b^4 d \log ^4(F)}-\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{b^3 d \log ^3(F) (c+d x)^3}+\frac {F^{a+\frac {b}{(c+d x)^3}}}{b^2 d \log ^2(F) (c+d x)^6}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^9} \]
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Rubi [A] time = 0.19, antiderivative size = 123, 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} \[ \frac {F^{a+\frac {b}{(c+d x)^3}}}{b^2 d \log ^2(F) (c+d x)^6}-\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{b^3 d \log ^3(F) (c+d x)^3}+\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{b^4 d \log ^4(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^9} \]
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)^3}}}{(c+d x)^{13}} \, dx &=-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^9 \log (F)}-\frac {3 \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{10}} \, dx}{b \log (F)}\\ &=\frac {F^{a+\frac {b}{(c+d x)^3}}}{b^2 d (c+d x)^6 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^9 \log (F)}+\frac {6 \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^7} \, dx}{b^2 \log ^2(F)}\\ &=-\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{b^3 d (c+d x)^3 \log ^3(F)}+\frac {F^{a+\frac {b}{(c+d x)^3}}}{b^2 d (c+d x)^6 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^9 \log (F)}-\frac {6 \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^4} \, dx}{b^3 \log ^3(F)}\\ &=\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{b^4 d \log ^4(F)}-\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{b^3 d (c+d x)^3 \log ^3(F)}+\frac {F^{a+\frac {b}{(c+d x)^3}}}{b^2 d (c+d x)^6 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^9 \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 73, normalized size = 0.59 \[ \frac {F^{a+\frac {b}{(c+d x)^3}} \left (-\frac {b^3 \log ^3(F)}{(c+d x)^9}+\frac {3 b^2 \log ^2(F)}{(c+d x)^6}-\frac {6 b \log (F)}{(c+d x)^3}+6\right )}{3 b^4 d \log ^4(F)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 423, normalized size = 3.44 \[ \frac {{\left (6 \, d^{9} x^{9} + 54 \, c d^{8} x^{8} + 216 \, c^{2} d^{7} x^{7} + 504 \, c^{3} d^{6} x^{6} + 756 \, c^{4} d^{5} x^{5} + 756 \, c^{5} d^{4} x^{4} + 504 \, c^{6} d^{3} x^{3} + 216 \, c^{7} d^{2} x^{2} + 54 \, c^{8} d x + 6 \, c^{9} - b^{3} \log \relax (F)^{3} + 3 \, {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \relax (F)^{2} - 6 \, {\left (b d^{6} x^{6} + 6 \, b c d^{5} x^{5} + 15 \, b c^{2} d^{4} x^{4} + 20 \, b c^{3} d^{3} x^{3} + 15 \, b c^{4} d^{2} x^{2} + 6 \, b c^{5} d x + b c^{6}\right )} \log \relax (F)\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, {\left (b^{4} d^{10} x^{9} + 9 \, b^{4} c d^{9} x^{8} + 36 \, b^{4} c^{2} d^{8} x^{7} + 84 \, b^{4} c^{3} d^{7} x^{6} + 126 \, b^{4} c^{4} d^{6} x^{5} + 126 \, b^{4} c^{5} d^{5} x^{4} + 84 \, b^{4} c^{6} d^{4} x^{3} + 36 \, b^{4} c^{7} d^{3} x^{2} + 9 \, b^{4} c^{8} d^{2} x + b^{4} c^{9} d\right )} \log \relax (F)^{4}} \]
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 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{13}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 641, normalized size = 5.21 \[ \frac {\frac {2 d^{11} x^{12} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}+\frac {24 c \,d^{10} x^{11} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}+\frac {132 c^{2} d^{9} x^{10} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}-\frac {2 \left (-220 c^{3}+b \ln \relax (F )\right ) d^{8} x^{9} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}-\frac {18 \left (-55 c^{3}+b \ln \relax (F )\right ) c \,d^{7} x^{8} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}-\frac {72 \left (-22 c^{3}+b \ln \relax (F )\right ) c^{2} d^{6} x^{7} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}+\frac {\left (1848 c^{6}-168 b \,c^{3} \ln \relax (F )+b^{2} \ln \relax (F )^{2}\right ) d^{5} x^{6} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}+\frac {6 \left (264 c^{6}-42 b \,c^{3} \ln \relax (F )+b^{2} \ln \relax (F )^{2}\right ) c \,d^{4} x^{5} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}+\frac {3 \left (330 c^{6}-84 b \,c^{3} \ln \relax (F )+5 b^{2} \ln \relax (F )^{2}\right ) c^{2} d^{3} x^{4} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}-\frac {\left (-1320 c^{9}+504 b \,c^{6} \ln \relax (F )-60 b^{2} c^{3} \ln \relax (F )^{2}+b^{3} \ln \relax (F )^{3}\right ) d^{2} x^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{3 b^{4} \ln \relax (F )^{4}}-\frac {\left (-132 c^{9}+72 b \,c^{6} \ln \relax (F )-15 b^{2} c^{3} \ln \relax (F )^{2}+b^{3} \ln \relax (F )^{3}\right ) c d \,x^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}-\frac {\left (-24 c^{9}+18 b \,c^{6} \ln \relax (F )-6 b^{2} c^{3} \ln \relax (F )^{2}+b^{3} \ln \relax (F )^{3}\right ) c^{2} x \,{\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}-\frac {\left (-6 c^{9}+6 b \,c^{6} \ln \relax (F )-3 b^{2} c^{3} \ln \relax (F )^{2}+b^{3} \ln \relax (F )^{3}\right ) c^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{3 b^{4} d \ln \relax (F )^{4}}}{\left (d x +c \right )^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.74, size = 507, normalized size = 4.12 \[ \frac {{\left (6 \, F^{a} d^{9} x^{9} + 54 \, F^{a} c d^{8} x^{8} + 216 \, F^{a} c^{2} d^{7} x^{7} + 6 \, F^{a} c^{9} - 6 \, F^{a} b c^{6} \log \relax (F) + 3 \, F^{a} b^{2} c^{3} \log \relax (F)^{2} + 6 \, {\left (84 \, F^{a} c^{3} d^{6} - F^{a} b d^{6} \log \relax (F)\right )} x^{6} - F^{a} b^{3} \log \relax (F)^{3} + 36 \, {\left (21 \, F^{a} c^{4} d^{5} - F^{a} b c d^{5} \log \relax (F)\right )} x^{5} + 18 \, {\left (42 \, F^{a} c^{5} d^{4} - 5 \, F^{a} b c^{2} d^{4} \log \relax (F)\right )} x^{4} + 3 \, {\left (168 \, F^{a} c^{6} d^{3} - 40 \, F^{a} b c^{3} d^{3} \log \relax (F) + F^{a} b^{2} d^{3} \log \relax (F)^{2}\right )} x^{3} + 9 \, {\left (24 \, F^{a} c^{7} d^{2} - 10 \, F^{a} b c^{4} d^{2} \log \relax (F) + F^{a} b^{2} c d^{2} \log \relax (F)^{2}\right )} x^{2} + 9 \, {\left (6 \, F^{a} c^{8} d - 4 \, F^{a} b c^{5} d \log \relax (F) + F^{a} b^{2} c^{2} d \log \relax (F)^{2}\right )} x\right )} F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, {\left (b^{4} d^{10} x^{9} \log \relax (F)^{4} + 9 \, b^{4} c d^{9} x^{8} \log \relax (F)^{4} + 36 \, b^{4} c^{2} d^{8} x^{7} \log \relax (F)^{4} + 84 \, b^{4} c^{3} d^{7} x^{6} \log \relax (F)^{4} + 126 \, b^{4} c^{4} d^{6} x^{5} \log \relax (F)^{4} + 126 \, b^{4} c^{5} d^{5} x^{4} \log \relax (F)^{4} + 84 \, b^{4} c^{6} d^{4} x^{3} \log \relax (F)^{4} + 36 \, b^{4} c^{7} d^{3} x^{2} \log \relax (F)^{4} + 9 \, b^{4} c^{8} d^{2} x \log \relax (F)^{4} + b^{4} c^{9} d \log \relax (F)^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.59, size = 422, normalized size = 3.43 \[ \frac {F^a\,F^{\frac {b}{c^3+3\,c^2\,d\,x+3\,c\,d^2\,x^2+d^3\,x^3}}\,\left (\frac {2\,x^9}{b^4\,d\,{\ln \relax (F)}^4}-\frac {b^3\,{\ln \relax (F)}^3-3\,b^2\,c^3\,{\ln \relax (F)}^2+6\,b\,c^6\,\ln \relax (F)-6\,c^9}{3\,b^4\,d^{10}\,{\ln \relax (F)}^4}+\frac {18\,c\,x^8}{b^4\,d^2\,{\ln \relax (F)}^4}+\frac {72\,c^2\,x^7}{b^4\,d^3\,{\ln \relax (F)}^4}+\frac {x^3\,\left (b^2\,{\ln \relax (F)}^2-40\,b\,c^3\,\ln \relax (F)+168\,c^6\right )}{b^4\,d^7\,{\ln \relax (F)}^4}-\frac {2\,x^6\,\left (b\,\ln \relax (F)-84\,c^3\right )}{b^4\,d^4\,{\ln \relax (F)}^4}+\frac {3\,c^2\,x\,\left (b^2\,{\ln \relax (F)}^2-4\,b\,c^3\,\ln \relax (F)+6\,c^6\right )}{b^4\,d^9\,{\ln \relax (F)}^4}+\frac {3\,c\,x^2\,\left (b^2\,{\ln \relax (F)}^2-10\,b\,c^3\,\ln \relax (F)+24\,c^6\right )}{b^4\,d^8\,{\ln \relax (F)}^4}-\frac {12\,c\,x^5\,\left (b\,\ln \relax (F)-21\,c^3\right )}{b^4\,d^5\,{\ln \relax (F)}^4}-\frac {6\,c^2\,x^4\,\left (5\,b\,\ln \relax (F)-42\,c^3\right )}{b^4\,d^6\,{\ln \relax (F)}^4}\right )}{x^9+\frac {c^9}{d^9}+\frac {9\,c\,x^8}{d}+\frac {9\,c^8\,x}{d^8}+\frac {36\,c^2\,x^7}{d^2}+\frac {84\,c^3\,x^6}{d^3}+\frac {126\,c^4\,x^5}{d^4}+\frac {126\,c^5\,x^4}{d^5}+\frac {84\,c^6\,x^3}{d^6}+\frac {36\,c^7\,x^2}{d^7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.65, size = 484, normalized size = 3.93 \[ \frac {F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (- b^{3} \log {\relax (F )}^{3} + 3 b^{2} c^{3} \log {\relax (F )}^{2} + 9 b^{2} c^{2} d x \log {\relax (F )}^{2} + 9 b^{2} c d^{2} x^{2} \log {\relax (F )}^{2} + 3 b^{2} d^{3} x^{3} \log {\relax (F )}^{2} - 6 b c^{6} \log {\relax (F )} - 36 b c^{5} d x \log {\relax (F )} - 90 b c^{4} d^{2} x^{2} \log {\relax (F )} - 120 b c^{3} d^{3} x^{3} \log {\relax (F )} - 90 b c^{2} d^{4} x^{4} \log {\relax (F )} - 36 b c d^{5} x^{5} \log {\relax (F )} - 6 b d^{6} x^{6} \log {\relax (F )} + 6 c^{9} + 54 c^{8} d x + 216 c^{7} d^{2} x^{2} + 504 c^{6} d^{3} x^{3} + 756 c^{5} d^{4} x^{4} + 756 c^{4} d^{5} x^{5} + 504 c^{3} d^{6} x^{6} + 216 c^{2} d^{7} x^{7} + 54 c d^{8} x^{8} + 6 d^{9} x^{9}\right )}{3 b^{4} c^{9} d \log {\relax (F )}^{4} + 27 b^{4} c^{8} d^{2} x \log {\relax (F )}^{4} + 108 b^{4} c^{7} d^{3} x^{2} \log {\relax (F )}^{4} + 252 b^{4} c^{6} d^{4} x^{3} \log {\relax (F )}^{4} + 378 b^{4} c^{5} d^{5} x^{4} \log {\relax (F )}^{4} + 378 b^{4} c^{4} d^{6} x^{5} \log {\relax (F )}^{4} + 252 b^{4} c^{3} d^{7} x^{6} \log {\relax (F )}^{4} + 108 b^{4} c^{2} d^{8} x^{7} \log {\relax (F )}^{4} + 27 b^{4} c d^{9} x^{8} \log {\relax (F )}^{4} + 3 b^{4} d^{10} x^{9} \log {\relax (F )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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