Optimal. Leaf size=96 \[ -\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{3 b^3 d \log ^3(F)}+\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{3 b^2 d \log ^2(F) (c+d x)^3}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^6} \]
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Rubi [A] time = 0.13, antiderivative size = 96, 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} \[ \frac {2 F^{a+\frac {b}{(c+d x)^3}}}{3 b^2 d \log ^2(F) (c+d x)^3}-\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{3 b^3 d \log ^3(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^6} \]
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)^{10}} \, dx &=-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^6 \log (F)}-\frac {2 \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^7} \, dx}{b \log (F)}\\ &=\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{3 b^2 d (c+d x)^3 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^6 \log (F)}+\frac {2 \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^4} \, dx}{b^2 \log ^2(F)}\\ &=-\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{3 b^3 d \log ^3(F)}+\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{3 b^2 d (c+d x)^3 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^6 \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 64, normalized size = 0.67 \[ -\frac {F^{a+\frac {b}{(c+d x)^3}} \left (b^2 \log ^2(F)-2 b \log (F) (c+d x)^3+2 (c+d x)^6\right )}{3 b^3 d \log ^3(F) (c+d x)^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 265, normalized size = 2.76 \[ -\frac {{\left (2 \, d^{6} x^{6} + 12 \, c d^{5} x^{5} + 30 \, c^{2} d^{4} x^{4} + 40 \, c^{3} d^{3} x^{3} + 30 \, c^{4} d^{2} x^{2} + 12 \, c^{5} d x + 2 \, c^{6} + b^{2} \log \relax (F)^{2} - 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \relax (F)\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, {\left (b^{3} d^{7} x^{6} + 6 \, b^{3} c d^{6} x^{5} + 15 \, b^{3} c^{2} d^{5} x^{4} + 20 \, b^{3} c^{3} d^{4} x^{3} + 15 \, b^{3} c^{4} d^{3} x^{2} + 6 \, b^{3} c^{5} d^{2} x + b^{3} c^{6} d\right )} \log \relax (F)^{3}} \]
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 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{10}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 434, normalized size = 4.52 \[ \frac {-\frac {2 d^{8} x^{9} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{3 b^{3} \ln \relax (F )^{3}}-\frac {6 c \,d^{7} x^{8} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{3} \ln \relax (F )^{3}}-\frac {24 c^{2} d^{6} x^{7} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{3} \ln \relax (F )^{3}}+\frac {2 \left (-84 c^{3}+b \ln \relax (F )\right ) d^{5} x^{6} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{3 b^{3} \ln \relax (F )^{3}}+\frac {4 \left (-21 c^{3}+b \ln \relax (F )\right ) c \,d^{4} x^{5} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{3} \ln \relax (F )^{3}}+\frac {2 \left (-42 c^{3}+5 b \ln \relax (F )\right ) c^{2} d^{3} x^{4} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{3} \ln \relax (F )^{3}}-\frac {\left (168 c^{6}-40 b \,c^{3} \ln \relax (F )+b^{2} \ln \relax (F )^{2}\right ) d^{2} x^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{3 b^{3} \ln \relax (F )^{3}}-\frac {\left (24 c^{6}-10 b \,c^{3} \ln \relax (F )+b^{2} \ln \relax (F )^{2}\right ) c d \,x^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{3} \ln \relax (F )^{3}}-\frac {\left (6 c^{6}-4 b \,c^{3} \ln \relax (F )+b^{2} \ln \relax (F )^{2}\right ) c^{2} x \,{\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{3} \ln \relax (F )^{3}}-\frac {\left (2 c^{6}-2 b \,c^{3} \ln \relax (F )+b^{2} \ln \relax (F )^{2}\right ) c^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{3 b^{3} d \ln \relax (F )^{3}}}{\left (d x +c \right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.89, size = 300, normalized size = 3.12 \[ -\frac {{\left (2 \, F^{a} d^{6} x^{6} + 12 \, F^{a} c d^{5} x^{5} + 30 \, F^{a} c^{2} d^{4} x^{4} + 2 \, F^{a} c^{6} - 2 \, F^{a} b c^{3} \log \relax (F) + F^{a} b^{2} \log \relax (F)^{2} + 2 \, {\left (20 \, F^{a} c^{3} d^{3} - F^{a} b d^{3} \log \relax (F)\right )} x^{3} + 6 \, {\left (5 \, F^{a} c^{4} d^{2} - F^{a} b c d^{2} \log \relax (F)\right )} x^{2} + 6 \, {\left (2 \, F^{a} c^{5} d - F^{a} b c^{2} d \log \relax (F)\right )} x\right )} F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, {\left (b^{3} d^{7} x^{6} \log \relax (F)^{3} + 6 \, b^{3} c d^{6} x^{5} \log \relax (F)^{3} + 15 \, b^{3} c^{2} d^{5} x^{4} \log \relax (F)^{3} + 20 \, b^{3} c^{3} d^{4} x^{3} \log \relax (F)^{3} + 15 \, b^{3} c^{4} d^{3} x^{2} \log \relax (F)^{3} + 6 \, b^{3} c^{5} d^{2} x \log \relax (F)^{3} + b^{3} c^{6} d \log \relax (F)^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.08, size = 263, normalized size = 2.74 \[ -\frac {F^a\,F^{\frac {b}{c^3+3\,c^2\,d\,x+3\,c\,d^2\,x^2+d^3\,x^3}}\,\left (\frac {2\,x^6}{3\,b^3\,d\,{\ln \relax (F)}^3}+\frac {b^2\,{\ln \relax (F)}^2-2\,b\,c^3\,\ln \relax (F)+2\,c^6}{3\,b^3\,d^7\,{\ln \relax (F)}^3}+\frac {4\,c\,x^5}{b^3\,d^2\,{\ln \relax (F)}^3}+\frac {10\,c^2\,x^4}{b^3\,d^3\,{\ln \relax (F)}^3}-\frac {2\,x^3\,\left (b\,\ln \relax (F)-20\,c^3\right )}{3\,b^3\,d^4\,{\ln \relax (F)}^3}-\frac {2\,c^2\,x\,\left (b\,\ln \relax (F)-2\,c^3\right )}{b^3\,d^6\,{\ln \relax (F)}^3}-\frac {2\,c\,x^2\,\left (b\,\ln \relax (F)-5\,c^3\right )}{b^3\,d^5\,{\ln \relax (F)}^3}\right )}{x^6+\frac {c^6}{d^6}+\frac {6\,c\,x^5}{d}+\frac {6\,c^5\,x}{d^5}+\frac {15\,c^2\,x^4}{d^2}+\frac {20\,c^3\,x^3}{d^3}+\frac {15\,c^4\,x^2}{d^4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.49, size = 270, normalized size = 2.81 \[ \frac {F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (- b^{2} \log {\relax (F )}^{2} + 2 b c^{3} \log {\relax (F )} + 6 b c^{2} d x \log {\relax (F )} + 6 b c d^{2} x^{2} \log {\relax (F )} + 2 b d^{3} x^{3} \log {\relax (F )} - 2 c^{6} - 12 c^{5} d x - 30 c^{4} d^{2} x^{2} - 40 c^{3} d^{3} x^{3} - 30 c^{2} d^{4} x^{4} - 12 c d^{5} x^{5} - 2 d^{6} x^{6}\right )}{3 b^{3} c^{6} d \log {\relax (F )}^{3} + 18 b^{3} c^{5} d^{2} x \log {\relax (F )}^{3} + 45 b^{3} c^{4} d^{3} x^{2} \log {\relax (F )}^{3} + 60 b^{3} c^{3} d^{4} x^{3} \log {\relax (F )}^{3} + 45 b^{3} c^{2} d^{5} x^{4} \log {\relax (F )}^{3} + 18 b^{3} c d^{6} x^{5} \log {\relax (F )}^{3} + 3 b^{3} d^{7} x^{6} \log {\relax (F )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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