Optimal. Leaf size=96 \[ -\frac {F^{a+\frac {b}{(c+d x)^2}} \left (b^4 \log ^4(F)-4 b^3 \log ^3(F) (c+d x)^2+12 b^2 \log ^2(F) (c+d x)^4-24 b \log (F) (c+d x)^6+24 (c+d x)^8\right )}{2 b^5 d \log ^5(F) (c+d x)^8} \]
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Rubi [C] time = 0.04, antiderivative size = 31, normalized size of antiderivative = 0.32, 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} \[ -\frac {F^a \text {Gamma}\left (5,-\frac {b \log (F)}{(c+d x)^2}\right )}{2 b^5 d \log ^5(F)} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{11}} \, dx &=-\frac {F^a \Gamma \left (5,-\frac {b \log (F)}{(c+d x)^2}\right )}{2 b^5 d \log ^5(F)}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 31, normalized size = 0.32 \[ -\frac {F^a \Gamma \left (5,-\frac {b \log (F)}{(c+d x)^2}\right )}{2 b^5 d \log ^5(F)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 420, normalized size = 4.38 \[ -\frac {{\left (24 \, d^{8} x^{8} + 192 \, c d^{7} x^{7} + 672 \, c^{2} d^{6} x^{6} + 1344 \, c^{3} d^{5} x^{5} + 1680 \, c^{4} d^{4} x^{4} + 1344 \, c^{5} d^{3} x^{3} + 672 \, c^{6} d^{2} x^{2} + 192 \, c^{7} d x + 24 \, c^{8} + b^{4} \log \relax (F)^{4} - 4 \, {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \log \relax (F)^{3} + 12 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \relax (F)^{2} - 24 \, {\left (b d^{6} x^{6} + 6 \, b c d^{5} x^{5} + 15 \, b c^{2} d^{4} x^{4} + 20 \, b c^{3} d^{3} x^{3} + 15 \, b c^{4} d^{2} x^{2} + 6 \, b c^{5} d x + b c^{6}\right )} \log \relax (F)\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{5} d^{9} x^{8} + 8 \, b^{5} c d^{8} x^{7} + 28 \, b^{5} c^{2} d^{7} x^{6} + 56 \, b^{5} c^{3} d^{6} x^{5} + 70 \, b^{5} c^{4} d^{5} x^{4} + 56 \, b^{5} c^{5} d^{4} x^{3} + 28 \, b^{5} c^{6} d^{3} x^{2} + 8 \, b^{5} c^{7} d^{2} x + b^{5} c^{8} d\right )} \log \relax (F)^{5}} \]
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 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{11}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 609, normalized size = 6.34 \[ \frac {-\frac {12 d^{9} x^{10} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{5} \ln \relax (F )^{5}}-\frac {120 c \,d^{8} x^{9} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{5} \ln \relax (F )^{5}}+\frac {12 \left (b \ln \relax (F )-45 c^{2}\right ) d^{7} x^{8} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{5} \ln \relax (F )^{5}}+\frac {96 \left (b \ln \relax (F )-15 c^{2}\right ) c \,d^{6} x^{7} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{5} \ln \relax (F )^{5}}-\frac {6 \left (b^{2} \ln \relax (F )^{2}-56 b \,c^{2} \ln \relax (F )+420 c^{4}\right ) d^{5} x^{6} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{5} \ln \relax (F )^{5}}-\frac {12 \left (3 b^{2} \ln \relax (F )^{2}-56 b \,c^{2} \ln \relax (F )+252 c^{4}\right ) c \,d^{4} x^{5} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{5} \ln \relax (F )^{5}}+\frac {2 \left (b^{3} \ln \relax (F )^{3}-45 b^{2} c^{2} \ln \relax (F )^{2}+420 b \,c^{4} \ln \relax (F )-1260 c^{6}\right ) d^{3} x^{4} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{5} \ln \relax (F )^{5}}+\frac {8 \left (b^{3} \ln \relax (F )^{3}-15 b^{2} c^{2} \ln \relax (F )^{2}+84 b \,c^{4} \ln \relax (F )-180 c^{6}\right ) c \,d^{2} x^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{5} \ln \relax (F )^{5}}-\frac {\left (b^{4} \ln \relax (F )^{4}-24 b^{3} c^{2} \ln \relax (F )^{3}+180 b^{2} c^{4} \ln \relax (F )^{2}-672 b \,c^{6} \ln \relax (F )+1080 c^{8}\right ) d \,x^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{2 b^{5} \ln \relax (F )^{5}}-\frac {\left (b^{4} \ln \relax (F )^{4}-8 b^{3} c^{2} \ln \relax (F )^{3}+36 b^{2} c^{4} \ln \relax (F )^{2}-96 b \,c^{6} \ln \relax (F )+120 c^{8}\right ) c x \,{\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{5} \ln \relax (F )^{5}}-\frac {\left (b^{4} \ln \relax (F )^{4}-4 b^{3} c^{2} \ln \relax (F )^{3}+12 b^{2} c^{4} \ln \relax (F )^{2}-24 b \,c^{6} \ln \relax (F )+24 c^{8}\right ) c^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{2 b^{5} d \ln \relax (F )^{5}}}{\left (d x +c \right )^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.14, size = 526, normalized size = 5.48 \[ -\frac {{\left (24 \, F^{a} d^{8} x^{8} + 192 \, F^{a} c d^{7} x^{7} + 24 \, F^{a} c^{8} - 24 \, F^{a} b c^{6} \log \relax (F) + 12 \, F^{a} b^{2} c^{4} \log \relax (F)^{2} - 4 \, F^{a} b^{3} c^{2} \log \relax (F)^{3} + F^{a} b^{4} \log \relax (F)^{4} + 24 \, {\left (28 \, F^{a} c^{2} d^{6} - F^{a} b d^{6} \log \relax (F)\right )} x^{6} + 48 \, {\left (28 \, F^{a} c^{3} d^{5} - 3 \, F^{a} b c d^{5} \log \relax (F)\right )} x^{5} + 12 \, {\left (140 \, F^{a} c^{4} d^{4} - 30 \, F^{a} b c^{2} d^{4} \log \relax (F) + F^{a} b^{2} d^{4} \log \relax (F)^{2}\right )} x^{4} + 48 \, {\left (28 \, F^{a} c^{5} d^{3} - 10 \, F^{a} b c^{3} d^{3} \log \relax (F) + F^{a} b^{2} c d^{3} \log \relax (F)^{2}\right )} x^{3} + 4 \, {\left (168 \, F^{a} c^{6} d^{2} - 90 \, F^{a} b c^{4} d^{2} \log \relax (F) + 18 \, F^{a} b^{2} c^{2} d^{2} \log \relax (F)^{2} - F^{a} b^{3} d^{2} \log \relax (F)^{3}\right )} x^{2} + 8 \, {\left (24 \, F^{a} c^{7} d - 18 \, F^{a} b c^{5} d \log \relax (F) + 6 \, F^{a} b^{2} c^{3} d \log \relax (F)^{2} - F^{a} b^{3} c d \log \relax (F)^{3}\right )} x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{5} d^{9} x^{8} \log \relax (F)^{5} + 8 \, b^{5} c d^{8} x^{7} \log \relax (F)^{5} + 28 \, b^{5} c^{2} d^{7} x^{6} \log \relax (F)^{5} + 56 \, b^{5} c^{3} d^{6} x^{5} \log \relax (F)^{5} + 70 \, b^{5} c^{4} d^{5} x^{4} \log \relax (F)^{5} + 56 \, b^{5} c^{5} d^{4} x^{3} \log \relax (F)^{5} + 28 \, b^{5} c^{6} d^{3} x^{2} \log \relax (F)^{5} + 8 \, b^{5} c^{7} d^{2} x \log \relax (F)^{5} + b^{5} c^{8} d \log \relax (F)^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.57, size = 427, normalized size = 4.45 \[ -\frac {F^a\,F^{\frac {b}{c^2+2\,c\,d\,x+d^2\,x^2}}\,\left (\frac {b^4\,{\ln \relax (F)}^4-4\,b^3\,c^2\,{\ln \relax (F)}^3+12\,b^2\,c^4\,{\ln \relax (F)}^2-24\,b\,c^6\,\ln \relax (F)+24\,c^8}{2\,b^5\,d^9\,{\ln \relax (F)}^5}+\frac {12\,x^8}{b^5\,d\,{\ln \relax (F)}^5}+\frac {96\,c\,x^7}{b^5\,d^2\,{\ln \relax (F)}^5}-\frac {2\,x^2\,\left (b^3\,{\ln \relax (F)}^3-18\,b^2\,c^2\,{\ln \relax (F)}^2+90\,b\,c^4\,\ln \relax (F)-168\,c^6\right )}{b^5\,d^7\,{\ln \relax (F)}^5}+\frac {6\,x^4\,\left (b^2\,{\ln \relax (F)}^2-30\,b\,c^2\,\ln \relax (F)+140\,c^4\right )}{b^5\,d^5\,{\ln \relax (F)}^5}-\frac {12\,x^6\,\left (b\,\ln \relax (F)-28\,c^2\right )}{b^5\,d^3\,{\ln \relax (F)}^5}+\frac {24\,c\,x^3\,\left (b^2\,{\ln \relax (F)}^2-10\,b\,c^2\,\ln \relax (F)+28\,c^4\right )}{b^5\,d^6\,{\ln \relax (F)}^5}-\frac {24\,c\,x^5\,\left (3\,b\,\ln \relax (F)-28\,c^2\right )}{b^5\,d^4\,{\ln \relax (F)}^5}-\frac {4\,c\,x\,\left (b^3\,{\ln \relax (F)}^3-6\,b^2\,c^2\,{\ln \relax (F)}^2+18\,b\,c^4\,\ln \relax (F)-24\,c^6\right )}{b^5\,d^8\,{\ln \relax (F)}^5}\right )}{x^8+\frac {c^8}{d^8}+\frac {8\,c\,x^7}{d}+\frac {8\,c^7\,x}{d^7}+\frac {28\,c^2\,x^6}{d^2}+\frac {56\,c^3\,x^5}{d^3}+\frac {70\,c^4\,x^4}{d^4}+\frac {56\,c^5\,x^3}{d^5}+\frac {28\,c^6\,x^2}{d^6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.60, size = 518, normalized size = 5.40 \[ \frac {F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (- b^{4} \log {\relax (F )}^{4} + 4 b^{3} c^{2} \log {\relax (F )}^{3} + 8 b^{3} c d x \log {\relax (F )}^{3} + 4 b^{3} d^{2} x^{2} \log {\relax (F )}^{3} - 12 b^{2} c^{4} \log {\relax (F )}^{2} - 48 b^{2} c^{3} d x \log {\relax (F )}^{2} - 72 b^{2} c^{2} d^{2} x^{2} \log {\relax (F )}^{2} - 48 b^{2} c d^{3} x^{3} \log {\relax (F )}^{2} - 12 b^{2} d^{4} x^{4} \log {\relax (F )}^{2} + 24 b c^{6} \log {\relax (F )} + 144 b c^{5} d x \log {\relax (F )} + 360 b c^{4} d^{2} x^{2} \log {\relax (F )} + 480 b c^{3} d^{3} x^{3} \log {\relax (F )} + 360 b c^{2} d^{4} x^{4} \log {\relax (F )} + 144 b c d^{5} x^{5} \log {\relax (F )} + 24 b d^{6} x^{6} \log {\relax (F )} - 24 c^{8} - 192 c^{7} d x - 672 c^{6} d^{2} x^{2} - 1344 c^{5} d^{3} x^{3} - 1680 c^{4} d^{4} x^{4} - 1344 c^{3} d^{5} x^{5} - 672 c^{2} d^{6} x^{6} - 192 c d^{7} x^{7} - 24 d^{8} x^{8}\right )}{2 b^{5} c^{8} d \log {\relax (F )}^{5} + 16 b^{5} c^{7} d^{2} x \log {\relax (F )}^{5} + 56 b^{5} c^{6} d^{3} x^{2} \log {\relax (F )}^{5} + 112 b^{5} c^{5} d^{4} x^{3} \log {\relax (F )}^{5} + 140 b^{5} c^{4} d^{5} x^{4} \log {\relax (F )}^{5} + 112 b^{5} c^{3} d^{6} x^{5} \log {\relax (F )}^{5} + 56 b^{5} c^{2} d^{7} x^{6} \log {\relax (F )}^{5} + 16 b^{5} c d^{8} x^{7} \log {\relax (F )}^{5} + 2 b^{5} d^{9} x^{8} \log {\relax (F )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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