Optimal. Leaf size=49 \[ \frac {F^a (c+d x)^{11} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2} \Gamma \left (-\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, 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} \[ \frac {F^a (c+d x)^{11} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2} \text {Gamma}\left (-\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^{10} \, dx &=\frac {F^a (c+d x)^{11} \Gamma \left (-\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right ) \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2}}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 1.00 \[ \frac {F^a (c+d x)^{11} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2} \Gamma \left (-\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 561, normalized size = 11.45 \[ \frac {32 \, \sqrt {\pi } F^{a} b^{5} d \sqrt {-\frac {b \log \relax (F)}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {b \log \relax (F)}{d^{2}}}}{d x + c}\right ) \log \relax (F)^{5} + {\left (945 \, d^{11} x^{11} + 10395 \, c d^{10} x^{10} + 51975 \, c^{2} d^{9} x^{9} + 155925 \, c^{3} d^{8} x^{8} + 311850 \, c^{4} d^{7} x^{7} + 436590 \, c^{5} d^{6} x^{6} + 436590 \, c^{6} d^{5} x^{5} + 311850 \, c^{7} d^{4} x^{4} + 155925 \, c^{8} d^{3} x^{3} + 51975 \, c^{9} d^{2} x^{2} + 10395 \, c^{10} d x + 945 \, c^{11} + 32 \, {\left (b^{5} d x + b^{5} c\right )} \log \relax (F)^{5} + 16 \, {\left (b^{4} d^{3} x^{3} + 3 \, b^{4} c d^{2} x^{2} + 3 \, b^{4} c^{2} d x + b^{4} c^{3}\right )} \log \relax (F)^{4} + 24 \, {\left (b^{3} d^{5} x^{5} + 5 \, b^{3} c d^{4} x^{4} + 10 \, b^{3} c^{2} d^{3} x^{3} + 10 \, b^{3} c^{3} d^{2} x^{2} + 5 \, b^{3} c^{4} d x + b^{3} c^{5}\right )} \log \relax (F)^{3} + 60 \, {\left (b^{2} d^{7} x^{7} + 7 \, b^{2} c d^{6} x^{6} + 21 \, b^{2} c^{2} d^{5} x^{5} + 35 \, b^{2} c^{3} d^{4} x^{4} + 35 \, b^{2} c^{4} d^{3} x^{3} + 21 \, b^{2} c^{5} d^{2} x^{2} + 7 \, b^{2} c^{6} d x + b^{2} c^{7}\right )} \log \relax (F)^{2} + 210 \, {\left (b d^{9} x^{9} + 9 \, b c d^{8} x^{8} + 36 \, b c^{2} d^{7} x^{7} + 84 \, b c^{3} d^{6} x^{6} + 126 \, b c^{4} d^{5} x^{5} + 126 \, b c^{5} d^{4} x^{4} + 84 \, b c^{6} d^{3} x^{3} + 36 \, b c^{7} d^{2} x^{2} + 9 \, b c^{8} d x + b c^{9}\right )} \log \relax (F)\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{10395 \, d} \]
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 \[ \int {\left (d x + c\right )}^{10} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 1173, normalized size = 23.94 \[ \text {result too large to display} \]
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 \[ \frac {1}{10395} \, {\left (945 \, F^{a} d^{10} x^{11} + 10395 \, F^{a} c d^{9} x^{10} + 105 \, {\left (495 \, F^{a} c^{2} d^{8} + 2 \, F^{a} b d^{8} \log \relax (F)\right )} x^{9} + 945 \, {\left (165 \, F^{a} c^{3} d^{7} + 2 \, F^{a} b c d^{7} \log \relax (F)\right )} x^{8} + 30 \, {\left (10395 \, F^{a} c^{4} d^{6} + 252 \, F^{a} b c^{2} d^{6} \log \relax (F) + 2 \, F^{a} b^{2} d^{6} \log \relax (F)^{2}\right )} x^{7} + 210 \, {\left (2079 \, F^{a} c^{5} d^{5} + 84 \, F^{a} b c^{3} d^{5} \log \relax (F) + 2 \, F^{a} b^{2} c d^{5} \log \relax (F)^{2}\right )} x^{6} + 6 \, {\left (72765 \, F^{a} c^{6} d^{4} + 4410 \, F^{a} b c^{4} d^{4} \log \relax (F) + 210 \, F^{a} b^{2} c^{2} d^{4} \log \relax (F)^{2} + 4 \, F^{a} b^{3} d^{4} \log \relax (F)^{3}\right )} x^{5} + 30 \, {\left (10395 \, F^{a} c^{7} d^{3} + 882 \, F^{a} b c^{5} d^{3} \log \relax (F) + 70 \, F^{a} b^{2} c^{3} d^{3} \log \relax (F)^{2} + 4 \, F^{a} b^{3} c d^{3} \log \relax (F)^{3}\right )} x^{4} + {\left (155925 \, F^{a} c^{8} d^{2} + 17640 \, F^{a} b c^{6} d^{2} \log \relax (F) + 2100 \, F^{a} b^{2} c^{4} d^{2} \log \relax (F)^{2} + 240 \, F^{a} b^{3} c^{2} d^{2} \log \relax (F)^{3} + 16 \, F^{a} b^{4} d^{2} \log \relax (F)^{4}\right )} x^{3} + 3 \, {\left (17325 \, F^{a} c^{9} d + 2520 \, F^{a} b c^{7} d \log \relax (F) + 420 \, F^{a} b^{2} c^{5} d \log \relax (F)^{2} + 80 \, F^{a} b^{3} c^{3} d \log \relax (F)^{3} + 16 \, F^{a} b^{4} c d \log \relax (F)^{4}\right )} x^{2} + {\left (10395 \, F^{a} c^{10} + 1890 \, F^{a} b c^{8} \log \relax (F) + 420 \, F^{a} b^{2} c^{6} \log \relax (F)^{2} + 120 \, F^{a} b^{3} c^{4} \log \relax (F)^{3} + 48 \, F^{a} b^{4} c^{2} \log \relax (F)^{4} + 32 \, F^{a} b^{5} \log \relax (F)^{5}\right )} x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \int \frac {2 \, {\left (32 \, F^{a} b^{6} d x \log \relax (F)^{6} - 945 \, F^{a} b c^{11} \log \relax (F) - 210 \, F^{a} b^{2} c^{9} \log \relax (F)^{2} - 60 \, F^{a} b^{3} c^{7} \log \relax (F)^{3} - 24 \, F^{a} b^{4} c^{5} \log \relax (F)^{4} - 16 \, F^{a} b^{5} c^{3} \log \relax (F)^{5}\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{10395 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.35, size = 265, normalized size = 5.41 \[ \frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,{\left (c+d\,x\right )}^{11}}{11\,d}-\frac {32\,F^a\,\sqrt {\pi }\,{\left (c+d\,x\right )}^{11}\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )}^{11/2}}{10395\,d}+\frac {4\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^2\,{\ln \relax (F)}^2\,{\left (c+d\,x\right )}^7}{693\,d}+\frac {8\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^3\,{\ln \relax (F)}^3\,{\left (c+d\,x\right )}^5}{3465\,d}+\frac {16\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^4\,{\ln \relax (F)}^4\,{\left (c+d\,x\right )}^3}{10395\,d}+\frac {2\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b\,\ln \relax (F)\,{\left (c+d\,x\right )}^9}{99\,d}+\frac {32\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^5\,{\ln \relax (F)}^5\,\left (c+d\,x\right )}{10395\,d}+\frac {32\,F^a\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}}\right )\,{\left (c+d\,x\right )}^{11}\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )}^{11/2}}{10395\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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