Optimal. Leaf size=49 \[ -\frac {F^a \left (-b \log (F) (c+d x)^2\right )^{11/2} \Gamma \left (-\frac {11}{2},-b (c+d x)^2 \log (F)\right )}{2 d (c+d x)^{11}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, 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} \[ -\frac {F^a \left (-b \log (F) (c+d x)^2\right )^{11/2} \text {Gamma}\left (-\frac {11}{2},-b \log (F) (c+d x)^2\right )}{2 d (c+d x)^{11}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2218
Rubi steps
\begin {align*} \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^{12}} \, dx &=-\frac {F^a \Gamma \left (-\frac {11}{2},-b (c+d x)^2 \log (F)\right ) \left (-b (c+d x)^2 \log (F)\right )^{11/2}}{2 d (c+d x)^{11}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 49, normalized size = 1.00 \[ -\frac {F^a \left (-b \log (F) (c+d x)^2\right )^{11/2} \Gamma \left (-\frac {11}{2},-b (c+d x)^2 \log (F)\right )}{2 d (c+d x)^{11}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 795, normalized size = 16.22 \[ -\frac {32 \, \sqrt {\pi } {\left (b^{5} d^{11} x^{11} + 11 \, b^{5} c d^{10} x^{10} + 55 \, b^{5} c^{2} d^{9} x^{9} + 165 \, b^{5} c^{3} d^{8} x^{8} + 330 \, b^{5} c^{4} d^{7} x^{7} + 462 \, b^{5} c^{5} d^{6} x^{6} + 462 \, b^{5} c^{6} d^{5} x^{5} + 330 \, b^{5} c^{7} d^{4} x^{4} + 165 \, b^{5} c^{8} d^{3} x^{3} + 55 \, b^{5} c^{9} d^{2} x^{2} + 11 \, b^{5} c^{10} d x + b^{5} c^{11}\right )} \sqrt {-b d^{2} \log \relax (F)} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \relax (F)} {\left (d x + c\right )}}{d}\right ) \log \relax (F)^{5} + {\left (32 \, {\left (b^{5} d^{11} x^{10} + 10 \, b^{5} c d^{10} x^{9} + 45 \, b^{5} c^{2} d^{9} x^{8} + 120 \, b^{5} c^{3} d^{8} x^{7} + 210 \, b^{5} c^{4} d^{7} x^{6} + 252 \, b^{5} c^{5} d^{6} x^{5} + 210 \, b^{5} c^{6} d^{5} x^{4} + 120 \, b^{5} c^{7} d^{4} x^{3} + 45 \, b^{5} c^{8} d^{3} x^{2} + 10 \, b^{5} c^{9} d^{2} x + b^{5} c^{10} d\right )} \log \relax (F)^{5} + 16 \, {\left (b^{4} d^{9} x^{8} + 8 \, b^{4} c d^{8} x^{7} + 28 \, b^{4} c^{2} d^{7} x^{6} + 56 \, b^{4} c^{3} d^{6} x^{5} + 70 \, b^{4} c^{4} d^{5} x^{4} + 56 \, b^{4} c^{5} d^{4} x^{3} + 28 \, b^{4} c^{6} d^{3} x^{2} + 8 \, b^{4} c^{7} d^{2} x + b^{4} c^{8} d\right )} \log \relax (F)^{4} + 24 \, {\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} + 60 \, {\left (b^{2} d^{5} x^{4} + 4 \, b^{2} c d^{4} x^{3} + 6 \, b^{2} c^{2} d^{3} x^{2} + 4 \, b^{2} c^{3} d^{2} x + b^{2} c^{4} d\right )} \log \relax (F)^{2} + 210 \, {\left (b d^{3} x^{2} + 2 \, b c d^{2} x + b c^{2} d\right )} \log \relax (F) + 945 \, d\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{10395 \, {\left (d^{13} x^{11} + 11 \, c d^{12} x^{10} + 55 \, c^{2} d^{11} x^{9} + 165 \, c^{3} d^{10} x^{8} + 330 \, c^{4} d^{9} x^{7} + 462 \, c^{5} d^{8} x^{6} + 462 \, c^{6} d^{7} x^{5} + 330 \, c^{7} d^{6} x^{4} + 165 \, c^{8} d^{5} x^{3} + 55 \, c^{9} d^{4} x^{2} + 11 \, c^{10} d^{3} x + c^{11} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{12}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.22, size = 228, normalized size = 4.65 \[ \frac {32 \sqrt {\pi }\, b^{6} F^{a} \erf \left (\sqrt {-b \ln \relax (F )}\, \left (d x +c \right )\right ) \ln \relax (F )^{6}}{10395 \sqrt {-b \ln \relax (F )}\, d}-\frac {32 b^{5} F^{a} F^{\left (d x +c \right )^{2} b} \ln \relax (F )^{5}}{10395 \left (d x +c \right ) d}-\frac {16 b^{4} F^{a} F^{\left (d x +c \right )^{2} b} \ln \relax (F )^{4}}{10395 \left (d x +c \right )^{3} d}-\frac {8 b^{3} F^{a} F^{\left (d x +c \right )^{2} b} \ln \relax (F )^{3}}{3465 \left (d x +c \right )^{5} d}-\frac {4 b^{2} F^{a} F^{\left (d x +c \right )^{2} b} \ln \relax (F )^{2}}{693 \left (d x +c \right )^{7} d}-\frac {2 b \,F^{a} F^{\left (d x +c \right )^{2} b} \ln \relax (F )}{99 \left (d x +c \right )^{9} d}-\frac {F^{a} F^{\left (d x +c \right )^{2} b}}{11 \left (d x +c \right )^{11} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{12}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.15, size = 267, normalized size = 5.45 \[ \frac {32\,F^a\,\sqrt {\pi }\,{\left (-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2\right )}^{11/2}}{10395\,d\,{\left (c+d\,x\right )}^{11}}-\frac {F^a\,F^{b\,{\left (c+d\,x\right )}^2}}{11\,d\,{\left (c+d\,x\right )}^{11}}-\frac {4\,F^a\,F^{b\,{\left (c+d\,x\right )}^2}\,b^2\,{\ln \relax (F)}^2}{693\,d\,{\left (c+d\,x\right )}^7}-\frac {8\,F^a\,F^{b\,{\left (c+d\,x\right )}^2}\,b^3\,{\ln \relax (F)}^3}{3465\,d\,{\left (c+d\,x\right )}^5}-\frac {16\,F^a\,F^{b\,{\left (c+d\,x\right )}^2}\,b^4\,{\ln \relax (F)}^4}{10395\,d\,{\left (c+d\,x\right )}^3}-\frac {32\,F^a\,F^{b\,{\left (c+d\,x\right )}^2}\,b^5\,{\ln \relax (F)}^5}{10395\,d\,\left (c+d\,x\right )}-\frac {2\,F^a\,F^{b\,{\left (c+d\,x\right )}^2}\,b\,\ln \relax (F)}{99\,d\,{\left (c+d\,x\right )}^9}-\frac {32\,F^a\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2}\right )\,{\left (-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2\right )}^{11/2}}{10395\,d\,{\left (c+d\,x\right )}^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________