Integrand size = 21, antiderivative size = 49 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx=\frac {F^a \Gamma \left (\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d (c+d x)^{11} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2}} \]
1/2*F^a*(1048576/61836869254970658257624840625*GAMMA(51/2,-b*ln(F)/(d*x+c) ^2)-1048576/61836869254970658257624840625*(-b*ln(F)/(d*x+c)^2)^(49/2)*exp( b*ln(F)/(d*x+c)^2)-524288/1261976923570829760359690625*(-b*ln(F)/(d*x+c)^2 )^(47/2)*exp(b*ln(F)/(d*x+c)^2)-262144/26850572841932548092759375*(-b*ln(F )/(d*x+c)^2)^(45/2)*exp(b*ln(F)/(d*x+c)^2)-131072/596679396487389957616875 *(-b*ln(F)/(d*x+c)^2)^(43/2)*exp(b*ln(F)/(d*x+c)^2)-65536/1387626503459046 4130625*(-b*ln(F)/(d*x+c)^2)^(41/2)*exp(b*ln(F)/(d*x+c)^2)-32768/338445488 648547905625*(-b*ln(F)/(d*x+c)^2)^(39/2)*exp(b*ln(F)/(d*x+c)^2)-16384/8678 089452526869375*(-b*ln(F)/(d*x+c)^2)^(37/2)*exp(b*ln(F)/(d*x+c)^2)-8192/23 4542958176401875*(-b*ln(F)/(d*x+c)^2)^(35/2)*exp(b*ln(F)/(d*x+c)^2)-4096/6 701227376468625*(-b*ln(F)/(d*x+c)^2)^(33/2)*exp(b*ln(F)/(d*x+c)^2)-2048/20 3067496256625*(-b*ln(F)/(d*x+c)^2)^(31/2)*exp(b*ln(F)/(d*x+c)^2)-1024/6550 564395375*(-b*ln(F)/(d*x+c)^2)^(29/2)*exp(b*ln(F)/(d*x+c)^2)-512/225881530 875*(-b*ln(F)/(d*x+c)^2)^(27/2)*exp(b*ln(F)/(d*x+c)^2)-256/8365982625*(-b* ln(F)/(d*x+c)^2)^(25/2)*exp(b*ln(F)/(d*x+c)^2)-128/334639305*(-b*ln(F)/(d* x+c)^2)^(23/2)*exp(b*ln(F)/(d*x+c)^2)-64/14549535*(-b*ln(F)/(d*x+c)^2)^(21 /2)*exp(b*ln(F)/(d*x+c)^2)-32/692835*(-b*ln(F)/(d*x+c)^2)^(19/2)*exp(b*ln( F)/(d*x+c)^2)-16/36465*(-b*ln(F)/(d*x+c)^2)^(17/2)*exp(b*ln(F)/(d*x+c)^2)- 8/2145*(-b*ln(F)/(d*x+c)^2)^(15/2)*exp(b*ln(F)/(d*x+c)^2)-4/143*(-b*ln(F)/ (d*x+c)^2)^(13/2)*exp(b*ln(F)/(d*x+c)^2)-2/11*(-b*ln(F)/(d*x+c)^2)^(11/...
Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx=\frac {F^a \Gamma \left (\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d (c+d x)^{11} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2}} \]
(F^a*Gamma[11/2, -((b*Log[F])/(c + d*x)^2)])/(2*d*(c + d*x)^11*(-((b*Log[F ])/(c + d*x)^2))^(11/2))
Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2648}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx\) |
\(\Big \downarrow \) 2648 |
\(\displaystyle \frac {F^a \Gamma \left (\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d (c+d x)^{11} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2}}\) |
(F^a*Gamma[11/2, -((b*Log[F])/(c + d*x)^2)])/(2*d*(c + d*x)^11*(-((b*Log[F ])/(c + d*x)^2))^(11/2))
3.4.38.3.1 Defintions of rubi rules used
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_ .), x_Symbol] :> Simp[(-F^a)*((e + f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[ F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; FreeQ[{F , a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]
Time = 7.00 (sec) , antiderivative size = 208, normalized size of antiderivative = 4.24
method | result | size |
risch | \(-\frac {F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{2 d \left (d x +c \right )^{9} b \ln \left (F \right )}+\frac {9 F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{4 d \,b^{2} \ln \left (F \right )^{2} \left (d x +c \right )^{7}}-\frac {63 F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{8 d \,b^{3} \ln \left (F \right )^{3} \left (d x +c \right )^{5}}+\frac {315 F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{16 d \,b^{4} \ln \left (F \right )^{4} \left (d x +c \right )^{3}}-\frac {945 F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{32 d \,b^{5} \ln \left (F \right )^{5} \left (d x +c \right )}+\frac {945 F^{a} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (F \right )}}{d x +c}\right )}{64 d \,b^{5} \ln \left (F \right )^{5} \sqrt {-b \ln \left (F \right )}}\) | \(208\) |
-1/2*F^a/d*F^(b/(d*x+c)^2)/(d*x+c)^9/b/ln(F)+9/4*F^a/d/b^2/ln(F)^2*F^(b/(d *x+c)^2)/(d*x+c)^7-63/8*F^a/d/b^3/ln(F)^3*F^(b/(d*x+c)^2)/(d*x+c)^5+315/16 *F^a/d/b^4/ln(F)^4*F^(b/(d*x+c)^2)/(d*x+c)^3-945/32*F^a/d/b^5/ln(F)^5*F^(b /(d*x+c)^2)/(d*x+c)+945/64*F^a/d/b^5/ln(F)^5*Pi^(1/2)/(-b*ln(F))^(1/2)*erf ((-b*ln(F))^(1/2)/(d*x+c))
Time = 0.13 (sec) , antiderivative size = 601, normalized size of antiderivative = 12.27 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx=-\frac {945 \, \sqrt {\pi } {\left (d^{10} x^{9} + 9 \, c d^{9} x^{8} + 36 \, c^{2} d^{8} x^{7} + 84 \, c^{3} d^{7} x^{6} + 126 \, c^{4} d^{6} x^{5} + 126 \, c^{5} d^{5} x^{4} + 84 \, c^{6} d^{4} x^{3} + 36 \, c^{7} d^{3} x^{2} + 9 \, c^{8} d^{2} x + c^{9} d\right )} F^{a} \sqrt {-\frac {b \log \left (F\right )}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {b \log \left (F\right )}{d^{2}}}}{d x + c}\right ) + 2 \, {\left (16 \, b^{5} \log \left (F\right )^{5} - 72 \, {\left (b^{4} d^{2} x^{2} + 2 \, b^{4} c d x + b^{4} c^{2}\right )} \log \left (F\right )^{4} + 252 \, {\left (b^{3} d^{4} x^{4} + 4 \, b^{3} c d^{3} x^{3} + 6 \, b^{3} c^{2} d^{2} x^{2} + 4 \, b^{3} c^{3} d x + b^{3} c^{4}\right )} \log \left (F\right )^{3} - 630 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} + 945 \, {\left (b d^{8} x^{8} + 8 \, b c d^{7} x^{7} + 28 \, b c^{2} d^{6} x^{6} + 56 \, b c^{3} d^{5} x^{5} + 70 \, b c^{4} d^{4} x^{4} + 56 \, b c^{5} d^{3} x^{3} + 28 \, b c^{6} d^{2} x^{2} + 8 \, b c^{7} d x + b c^{8}\right )} \log \left (F\right )\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{64 \, {\left (b^{6} d^{10} x^{9} + 9 \, b^{6} c d^{9} x^{8} + 36 \, b^{6} c^{2} d^{8} x^{7} + 84 \, b^{6} c^{3} d^{7} x^{6} + 126 \, b^{6} c^{4} d^{6} x^{5} + 126 \, b^{6} c^{5} d^{5} x^{4} + 84 \, b^{6} c^{6} d^{4} x^{3} + 36 \, b^{6} c^{7} d^{3} x^{2} + 9 \, b^{6} c^{8} d^{2} x + b^{6} c^{9} d\right )} \log \left (F\right )^{6}} \]
-1/64*(945*sqrt(pi)*(d^10*x^9 + 9*c*d^9*x^8 + 36*c^2*d^8*x^7 + 84*c^3*d^7* x^6 + 126*c^4*d^6*x^5 + 126*c^5*d^5*x^4 + 84*c^6*d^4*x^3 + 36*c^7*d^3*x^2 + 9*c^8*d^2*x + c^9*d)*F^a*sqrt(-b*log(F)/d^2)*erf(d*sqrt(-b*log(F)/d^2)/( d*x + c)) + 2*(16*b^5*log(F)^5 - 72*(b^4*d^2*x^2 + 2*b^4*c*d*x + b^4*c^2)* log(F)^4 + 252*(b^3*d^4*x^4 + 4*b^3*c*d^3*x^3 + 6*b^3*c^2*d^2*x^2 + 4*b^3* c^3*d*x + b^3*c^4)*log(F)^3 - 630*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2* c^2*d^4*x^4 + 20*b^2*c^3*d^3*x^3 + 15*b^2*c^4*d^2*x^2 + 6*b^2*c^5*d*x + b^ 2*c^6)*log(F)^2 + 945*(b*d^8*x^8 + 8*b*c*d^7*x^7 + 28*b*c^2*d^6*x^6 + 56*b *c^3*d^5*x^5 + 70*b*c^4*d^4*x^4 + 56*b*c^5*d^3*x^3 + 28*b*c^6*d^2*x^2 + 8* b*c^7*d*x + b*c^8)*log(F))*F^((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b)/(d^2*x^2 + 2*c*d*x + c^2)))/((b^6*d^10*x^9 + 9*b^6*c*d^9*x^8 + 36*b^6*c^2*d^8*x^7 + 84*b^6*c^3*d^7*x^6 + 126*b^6*c^4*d^6*x^5 + 126*b^6*c^5*d^5*x^4 + 84*b^6* c^6*d^4*x^3 + 36*b^6*c^7*d^3*x^2 + 9*b^6*c^8*d^2*x + b^6*c^9*d)*log(F)^6)
Timed out. \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx=\text {Timed out} \]
\[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx=\int { \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{12}} \,d x } \]
\[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx=\int { \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{12}} \,d x } \]
Time = 1.50 (sec) , antiderivative size = 189, normalized size of antiderivative = 3.86 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx=\frac {\frac {F^a\,\left (945\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \left (F\right )}{\sqrt {b\,\ln \left (F\right )}\,\left (c+d\,x\right )}\right )-\frac {1890\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,\sqrt {b\,\ln \left (F\right )}}{c+d\,x}\right )}{64\,\sqrt {b\,\ln \left (F\right )}}-\frac {63\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^2\,{\ln \left (F\right )}^2}{8\,{\left (c+d\,x\right )}^5}+\frac {9\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^3\,{\ln \left (F\right )}^3}{4\,{\left (c+d\,x\right )}^7}-\frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^4\,{\ln \left (F\right )}^4}{2\,{\left (c+d\,x\right )}^9}+\frac {315\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b\,\ln \left (F\right )}{16\,{\left (c+d\,x\right )}^3}}{b^5\,d\,{\ln \left (F\right )}^5} \]
((F^a*(945*pi^(1/2)*erfi((b*log(F))/((b*log(F))^(1/2)*(c + d*x))) - (1890* F^(b/(c + d*x)^2)*(b*log(F))^(1/2))/(c + d*x)))/(64*(b*log(F))^(1/2)) - (6 3*F^a*F^(b/(c + d*x)^2)*b^2*log(F)^2)/(8*(c + d*x)^5) + (9*F^a*F^(b/(c + d *x)^2)*b^3*log(F)^3)/(4*(c + d*x)^7) - (F^a*F^(b/(c + d*x)^2)*b^4*log(F)^4 )/(2*(c + d*x)^9) + (315*F^a*F^(b/(c + d*x)^2)*b*log(F))/(16*(c + d*x)^3)) /(b^5*d*log(F)^5)