Integrand size = 13, antiderivative size = 34 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{14}} \, dx=\frac {f^a \Gamma \left (\frac {13}{2},-\frac {b \log (f)}{x^2}\right )}{2 x^{13} \left (-\frac {b \log (f)}{x^2}\right )^{13/2}} \]
1/2*f^a*(524288/5621533568633696205238621875*GAMMA(51/2,-b*ln(f)/x^2)-5242 88/5621533568633696205238621875*(-b*ln(f)/x^2)^(49/2)*exp(b*ln(f)/x^2)-262 144/114725174870075432759971875*(-b*ln(f)/x^2)^(47/2)*exp(b*ln(f)/x^2)-131 072/2440961167448413462978125*(-b*ln(f)/x^2)^(45/2)*exp(b*ln(f)/x^2)-65536 /54243581498853632510625*(-b*ln(f)/x^2)^(43/2)*exp(b*ln(f)/x^2)-32768/1261 478639508224011875*(-b*ln(f)/x^2)^(41/2)*exp(b*ln(f)/x^2)-16384/3076777169 5322536875*(-b*ln(f)/x^2)^(39/2)*exp(b*ln(f)/x^2)-8192/788917222956988125* (-b*ln(f)/x^2)^(37/2)*exp(b*ln(f)/x^2)-4096/21322087106945625*(-b*ln(f)/x^ 2)^(35/2)*exp(b*ln(f)/x^2)-2048/609202488769875*(-b*ln(f)/x^2)^(33/2)*exp( b*ln(f)/x^2)-1024/18460681477875*(-b*ln(f)/x^2)^(31/2)*exp(b*ln(f)/x^2)-51 2/595505854125*(-b*ln(f)/x^2)^(29/2)*exp(b*ln(f)/x^2)-256/20534684625*(-b* ln(f)/x^2)^(27/2)*exp(b*ln(f)/x^2)-128/760543875*(-b*ln(f)/x^2)^(25/2)*exp (b*ln(f)/x^2)-64/30421755*(-b*ln(f)/x^2)^(23/2)*exp(b*ln(f)/x^2)-32/132268 5*(-b*ln(f)/x^2)^(21/2)*exp(b*ln(f)/x^2)-16/62985*(-b*ln(f)/x^2)^(19/2)*ex p(b*ln(f)/x^2)-8/3315*(-b*ln(f)/x^2)^(17/2)*exp(b*ln(f)/x^2)-4/195*(-b*ln( f)/x^2)^(15/2)*exp(b*ln(f)/x^2)-2/13*(-b*ln(f)/x^2)^(13/2)*exp(b*ln(f)/x^2 ))/x^13/(-b*ln(f)/x^2)^(13/2)
Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{14}} \, dx=\frac {f^a \Gamma \left (\frac {13}{2},-\frac {b \log (f)}{x^2}\right )}{2 x^{13} \left (-\frac {b \log (f)}{x^2}\right )^{13/2}} \]
Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.
\(\displaystyle \int \frac {f^{a+\frac {b}{x^2}}}{x^{14}} \, dx\) |
\(\Big \downarrow \) 2648 |
\(\displaystyle \frac {f^a \Gamma \left (\frac {13}{2},-\frac {b \log (f)}{x^2}\right )}{2 x^{13} \left (-\frac {b \log (f)}{x^2}\right )^{13/2}}\) |
3.2.53.3.1 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 = 0.70 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.74
method | result | size |
meijerg | \(\frac {f^{a} \sqrt {-b}\, \left (-\frac {\left (-b \right )^{\frac {13}{2}} \sqrt {\ln \left (f \right )}\, \left (-\frac {416 b^{5} \ln \left (f \right )^{5}}{x^{10}}+\frac {2288 b^{4} \ln \left (f \right )^{4}}{x^{8}}-\frac {10296 b^{3} \ln \left (f \right )^{3}}{x^{6}}+\frac {36036 b^{2} \ln \left (f \right )^{2}}{x^{4}}-\frac {90090 b \ln \left (f \right )}{x^{2}}+135135\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{416 x \,b^{6}}+\frac {10395 \left (-b \right )^{\frac {13}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {b}\, \sqrt {\ln \left (f \right )}}{x}\right )}{64 b^{\frac {13}{2}}}\right )}{2 b^{7} \ln \left (f \right )^{\frac {13}{2}}}\) | \(127\) |
risch | \(-\frac {f^{a} f^{\frac {b}{x^{2}}}}{2 x^{11} b \ln \left (f \right )}+\frac {11 f^{a} f^{\frac {b}{x^{2}}}}{4 \ln \left (f \right )^{2} b^{2} x^{9}}-\frac {99 f^{a} f^{\frac {b}{x^{2}}}}{8 \ln \left (f \right )^{3} b^{3} x^{7}}+\frac {693 f^{a} f^{\frac {b}{x^{2}}}}{16 \ln \left (f \right )^{4} b^{4} x^{5}}-\frac {3465 f^{a} f^{\frac {b}{x^{2}}}}{32 \ln \left (f \right )^{5} b^{5} x^{3}}+\frac {10395 f^{a} f^{\frac {b}{x^{2}}}}{64 \ln \left (f \right )^{6} b^{6} x}-\frac {10395 f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (f \right )}}{x}\right )}{128 \ln \left (f \right )^{6} b^{6} \sqrt {-b \ln \left (f \right )}}\) | \(168\) |
1/2*f^a/b^7/ln(f)^(13/2)*(-b)^(1/2)*(-1/416/x*(-b)^(13/2)*ln(f)^(1/2)*(-41 6*b^5*ln(f)^5/x^10+2288*b^4*ln(f)^4/x^8-10296*b^3*ln(f)^3/x^6+36036*b^2*ln (f)^2/x^4-90090*b*ln(f)/x^2+135135)/b^6*exp(b*ln(f)/x^2)+10395/64*(-b)^(13 /2)/b^(13/2)*Pi^(1/2)*erfi(b^(1/2)*ln(f)^(1/2)/x))
Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 3.65 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{14}} \, dx=\frac {10395 \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} x^{11} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (f\right )}}{x}\right ) + 2 \, {\left (10395 \, b x^{10} \log \left (f\right ) - 6930 \, b^{2} x^{8} \log \left (f\right )^{2} + 2772 \, b^{3} x^{6} \log \left (f\right )^{3} - 792 \, b^{4} x^{4} \log \left (f\right )^{4} + 176 \, b^{5} x^{2} \log \left (f\right )^{5} - 32 \, b^{6} \log \left (f\right )^{6}\right )} f^{\frac {a x^{2} + b}{x^{2}}}}{128 \, b^{7} x^{11} \log \left (f\right )^{7}} \]
1/128*(10395*sqrt(pi)*sqrt(-b*log(f))*f^a*x^11*erf(sqrt(-b*log(f))/x) + 2* (10395*b*x^10*log(f) - 6930*b^2*x^8*log(f)^2 + 2772*b^3*x^6*log(f)^3 - 792 *b^4*x^4*log(f)^4 + 176*b^5*x^2*log(f)^5 - 32*b^6*log(f)^6)*f^((a*x^2 + b) /x^2))/(b^7*x^11*log(f)^7)
Timed out. \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{14}} \, dx=\text {Timed out} \]
Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{14}} \, dx=\frac {f^{a} \Gamma \left (\frac {13}{2}, -\frac {b \log \left (f\right )}{x^{2}}\right )}{2 \, x^{13} \left (-\frac {b \log \left (f\right )}{x^{2}}\right )^{\frac {13}{2}}} \]
\[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{14}} \, dx=\int { \frac {f^{a + \frac {b}{x^{2}}}}{x^{14}} \,d x } \]
Time = 0.40 (sec) , antiderivative size = 159, normalized size of antiderivative = 4.68 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{14}} \, dx=-\frac {\frac {f^a\,\left (\frac {10395\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \left (f\right )}{x\,\sqrt {b\,\ln \left (f\right )}}\right )}{128}-\frac {10395\,f^{\frac {b}{x^2}}\,\sqrt {b\,\ln \left (f\right )}}{64\,x}\right )}{\sqrt {b\,\ln \left (f\right )}}-\frac {693\,b^2\,f^{a+\frac {b}{x^2}}\,{\ln \left (f\right )}^2}{16\,x^5}+\frac {99\,b^3\,f^{a+\frac {b}{x^2}}\,{\ln \left (f\right )}^3}{8\,x^7}-\frac {11\,b^4\,f^{a+\frac {b}{x^2}}\,{\ln \left (f\right )}^4}{4\,x^9}+\frac {b^5\,f^{a+\frac {b}{x^2}}\,{\ln \left (f\right )}^5}{2\,x^{11}}+\frac {3465\,b\,f^{a+\frac {b}{x^2}}\,\ln \left (f\right )}{32\,x^3}}{b^6\,{\ln \left (f\right )}^6} \]
-((f^a*((10395*pi^(1/2)*erfi((b*log(f))/(x*(b*log(f))^(1/2))))/128 - (1039 5*f^(b/x^2)*(b*log(f))^(1/2))/(64*x)))/(b*log(f))^(1/2) - (693*b^2*f^(a + b/x^2)*log(f)^2)/(16*x^5) + (99*b^3*f^(a + b/x^2)*log(f)^3)/(8*x^7) - (11* b^4*f^(a + b/x^2)*log(f)^4)/(4*x^9) + (b^5*f^(a + b/x^2)*log(f)^5)/(2*x^11 ) + (3465*b*f^(a + b/x^2)*log(f))/(32*x^3))/(b^6*log(f)^6)