\(\int f^{a+\frac {b}{x^2}} x^{10} \, dx\) [141]

Optimal result

Integrand size = 13, antiderivative size = 34 \[ \int f^{a+\frac {b}{x^2}} x^{10} \, dx=\frac {1}{2} f^a x^{11} \Gamma \left (-\frac {11}{2},-\frac {b \log (f)}{x^2}\right ) \left (-\frac {b \log (f)}{x^2}\right )^{11/2} \]

[Out]

1/2*f^a*x^11*(64/10395*Pi^(1/2)*erfc((-b*ln(f)/x^2)^(1/2))-64/10395/(-b*ln(f)/x^2)^(1/2)*exp(b*ln(f)/x^2)+32/1
0395/(-b*ln(f)/x^2)^(3/2)*exp(b*ln(f)/x^2)-16/3465/(-b*ln(f)/x^2)^(5/2)*exp(b*ln(f)/x^2)+8/693/(-b*ln(f)/x^2)^
(7/2)*exp(b*ln(f)/x^2)-4/99/(-b*ln(f)/x^2)^(9/2)*exp(b*ln(f)/x^2)+2/11/(-b*ln(f)/x^2)^(11/2)*exp(b*ln(f)/x^2))
*(-b*ln(f)/x^2)^(11/2)

Rubi [A] (verified)

Time = 0.02 (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 = {2250} \[ \int f^{a+\frac {b}{x^2}} x^{10} \, dx=\frac {1}{2} x^{11} f^a \left (-\frac {b \log (f)}{x^2}\right )^{11/2} \Gamma \left (-\frac {11}{2},-\frac {b \log (f)}{x^2}\right ) \]

[In]

Int[f^(a + b/x^2)*x^10,x]

[Out]

(f^a*x^11*Gamma[-11/2, -((b*Log[f])/x^2)]*(-((b*Log[f])/x^2))^(11/2))/2

Rule 2250

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] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} f^a x^{11} \Gamma \left (-\frac {11}{2},-\frac {b \log (f)}{x^2}\right ) \left (-\frac {b \log (f)}{x^2}\right )^{11/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int f^{a+\frac {b}{x^2}} x^{10} \, dx=\frac {1}{2} f^a x^{11} \Gamma \left (-\frac {11}{2},-\frac {b \log (f)}{x^2}\right ) \left (-\frac {b \log (f)}{x^2}\right )^{11/2} \]

[In]

Integrate[f^(a + b/x^2)*x^10,x]

[Out]

(f^a*x^11*Gamma[-11/2, -((b*Log[f])/x^2)]*(-((b*Log[f])/x^2))^(11/2))/2

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 124, normalized size of antiderivative = 3.65

method	result	size
meijerg	\(\frac {f^{a} b^{5} \ln \left (f \right )^{\frac {11}{2}} \sqrt {-b}\, \left (-\frac {2 x^{11} \left (\frac {32 b^{5} \ln \left (f \right )^{5}}{945 x^{10}}+\frac {16 b^{4} \ln \left (f \right )^{4}}{945 x^{8}}+\frac {8 b^{3} \ln \left (f \right )^{3}}{315 x^{6}}+\frac {4 b^{2} \ln \left (f \right )^{2}}{63 x^{4}}+\frac {2 b \ln \left (f \right )}{9 x^{2}}+1\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{11 \left (-b \right )^{\frac {11}{2}} \ln \left (f \right )^{\frac {11}{2}}}+\frac {64 b^{\frac {11}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {b}\, \sqrt {\ln \left (f \right )}}{x}\right )}{10395 \left (-b \right )^{\frac {11}{2}}}\right )}{2}\)	\(124\)
risch	\(\frac {f^{a} x^{11} f^{\frac {b}{x^{2}}}}{11}+\frac {2 f^{a} \ln \left (f \right ) b \,x^{9} f^{\frac {b}{x^{2}}}}{99}+\frac {4 f^{a} \ln \left (f \right )^{2} b^{2} x^{7} f^{\frac {b}{x^{2}}}}{693}+\frac {8 f^{a} \ln \left (f \right )^{3} b^{3} x^{5} f^{\frac {b}{x^{2}}}}{3465}+\frac {16 f^{a} \ln \left (f \right )^{4} b^{4} x^{3} f^{\frac {b}{x^{2}}}}{10395}+\frac {32 f^{a} \ln \left (f \right )^{5} b^{5} x \,f^{\frac {b}{x^{2}}}}{10395}-\frac {32 f^{a} \ln \left (f \right )^{6} b^{6} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (f \right )}}{x}\right )}{10395 \sqrt {-b \ln \left (f \right )}}\)	\(155\)

[In]

int(f^(a+b/x^2)*x^10,x,method=_RETURNVERBOSE)

[Out]

1/2*f^a*b^5*ln(f)^(11/2)*(-b)^(1/2)*(-2/11*x^11/(-b)^(11/2)/ln(f)^(11/2)*(32/945*b^5*ln(f)^5/x^10+16/945*b^4*l
n(f)^4/x^8+8/315*b^3*ln(f)^3/x^6+4/63*b^2*ln(f)^2/x^4+2/9*b*ln(f)/x^2+1)*exp(b*ln(f)/x^2)+64/10395/(-b)^(11/2)
*b^(11/2)*Pi^(1/2)*erfi(b^(1/2)*ln(f)^(1/2)/x))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.24 \[ \int f^{a+\frac {b}{x^2}} x^{10} \, dx=\frac {32}{10395} \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} b^{5} f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (f\right )}}{x}\right ) \log \left (f\right )^{5} + \frac {1}{10395} \, {\left (945 \, x^{11} + 210 \, b x^{9} \log \left (f\right ) + 60 \, b^{2} x^{7} \log \left (f\right )^{2} + 24 \, b^{3} x^{5} \log \left (f\right )^{3} + 16 \, b^{4} x^{3} \log \left (f\right )^{4} + 32 \, b^{5} x \log \left (f\right )^{5}\right )} f^{\frac {a x^{2} + b}{x^{2}}} \]

[In]

integrate(f^(a+b/x^2)*x^10,x, algorithm="fricas")

[Out]

32/10395*sqrt(pi)*sqrt(-b*log(f))*b^5*f^a*erf(sqrt(-b*log(f))/x)*log(f)^5 + 1/10395*(945*x^11 + 210*b*x^9*log(
f) + 60*b^2*x^7*log(f)^2 + 24*b^3*x^5*log(f)^3 + 16*b^4*x^3*log(f)^4 + 32*b^5*x*log(f)^5)*f^((a*x^2 + b)/x^2)

Sympy [F]

\[ \int f^{a+\frac {b}{x^2}} x^{10} \, dx=\int f^{a + \frac {b}{x^{2}}} x^{10}\, dx \]

[In]

integrate(f**(a+b/x**2)*x**10,x)

[Out]

Integral(f**(a + b/x**2)*x**10, x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int f^{a+\frac {b}{x^2}} x^{10} \, dx=\frac {1}{2} \, f^{a} x^{11} \left (-\frac {b \log \left (f\right )}{x^{2}}\right )^{\frac {11}{2}} \Gamma \left (-\frac {11}{2}, -\frac {b \log \left (f\right )}{x^{2}}\right ) \]

[In]

integrate(f^(a+b/x^2)*x^10,x, algorithm="maxima")

[Out]

1/2*f^a*x^11*(-b*log(f)/x^2)^(11/2)*gamma(-11/2, -b*log(f)/x^2)

Giac [F]

\[ \int f^{a+\frac {b}{x^2}} x^{10} \, dx=\int { f^{a + \frac {b}{x^{2}}} x^{10} \,d x } \]

[In]

integrate(f^(a+b/x^2)*x^10,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)*x^10, x)

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 173, normalized size of antiderivative = 5.09 \[ \int f^{a+\frac {b}{x^2}} x^{10} \, dx=\frac {f^a\,f^{\frac {b}{x^2}}\,x^{11}}{11}-\frac {32\,f^a\,x^{11}\,\sqrt {\pi }\,{\left (-\frac {b\,\ln \left (f\right )}{x^2}\right )}^{11/2}}{10395}+\frac {32\,f^a\,x^{11}\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\frac {b\,\ln \left (f\right )}{x^2}}\right )\,{\left (-\frac {b\,\ln \left (f\right )}{x^2}\right )}^{11/2}}{10395}+\frac {32\,b^5\,f^a\,f^{\frac {b}{x^2}}\,x\,{\ln \left (f\right )}^5}{10395}+\frac {4\,b^2\,f^a\,f^{\frac {b}{x^2}}\,x^7\,{\ln \left (f\right )}^2}{693}+\frac {8\,b^3\,f^a\,f^{\frac {b}{x^2}}\,x^5\,{\ln \left (f\right )}^3}{3465}+\frac {16\,b^4\,f^a\,f^{\frac {b}{x^2}}\,x^3\,{\ln \left (f\right )}^4}{10395}+\frac {2\,b\,f^a\,f^{\frac {b}{x^2}}\,x^9\,\ln \left (f\right )}{99} \]

[In]

int(f^(a + b/x^2)*x^10,x)

[Out]

(f^a*f^(b/x^2)*x^11)/11 - (32*f^a*x^11*pi^(1/2)*(-(b*log(f))/x^2)^(11/2))/10395 + (32*f^a*x^11*pi^(1/2)*erfc((
-(b*log(f))/x^2)^(1/2))*(-(b*log(f))/x^2)^(11/2))/10395 + (32*b^5*f^a*f^(b/x^2)*x*log(f)^5)/10395 + (4*b^2*f^a
*f^(b/x^2)*x^7*log(f)^2)/693 + (8*b^3*f^a*f^(b/x^2)*x^5*log(f)^3)/3465 + (16*b^4*f^a*f^(b/x^2)*x^3*log(f)^4)/1
0395 + (2*b*f^a*f^(b/x^2)*x^9*log(f))/99