\(\int f^{a+\frac {b}{x^2}} x^8 \, dx\) [142]

Optimal result

Integrand size = 13, antiderivative size = 34 \[ \int f^{a+\frac {b}{x^2}} x^8 \, dx=\frac {1}{2} f^a x^9 \Gamma \left (-\frac {9}{2},-\frac {b \log (f)}{x^2}\right ) \left (-\frac {b \log (f)}{x^2}\right )^{9/2} \]

[Out]

1/2*f^a*x^9*(-32/945*Pi^(1/2)*erfc((-b*ln(f)/x^2)^(1/2))+32/945/(-b*ln(f)/x^2)^(1/2)*exp(b*ln(f)/x^2)-16/945/(
-b*ln(f)/x^2)^(3/2)*exp(b*ln(f)/x^2)+8/315/(-b*ln(f)/x^2)^(5/2)*exp(b*ln(f)/x^2)-4/63/(-b*ln(f)/x^2)^(7/2)*exp
(b*ln(f)/x^2)+2/9/(-b*ln(f)/x^2)^(9/2)*exp(b*ln(f)/x^2))*(-b*ln(f)/x^2)^(9/2)

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} \[ \int f^{a+\frac {b}{x^2}} x^8 \, dx=\frac {1}{2} x^9 f^a \left (-\frac {b \log (f)}{x^2}\right )^{9/2} \Gamma \left (-\frac {9}{2},-\frac {b \log (f)}{x^2}\right ) \]

[In]

Int[f^(a + b/x^2)*x^8,x]

[Out]

(f^a*x^9*Gamma[-9/2, -((b*Log[f])/x^2)]*(-((b*Log[f])/x^2))^(9/2))/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^9 \Gamma \left (-\frac {9}{2},-\frac {b \log (f)}{x^2}\right ) \left (-\frac {b \log (f)}{x^2}\right )^{9/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int f^{a+\frac {b}{x^2}} x^8 \, dx=\frac {1}{2} f^a x^9 \Gamma \left (-\frac {9}{2},-\frac {b \log (f)}{x^2}\right ) \left (-\frac {b \log (f)}{x^2}\right )^{9/2} \]

[In]

Integrate[f^(a + b/x^2)*x^8,x]

[Out]

(f^a*x^9*Gamma[-9/2, -((b*Log[f])/x^2)]*(-((b*Log[f])/x^2))^(9/2))/2

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.29

method	result	size
meijerg	\(-\frac {f^{a} b^{4} \ln \left (f \right )^{\frac {9}{2}} \sqrt {-b}\, \left (-\frac {2 x^{9} \left (\frac {16 b^{4} \ln \left (f \right )^{4}}{105 x^{8}}+\frac {8 b^{3} \ln \left (f \right )^{3}}{105 x^{6}}+\frac {4 b^{2} \ln \left (f \right )^{2}}{35 x^{4}}+\frac {2 b \ln \left (f \right )}{7 x^{2}}+1\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{9 \left (-b \right )^{\frac {9}{2}} \ln \left (f \right )^{\frac {9}{2}}}+\frac {32 b^{\frac {9}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {b}\, \sqrt {\ln \left (f \right )}}{x}\right )}{945 \left (-b \right )^{\frac {9}{2}}}\right )}{2}\)	\(112\)
risch	\(\frac {f^{a} x^{9} f^{\frac {b}{x^{2}}}}{9}+\frac {2 f^{a} \ln \left (f \right ) b \,x^{7} f^{\frac {b}{x^{2}}}}{63}+\frac {4 f^{a} \ln \left (f \right )^{2} b^{2} x^{5} f^{\frac {b}{x^{2}}}}{315}+\frac {8 f^{a} \ln \left (f \right )^{3} b^{3} x^{3} f^{\frac {b}{x^{2}}}}{945}+\frac {16 f^{a} \ln \left (f \right )^{4} b^{4} x \,f^{\frac {b}{x^{2}}}}{945}-\frac {16 f^{a} \ln \left (f \right )^{5} b^{5} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (f \right )}}{x}\right )}{945 \sqrt {-b \ln \left (f \right )}}\)	\(133\)

[In]

int(f^(a+b/x^2)*x^8,x,method=_RETURNVERBOSE)

[Out]

-1/2*f^a*b^4*ln(f)^(9/2)*(-b)^(1/2)*(-2/9*x^9/(-b)^(9/2)/ln(f)^(9/2)*(16/105*b^4*ln(f)^4/x^8+8/105*b^3*ln(f)^3
/x^6+4/35*b^2*ln(f)^2/x^4+2/7*b*ln(f)/x^2+1)*exp(b*ln(f)/x^2)+32/945/(-b)^(9/2)*b^(9/2)*Pi^(1/2)*erfi(b^(1/2)*
ln(f)^(1/2)/x))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.88 \[ \int f^{a+\frac {b}{x^2}} x^8 \, dx=\frac {16}{945} \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} b^{4} f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (f\right )}}{x}\right ) \log \left (f\right )^{4} + \frac {1}{945} \, {\left (105 \, x^{9} + 30 \, b x^{7} \log \left (f\right ) + 12 \, b^{2} x^{5} \log \left (f\right )^{2} + 8 \, b^{3} x^{3} \log \left (f\right )^{3} + 16 \, b^{4} x \log \left (f\right )^{4}\right )} f^{\frac {a x^{2} + b}{x^{2}}} \]

[In]

integrate(f^(a+b/x^2)*x^8,x, algorithm="fricas")

[Out]

16/945*sqrt(pi)*sqrt(-b*log(f))*b^4*f^a*erf(sqrt(-b*log(f))/x)*log(f)^4 + 1/945*(105*x^9 + 30*b*x^7*log(f) + 1
2*b^2*x^5*log(f)^2 + 8*b^3*x^3*log(f)^3 + 16*b^4*x*log(f)^4)*f^((a*x^2 + b)/x^2)

Sympy [F]

\[ \int f^{a+\frac {b}{x^2}} x^8 \, dx=\int f^{a + \frac {b}{x^{2}}} x^{8}\, dx \]

[In]

integrate(f**(a+b/x**2)*x**8,x)

[Out]

Integral(f**(a + b/x**2)*x**8, x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int f^{a+\frac {b}{x^2}} x^8 \, dx=\frac {1}{2} \, f^{a} x^{9} \left (-\frac {b \log \left (f\right )}{x^{2}}\right )^{\frac {9}{2}} \Gamma \left (-\frac {9}{2}, -\frac {b \log \left (f\right )}{x^{2}}\right ) \]

[In]

integrate(f^(a+b/x^2)*x^8,x, algorithm="maxima")

[Out]

1/2*f^a*x^9*(-b*log(f)/x^2)^(9/2)*gamma(-9/2, -b*log(f)/x^2)

Giac [F]

\[ \int f^{a+\frac {b}{x^2}} x^8 \, dx=\int { f^{a + \frac {b}{x^{2}}} x^{8} \,d x } \]

[In]

integrate(f^(a+b/x^2)*x^8,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)*x^8, x)

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 151, normalized size of antiderivative = 4.44 \[ \int f^{a+\frac {b}{x^2}} x^8 \, dx=\frac {f^a\,f^{\frac {b}{x^2}}\,x^9}{9}+\frac {16\,f^a\,x^9\,\sqrt {\pi }\,{\left (-\frac {b\,\ln \left (f\right )}{x^2}\right )}^{9/2}}{945}-\frac {16\,f^a\,x^9\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\frac {b\,\ln \left (f\right )}{x^2}}\right )\,{\left (-\frac {b\,\ln \left (f\right )}{x^2}\right )}^{9/2}}{945}+\frac {16\,b^4\,f^a\,f^{\frac {b}{x^2}}\,x\,{\ln \left (f\right )}^4}{945}+\frac {4\,b^2\,f^a\,f^{\frac {b}{x^2}}\,x^5\,{\ln \left (f\right )}^2}{315}+\frac {8\,b^3\,f^a\,f^{\frac {b}{x^2}}\,x^3\,{\ln \left (f\right )}^3}{945}+\frac {2\,b\,f^a\,f^{\frac {b}{x^2}}\,x^7\,\ln \left (f\right )}{63} \]

[In]

int(f^(a + b/x^2)*x^8,x)

[Out]

(f^a*f^(b/x^2)*x^9)/9 + (16*f^a*x^9*pi^(1/2)*(-(b*log(f))/x^2)^(9/2))/945 - (16*f^a*x^9*pi^(1/2)*erfc((-(b*log
(f))/x^2)^(1/2))*(-(b*log(f))/x^2)^(9/2))/945 + (16*b^4*f^a*f^(b/x^2)*x*log(f)^4)/945 + (4*b^2*f^a*f^(b/x^2)*x
^5*log(f)^2)/315 + (8*b^3*f^a*f^(b/x^2)*x^3*log(f)^3)/945 + (2*b*f^a*f^(b/x^2)*x^7*log(f))/63