\(\int \frac {f^{a+b x^2}}{x^{10}} \, dx\) [93]

Optimal result

Integrand size = 13, antiderivative size = 34 \[ \int \frac {f^{a+b x^2}}{x^{10}} \, dx=-\frac {f^a \Gamma \left (-\frac {9}{2},-b x^2 \log (f)\right ) \left (-b x^2 \log (f)\right )^{9/2}}{2 x^9} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (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} \[ \int \frac {f^{a+b x^2}}{x^{10}} \, dx=-\frac {f^a \left (-b x^2 \log (f)\right )^{9/2} \Gamma \left (-\frac {9}{2},-b x^2 \log (f)\right )}{2 x^9} \]

[In]

Int[f^(a + b*x^2)/x^10,x]

[Out]

-1/2*(f^a*Gamma[-9/2, -(b*x^2*Log[f])]*(-(b*x^2*Log[f]))^(9/2))/x^9

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {f^{a+b x^2}}{x^{10}} \, dx=-\frac {f^a \Gamma \left (-\frac {9}{2},-b x^2 \log (f)\right ) \left (-b x^2 \log (f)\right )^{9/2}}{2 x^9} \]

[In]

Integrate[f^(a + b*x^2)/x^10,x]

[Out]

-1/2*(f^a*Gamma[-9/2, -(b*x^2*Log[f])]*(-(b*x^2*Log[f]))^(9/2))/x^9

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.24

method	result	size
meijerg	\(-\frac {f^{a} b^{5} \ln \left (f \right )^{\frac {9}{2}} \left (-\frac {2 \left (\frac {16 b^{4} x^{8} \ln \left (f \right )^{4}}{105}+\frac {8 b^{3} x^{6} \ln \left (f \right )^{3}}{105}+\frac {4 b^{2} x^{4} \ln \left (f \right )^{2}}{35}+\frac {2 b \,x^{2} \ln \left (f \right )}{7}+1\right ) {\mathrm e}^{b \,x^{2} \ln \left (f \right )}}{9 x^{9} \left (-b \right )^{\frac {9}{2}} \ln \left (f \right )^{\frac {9}{2}}}+\frac {32 b^{\frac {9}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (x \sqrt {b}\, \sqrt {\ln \left (f \right )}\right )}{945 \left (-b \right )^{\frac {9}{2}}}\right )}{2 \sqrt {-b}}\)	\(110\)
risch	\(-\frac {f^{a} f^{b \,x^{2}}}{9 x^{9}}-\frac {2 f^{a} \ln \left (f \right ) b \,f^{b \,x^{2}}}{63 x^{7}}-\frac {4 f^{a} \ln \left (f \right )^{2} b^{2} f^{b \,x^{2}}}{315 x^{5}}-\frac {8 f^{a} \ln \left (f \right )^{3} b^{3} f^{b \,x^{2}}}{945 x^{3}}-\frac {16 f^{a} \ln \left (f \right )^{4} b^{4} f^{b \,x^{2}}}{945 x}+\frac {16 f^{a} \ln \left (f \right )^{5} b^{5} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b \ln \left (f \right )}\, x \right )}{945 \sqrt {-b \ln \left (f \right )}}\)	\(133\)

[In]

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

[Out]

-1/2*f^a*b^5*ln(f)^(9/2)/(-b)^(1/2)*(-2/9/x^9/(-b)^(9/2)/ln(f)^(9/2)*(16/105*b^4*x^8*ln(f)^4+8/105*b^3*x^6*ln(
f)^3+4/35*b^2*x^4*ln(f)^2+2/7*b*x^2*ln(f)+1)*exp(b*x^2*ln(f))+32/945/(-b)^(9/2)*b^(9/2)*Pi^(1/2)*erfi(x*b^(1/2
)*ln(f)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.85 \[ \int \frac {f^{a+b x^2}}{x^{10}} \, dx=-\frac {16 \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} b^{4} f^{a} x^{9} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x\right ) \log \left (f\right )^{4} + {\left (16 \, b^{4} x^{8} \log \left (f\right )^{4} + 8 \, b^{3} x^{6} \log \left (f\right )^{3} + 12 \, b^{2} x^{4} \log \left (f\right )^{2} + 30 \, b x^{2} \log \left (f\right ) + 105\right )} f^{b x^{2} + a}}{945 \, x^{9}} \]

[In]

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

[Out]

-1/945*(16*sqrt(pi)*sqrt(-b*log(f))*b^4*f^a*x^9*erf(sqrt(-b*log(f))*x)*log(f)^4 + (16*b^4*x^8*log(f)^4 + 8*b^3
*x^6*log(f)^3 + 12*b^2*x^4*log(f)^2 + 30*b*x^2*log(f) + 105)*f^(b*x^2 + a))/x^9

Sympy [F]

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

[In]

integrate(f**(b*x**2+a)/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 \[ \int \frac {f^{a+b x^2}}{x^{10}} \, dx=-\frac {\left (-b x^{2} \log \left (f\right )\right )^{\frac {9}{2}} f^{a} \Gamma \left (-\frac {9}{2}, -b x^{2} \log \left (f\right )\right )}{2 \, x^{9}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(f^(b*x^2 + a)/x^10, x)

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 153, normalized size of antiderivative = 4.50 \[ \int \frac {f^{a+b x^2}}{x^{10}} \, dx=\frac {16\,f^a\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-b\,x^2\,\ln \left (f\right )}\right )\,{\left (-b\,x^2\,\ln \left (f\right )\right )}^{9/2}}{945\,x^9}-\frac {16\,f^a\,\sqrt {\pi }\,{\left (-b\,x^2\,\ln \left (f\right )\right )}^{9/2}}{945\,x^9}-\frac {f^a\,f^{b\,x^2}}{9\,x^9}-\frac {4\,b^2\,f^a\,f^{b\,x^2}\,{\ln \left (f\right )}^2}{315\,x^5}-\frac {8\,b^3\,f^a\,f^{b\,x^2}\,{\ln \left (f\right )}^3}{945\,x^3}-\frac {16\,b^4\,f^a\,f^{b\,x^2}\,{\ln \left (f\right )}^4}{945\,x}-\frac {2\,b\,f^a\,f^{b\,x^2}\,\ln \left (f\right )}{63\,x^7} \]

[In]

int(f^(a + b*x^2)/x^10,x)

[Out]

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