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

Optimal. Leaf size=132 \[ -\frac{105 \sqrt{\pi } f^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{32 b^{9/2} \log ^{\frac{9}{2}}(f)}+\frac{7 f^{a+\frac{b}{x^2}}}{4 b^2 x^5 \log ^2(f)}-\frac{35 f^{a+\frac{b}{x^2}}}{8 b^3 x^3 \log ^3(f)}+\frac{105 f^{a+\frac{b}{x^2}}}{16 b^4 x \log ^4(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^7 \log (f)} \]

[Out]

(-105*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x])/(32*b^(9/2)*Log[f]^(9/2)) + (105*f^(a + b/x^2))/(16*b^4*x*L
og[f]^4) - (35*f^(a + b/x^2))/(8*b^3*x^3*Log[f]^3) + (7*f^(a + b/x^2))/(4*b^2*x^5*Log[f]^2) - f^(a + b/x^2)/(2
*b*x^7*Log[f])

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Rubi [A]  time = 0.160033, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2212, 2211, 2204} \[ -\frac{105 \sqrt{\pi } f^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{32 b^{9/2} \log ^{\frac{9}{2}}(f)}+\frac{7 f^{a+\frac{b}{x^2}}}{4 b^2 x^5 \log ^2(f)}-\frac{35 f^{a+\frac{b}{x^2}}}{8 b^3 x^3 \log ^3(f)}+\frac{105 f^{a+\frac{b}{x^2}}}{16 b^4 x \log ^4(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^7 \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^2)/x^10,x]

[Out]

(-105*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x])/(32*b^(9/2)*Log[f]^(9/2)) + (105*f^(a + b/x^2))/(16*b^4*x*L
og[f]^4) - (35*f^(a + b/x^2))/(8*b^3*x^3*Log[f]^3) + (7*f^(a + b/x^2))/(4*b^2*x^5*Log[f]^2) - f^(a + b/x^2)/(2
*b*x^7*Log[f])

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2211

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/(d*(m + 1))
, Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(m + 1
)]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{f^{a+\frac{b}{x^2}}}{x^{10}} \, dx &=-\frac{f^{a+\frac{b}{x^2}}}{2 b x^7 \log (f)}-\frac{7 \int \frac{f^{a+\frac{b}{x^2}}}{x^8} \, dx}{2 b \log (f)}\\ &=\frac{7 f^{a+\frac{b}{x^2}}}{4 b^2 x^5 \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^7 \log (f)}+\frac{35 \int \frac{f^{a+\frac{b}{x^2}}}{x^6} \, dx}{4 b^2 \log ^2(f)}\\ &=-\frac{35 f^{a+\frac{b}{x^2}}}{8 b^3 x^3 \log ^3(f)}+\frac{7 f^{a+\frac{b}{x^2}}}{4 b^2 x^5 \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^7 \log (f)}-\frac{105 \int \frac{f^{a+\frac{b}{x^2}}}{x^4} \, dx}{8 b^3 \log ^3(f)}\\ &=\frac{105 f^{a+\frac{b}{x^2}}}{16 b^4 x \log ^4(f)}-\frac{35 f^{a+\frac{b}{x^2}}}{8 b^3 x^3 \log ^3(f)}+\frac{7 f^{a+\frac{b}{x^2}}}{4 b^2 x^5 \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^7 \log (f)}+\frac{105 \int \frac{f^{a+\frac{b}{x^2}}}{x^2} \, dx}{16 b^4 \log ^4(f)}\\ &=\frac{105 f^{a+\frac{b}{x^2}}}{16 b^4 x \log ^4(f)}-\frac{35 f^{a+\frac{b}{x^2}}}{8 b^3 x^3 \log ^3(f)}+\frac{7 f^{a+\frac{b}{x^2}}}{4 b^2 x^5 \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^7 \log (f)}-\frac{105 \operatorname{Subst}\left (\int f^{a+b x^2} \, dx,x,\frac{1}{x}\right )}{16 b^4 \log ^4(f)}\\ &=-\frac{105 f^a \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{32 b^{9/2} \log ^{\frac{9}{2}}(f)}+\frac{105 f^{a+\frac{b}{x^2}}}{16 b^4 x \log ^4(f)}-\frac{35 f^{a+\frac{b}{x^2}}}{8 b^3 x^3 \log ^3(f)}+\frac{7 f^{a+\frac{b}{x^2}}}{4 b^2 x^5 \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^7 \log (f)}\\ \end{align*}

Mathematica [A]  time = 0.0864489, size = 100, normalized size = 0.76 \[ \frac{f^a \left (\frac{2 \sqrt{b} \sqrt{\log (f)} f^{\frac{b}{x^2}} \left (28 b^2 x^2 \log ^2(f)-8 b^3 \log ^3(f)-70 b x^4 \log (f)+105 x^6\right )}{x^7}-105 \sqrt{\pi } \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )\right )}{32 b^{9/2} \log ^{\frac{9}{2}}(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^2)/x^10,x]

[Out]

(f^a*(-105*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x] + (2*Sqrt[b]*f^(b/x^2)*Sqrt[Log[f]]*(105*x^6 - 70*b*x^4*Log
[f] + 28*b^2*x^2*Log[f]^2 - 8*b^3*Log[f]^3))/x^7))/(32*b^(9/2)*Log[f]^(9/2))

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Maple [A]  time = 0.058, size = 124, normalized size = 0.9 \begin{align*} -{\frac{{f}^{a}}{2\,{x}^{7}b\ln \left ( f \right ) }{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{7\,{f}^{a}}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{x}^{5}}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{35\,{f}^{a}}{8\,{b}^{3}{x}^{3} \left ( \ln \left ( f \right ) \right ) ^{3}}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{105\,{f}^{a}}{16\,{b}^{4}x \left ( \ln \left ( f \right ) \right ) ^{4}}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{105\,{f}^{a}\sqrt{\pi }}{32\,{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^2)/x^10,x)

[Out]

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

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Maxima [A]  time = 1.2237, size = 38, normalized size = 0.29 \begin{align*} \frac{f^{a} \Gamma \left (\frac{9}{2}, -\frac{b \log \left (f\right )}{x^{2}}\right )}{2 \, x^{9} \left (-\frac{b \log \left (f\right )}{x^{2}}\right )^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.91263, size = 258, normalized size = 1.95 \begin{align*} \frac{105 \, \sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} x^{7} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) + 2 \,{\left (105 \, b x^{6} \log \left (f\right ) - 70 \, b^{2} x^{4} \log \left (f\right )^{2} + 28 \, b^{3} x^{2} \log \left (f\right )^{3} - 8 \, b^{4} \log \left (f\right )^{4}\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{32 \, b^{5} x^{7} \log \left (f\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(105*sqrt(pi)*sqrt(-b*log(f))*f^a*x^7*erf(sqrt(-b*log(f))/x) + 2*(105*b*x^6*log(f) - 70*b^2*x^4*log(f)^2
+ 28*b^3*x^2*log(f)^3 - 8*b^4*log(f)^4)*f^((a*x^2 + b)/x^2))/(b^5*x^7*log(f)^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{2}}}}{x^{10}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x^2)/x^10, x)