3.2.51 \(\int \frac {f^{a+\frac {b}{x^2}}}{x^{10}} \, dx\) [151]

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

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 132, normalized size of antiderivative = 1.00, 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 = {2243, 2242, 2235} \begin {gather*} -\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 {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 {gather*}

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 2235

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]

Rule 2242

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 2243

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])

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 \text {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*}

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Mathematica [A]
time = 0.06, size = 100, normalized size = 0.76 \begin {gather*} \frac {f^a \left (-105 \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )+\frac {2 \sqrt {b} f^{\frac {b}{x^2}} \sqrt {\log (f)} \left (105 x^6-70 b x^4 \log (f)+28 b^2 x^2 \log ^2(f)-8 b^3 \log ^3(f)\right )}{x^7}\right )}{32 b^{9/2} \log ^{\frac {9}{2}}(f)} \end {gather*}

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 = 0.05, size = 103, normalized size = 0.78

method	result	size
meijerg	\(\frac {f^{a} \sqrt {-b}\, \left (-\frac {\left (-b \right )^{\frac {9}{2}} \sqrt {\ln \left (f \right )}\, \left (-\frac {72 b^{3} \ln \left (f \right )^{3}}{x^{6}}+\frac {252 b^{2} \ln \left (f \right )^{2}}{x^{4}}-\frac {630 b \ln \left (f \right )}{x^{2}}+945\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{72 x \,b^{4}}+\frac {105 \left (-b \right )^{\frac {9}{2}} \sqrt {\pi }\, \erfi \left (\frac {\sqrt {b}\, \sqrt {\ln \left (f \right )}}{x}\right )}{16 b^{\frac {9}{2}}}\right )}{2 b^{5} \ln \left (f \right )^{\frac {9}{2}}}\)	\(103\)
risch	\(-\frac {f^{a} f^{\frac {b}{x^{2}}}}{2 x^{7} b \ln \left (f \right )}+\frac {7 f^{a} f^{\frac {b}{x^{2}}}}{4 \ln \left (f \right )^{2} b^{2} x^{5}}-\frac {35 f^{a} f^{\frac {b}{x^{2}}}}{8 \ln \left (f \right )^{3} b^{3} x^{3}}+\frac {105 f^{a} f^{\frac {b}{x^{2}}}}{16 \ln \left (f \right )^{4} b^{4} x}-\frac {105 f^{a} \sqrt {\pi }\, \erf \left (\frac {\sqrt {-b \ln \left (f \right )}}{x}\right )}{32 \ln \left (f \right )^{4} b^{4} \sqrt {-b \ln \left (f \right )}}\)	\(124\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*f^a/b^5/ln(f)^(9/2)*(-b)^(1/2)*(-1/72/x*(-b)^(9/2)*ln(f)^(1/2)*(-72*b^3*ln(f)^3/x^6+252*b^2*ln(f)^2/x^4-63
0*b*ln(f)/x^2+945)/b^4*exp(b*ln(f)/x^2)+105/16*(-b)^(9/2)/b^(9/2)*Pi^(1/2)*erfi(b^(1/2)*ln(f)^(1/2)/x))

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Maxima [A]
time = 0.32, size = 28, normalized size = 0.21 \begin {gather*} \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 {gather*}

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 = 0.36, size = 100, normalized size = 0.76 \begin {gather*} \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 {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 3.69, size = 121, normalized size = 0.92 \begin {gather*} -\frac {\frac {f^a\,\left (105\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \left (f\right )}{x\,\sqrt {b\,\ln \left (f\right )}}\right )-\frac {210\,f^{\frac {b}{x^2}}\,\sqrt {b\,\ln \left (f\right )}}{x}\right )}{32\,\sqrt {b\,\ln \left (f\right )}}-\frac {7\,b^2\,f^a\,f^{\frac {b}{x^2}}\,{\ln \left (f\right )}^2}{4\,x^5}+\frac {b^3\,f^a\,f^{\frac {b}{x^2}}\,{\ln \left (f\right )}^3}{2\,x^7}+\frac {35\,b\,f^a\,f^{\frac {b}{x^2}}\,\ln \left (f\right )}{8\,x^3}}{b^4\,{\ln \left (f\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x^2)/x^10,x)

[Out]

-((f^a*(105*pi^(1/2)*erfi((b*log(f))/(x*(b*log(f))^(1/2))) - (210*f^(b/x^2)*(b*log(f))^(1/2))/x))/(32*(b*log(f
))^(1/2)) - (7*b^2*f^a*f^(b/x^2)*log(f)^2)/(4*x^5) + (b^3*f^a*f^(b/x^2)*log(f)^3)/(2*x^7) + (35*b*f^a*f^(b/x^2
)*log(f))/(8*x^3))/(b^4*log(f)^4)

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