3.2.29 \(\int f^{a+\frac {b}{x^2}} x^9 \, dx\) [129]

Optimal. Leaf size=24 \[ -\frac {1}{2} b^5 f^a \Gamma \left (-5,-\frac {b \log (f)}{x^2}\right ) \log ^5(f) \]

[Out]

1/2*f^a*x^10*Ei(6,-b*ln(f)/x^2)

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Rubi [A]
time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2250} \begin {gather*} -\frac {1}{2} b^5 f^a \log ^5(f) \text {Gamma}\left (-5,-\frac {b \log (f)}{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^2)*x^9,x]

[Out]

-1/2*(b^5*f^a*Gamma[-5, -((b*Log[f])/x^2)]*Log[f]^5)

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*} \int f^{a+\frac {b}{x^2}} x^9 \, dx &=-\frac {1}{2} b^5 f^a \Gamma \left (-5,-\frac {b \log (f)}{x^2}\right ) \log ^5(f)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} -\frac {1}{2} b^5 f^a \Gamma \left (-5,-\frac {b \log (f)}{x^2}\right ) \log ^5(f) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^2)*x^9,x]

[Out]

-1/2*(b^5*f^a*Gamma[-5, -((b*Log[f])/x^2)]*Log[f]^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(122\) vs. \(2(18)=36\).
time = 0.05, size = 123, normalized size = 5.12

method	result	size
risch	\(\frac {f^{a} x^{10} f^{\frac {b}{x^{2}}}}{10}+\frac {f^{a} \ln \left (f \right ) b \,x^{8} f^{\frac {b}{x^{2}}}}{40}+\frac {f^{a} \ln \left (f \right )^{2} b^{2} x^{6} f^{\frac {b}{x^{2}}}}{120}+\frac {f^{a} \ln \left (f \right )^{3} b^{3} x^{4} f^{\frac {b}{x^{2}}}}{240}+\frac {f^{a} \ln \left (f \right )^{4} b^{4} x^{2} f^{\frac {b}{x^{2}}}}{240}+\frac {f^{a} \ln \left (f \right )^{5} b^{5} \expIntegral \left (1, -\frac {b \ln \left (f \right )}{x^{2}}\right )}{240}\)	\(123\)
meijerg	\(\frac {f^{a} b^{5} \ln \left (f \right )^{5} \left (-\frac {x^{10} \left (\frac {137 b^{5} \ln \left (f \right )^{5}}{x^{10}}+\frac {300 b^{4} \ln \left (f \right )^{4}}{x^{8}}+\frac {600 b^{3} \ln \left (f \right )^{3}}{x^{6}}+\frac {1200 b^{2} \ln \left (f \right )^{2}}{x^{4}}+\frac {1800 b \ln \left (f \right )}{x^{2}}+1440\right )}{7200 b^{5} \ln \left (f \right )^{5}}+\frac {x^{10} \left (\frac {6 b^{4} \ln \left (f \right )^{4}}{x^{8}}+\frac {6 b^{3} \ln \left (f \right )^{3}}{x^{6}}+\frac {12 b^{2} \ln \left (f \right )^{2}}{x^{4}}+\frac {36 b \ln \left (f \right )}{x^{2}}+144\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{720 b^{5} \ln \left (f \right )^{5}}+\frac {\ln \left (-\frac {b \ln \left (f \right )}{x^{2}}\right )}{120}+\frac {\expIntegral \left (1, -\frac {b \ln \left (f \right )}{x^{2}}\right )}{120}+\frac {137}{7200}+\frac {\ln \left (x \right )}{60}-\frac {\ln \left (-b \right )}{120}-\frac {\ln \left (\ln \left (f \right )\right )}{120}+\frac {x^{10}}{5 b^{5} \ln \left (f \right )^{5}}+\frac {x^{8}}{4 b^{4} \ln \left (f \right )^{4}}+\frac {x^{6}}{6 b^{3} \ln \left (f \right )^{3}}+\frac {x^{4}}{12 b^{2} \ln \left (f \right )^{2}}+\frac {x^{2}}{24 b \ln \left (f \right )}\right )}{2}\)	\(249\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^2)*x^9,x,method=_RETURNVERBOSE)

[Out]

1/10*f^a*x^10*f^(b/x^2)+1/40*f^a*ln(f)*b*x^8*f^(b/x^2)+1/120*f^a*ln(f)^2*b^2*x^6*f^(b/x^2)+1/240*f^a*ln(f)^3*b
^3*x^4*f^(b/x^2)+1/240*f^a*ln(f)^4*b^4*x^2*f^(b/x^2)+1/240*f^a*ln(f)^5*b^5*Ei(1,-b*ln(f)/x^2)

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Maxima [A]
time = 0.06, size = 22, normalized size = 0.92 \begin {gather*} -\frac {1}{2} \, b^{5} f^{a} \Gamma \left (-5, -\frac {b \log \left (f\right )}{x^{2}}\right ) \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^9,x, algorithm="maxima")

[Out]

-1/2*b^5*f^a*gamma(-5, -b*log(f)/x^2)*log(f)^5

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (18) = 36\).
time = 0.11, size = 84, normalized size = 3.50 \begin {gather*} -\frac {1}{240} \, b^{5} f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right )^{5} + \frac {1}{240} \, {\left (24 \, x^{10} + 6 \, b x^{8} \log \left (f\right ) + 2 \, b^{2} x^{6} \log \left (f\right )^{2} + b^{3} x^{4} \log \left (f\right )^{3} + b^{4} x^{2} \log \left (f\right )^{4}\right )} f^{\frac {a x^{2} + b}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)*x^9,x, algorithm="fricas")

[Out]

-1/240*b^5*f^a*Ei(b*log(f)/x^2)*log(f)^5 + 1/240*(24*x^10 + 6*b*x^8*log(f) + 2*b^2*x^6*log(f)^2 + b^3*x^4*log(
f)^3 + b^4*x^2*log(f)^4)*f^((a*x^2 + b)/x^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int f^{a + \frac {b}{x^{2}}} x^{9}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**2)*x**9,x)

[Out]

Integral(f**(a + b/x**2)*x**9, x)

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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^9,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)*x^9, x)

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Mupad [B]
time = 3.79, size = 102, normalized size = 4.25 \begin {gather*} \frac {b^5\,f^a\,{\ln \left (f\right )}^5\,\mathrm {expint}\left (-\frac {b\,\ln \left (f\right )}{x^2}\right )}{240}+\frac {b^5\,f^a\,f^{\frac {b}{x^2}}\,{\ln \left (f\right )}^5\,\left (\frac {x^2}{120\,b\,\ln \left (f\right )}+\frac {x^4}{120\,b^2\,{\ln \left (f\right )}^2}+\frac {x^6}{60\,b^3\,{\ln \left (f\right )}^3}+\frac {x^8}{20\,b^4\,{\ln \left (f\right )}^4}+\frac {x^{10}}{5\,b^5\,{\ln \left (f\right )}^5}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x^2)*x^9,x)

[Out]

(b^5*f^a*log(f)^5*expint(-(b*log(f))/x^2))/240 + (b^5*f^a*f^(b/x^2)*log(f)^5*(x^2/(120*b*log(f)) + x^4/(120*b^
2*log(f)^2) + x^6/(60*b^3*log(f)^3) + x^8/(20*b^4*log(f)^4) + x^10/(5*b^5*log(f)^5)))/2

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