3.130 \(\int f^{a+\frac {b}{x^2}} x^7 \, dx\)

Optimal. Leaf size=24 \[ \frac {1}{2} b^4 f^a \log ^4(f) \Gamma \left (-4,-\frac {b \log (f)}{x^2}\right ) \]

[Out]

1/2*f^a*x^8*Ei(5,-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 = {2218} \[ \frac {1}{2} b^4 f^a \log ^4(f) \text {Gamma}\left (-4,-\frac {b \log (f)}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^2)*x^7,x]

[Out]

(b^4*f^a*Gamma[-4, -((b*Log[f])/x^2)]*Log[f]^4)/2

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{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^7 \, dx &=\frac {1}{2} b^4 f^a \Gamma \left (-4,-\frac {b \log (f)}{x^2}\right ) \log ^4(f)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 24, normalized size = 1.00 \[ \frac {1}{2} b^4 f^a \log ^4(f) \Gamma \left (-4,-\frac {b \log (f)}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^2)*x^7,x]

[Out]

(b^4*f^a*Gamma[-4, -((b*Log[f])/x^2)]*Log[f]^4)/2

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fricas [B]  time = 0.42, size = 72, normalized size = 3.00 \[ -\frac {1}{48} \, b^{4} f^{a} {\rm Ei}\left (\frac {b \log \relax (f)}{x^{2}}\right ) \log \relax (f)^{4} + \frac {1}{48} \, {\left (6 \, x^{8} + 2 \, b x^{6} \log \relax (f) + b^{2} x^{4} \log \relax (f)^{2} + b^{3} x^{2} \log \relax (f)^{3}\right )} f^{\frac {a x^{2} + b}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)*x^7,x, algorithm="fricas")

[Out]

-1/48*b^4*f^a*Ei(b*log(f)/x^2)*log(f)^4 + 1/48*(6*x^8 + 2*b*x^6*log(f) + b^2*x^4*log(f)^2 + b^3*x^2*log(f)^3)*
f^((a*x^2 + b)/x^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x^{2}}} x^{7}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)*x^7,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)*x^7, x)

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maple [B]  time = 0.07, size = 101, normalized size = 4.21 \[ \frac {b^{4} f^{a} \Ei \left (1, -\frac {b \ln \relax (f )}{x^{2}}\right ) \ln \relax (f )^{4}}{48}+\frac {b^{3} x^{2} f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )^{3}}{48}+\frac {b^{2} x^{4} f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )^{2}}{48}+\frac {b \,x^{6} f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )}{24}+\frac {x^{8} f^{a} f^{\frac {b}{x^{2}}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^2)*x^7,x)

[Out]

1/8*f^a*x^8*f^(b/x^2)+1/24*f^a*ln(f)*b*x^6*f^(b/x^2)+1/48*f^a*ln(f)^2*b^2*x^4*f^(b/x^2)+1/48*f^a*ln(f)^3*b^3*x
^2*f^(b/x^2)+1/48*f^a*ln(f)^4*b^4*Ei(1,-b/x^2*ln(f))

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maxima [B]  time = 1.35, size = 22, normalized size = 0.92 \[ \frac {1}{2} \, b^{4} f^{a} \Gamma \left (-4, -\frac {b \log \relax (f)}{x^{2}}\right ) \log \relax (f)^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)*x^7,x, algorithm="maxima")

[Out]

1/2*b^4*f^a*gamma(-4, -b*log(f)/x^2)*log(f)^4

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mupad [B]  time = 3.77, size = 90, normalized size = 3.75 \[ \frac {b^4\,f^a\,{\ln \relax (f)}^4\,\mathrm {expint}\left (-\frac {b\,\ln \relax (f)}{x^2}\right )}{48}+\frac {b^4\,f^a\,f^{\frac {b}{x^2}}\,{\ln \relax (f)}^4\,\left (\frac {x^2}{24\,b\,\ln \relax (f)}+\frac {x^4}{24\,b^2\,{\ln \relax (f)}^2}+\frac {x^6}{12\,b^3\,{\ln \relax (f)}^3}+\frac {x^8}{4\,b^4\,{\ln \relax (f)}^4}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x^2)*x^7,x)

[Out]

(b^4*f^a*log(f)^4*expint(-(b*log(f))/x^2))/48 + (b^4*f^a*f^(b/x^2)*log(f)^4*(x^2/(24*b*log(f)) + x^4/(24*b^2*l
og(f)^2) + x^6/(12*b^3*log(f)^3) + x^8/(4*b^4*log(f)^4)))/2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x^{2}}} x^{7}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**2)*x**7,x)

[Out]

Integral(f**(a + b/x**2)*x**7, x)

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