3.117 \(\int f^{a+\frac {b}{x}} x^3 \, dx\)

Optimal. Leaf size=21 \[ b^4 f^a \log ^4(f) \Gamma \left (-4,-\frac {b \log (f)}{x}\right ) \]

[Out]

f^a*x^4*Ei(5,-b*ln(f)/x)

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Rubi [A]  time = 0.02, antiderivative size = 21, 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} \[ b^4 f^a \log ^4(f) \text {Gamma}\left (-4,-\frac {b \log (f)}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x)*x^3,x]

[Out]

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

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}} x^3 \, dx &=b^4 f^a \Gamma \left (-4,-\frac {b \log (f)}{x}\right ) \log ^4(f)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \[ b^4 f^a \log ^4(f) \Gamma \left (-4,-\frac {b \log (f)}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x)*x^3,x]

[Out]

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

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fricas [B]  time = 0.41, size = 68, normalized size = 3.24 \[ -\frac {1}{24} \, b^{4} f^{a} {\rm Ei}\left (\frac {b \log \relax (f)}{x}\right ) \log \relax (f)^{4} + \frac {1}{24} \, {\left (b^{3} x \log \relax (f)^{3} + b^{2} x^{2} \log \relax (f)^{2} + 2 \, b x^{3} \log \relax (f) + 6 \, x^{4}\right )} f^{\frac {a x + b}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x)*x^3,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x}} x^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x)*x^3,x, algorithm="giac")

[Out]

integrate(f^(a + b/x)*x^3, x)

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maple [B]  time = 0.09, size = 99, normalized size = 4.71 \[ \frac {b^{4} f^{a} \Ei \left (1, -\frac {b \ln \relax (f )}{x}\right ) \ln \relax (f )^{4}}{24}+\frac {b^{3} x \,f^{a} f^{\frac {b}{x}} \ln \relax (f )^{3}}{24}+\frac {b^{2} x^{2} f^{a} f^{\frac {b}{x}} \ln \relax (f )^{2}}{24}+\frac {b \,x^{3} f^{a} f^{\frac {b}{x}} \ln \relax (f )}{12}+\frac {x^{4} f^{a} f^{\frac {b}{x}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x)*x^3,x)

[Out]

1/4*f^a*f^(b/x)*x^4+1/12*b*ln(f)*f^a*f^(b/x)*x^3+1/24*b^2*ln(f)^2*f^a*f^(b/x)*x^2+1/24*b^3*ln(f)^3*f^a*f^(b/x)
*x+1/24*b^4*ln(f)^4*f^a*Ei(1,-b/x*ln(f))

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maxima [B]  time = 1.63, size = 21, normalized size = 1.00 \[ b^{4} f^{a} \Gamma \left (-4, -\frac {b \log \relax (f)}{x}\right ) \log \relax (f)^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x)*x^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.63, size = 87, normalized size = 4.14 \[ \frac {b^4\,f^a\,{\ln \relax (f)}^4\,\mathrm {expint}\left (-\frac {b\,\ln \relax (f)}{x}\right )}{24}+b^4\,f^a\,f^{b/x}\,{\ln \relax (f)}^4\,\left (\frac {x^2}{24\,b^2\,{\ln \relax (f)}^2}+\frac {x^3}{12\,b^3\,{\ln \relax (f)}^3}+\frac {x^4}{4\,b^4\,{\ln \relax (f)}^4}+\frac {x}{24\,b\,\ln \relax (f)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x)*x^3,x)

[Out]

(b^4*f^a*log(f)^4*expint(-(b*log(f))/x))/24 + b^4*f^a*f^(b/x)*log(f)^4*(x^2/(24*b^2*log(f)^2) + x^3/(12*b^3*lo
g(f)^3) + x^4/(4*b^4*log(f)^4) + x/(24*b*log(f)))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x}} x^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x)*x**3,x)

[Out]

Integral(f**(a + b/x)*x**3, x)

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