\(\int f^{a+\frac {b}{x}} x^3 \, dx\) [117]

Optimal result

Integrand size = 13, antiderivative size = 21 \[ \int f^{a+\frac {b}{x}} x^3 \, dx=b^4 f^a \Gamma \left (-4,-\frac {b \log (f)}{x}\right ) \log ^4(f) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2250} \[ \int f^{a+\frac {b}{x}} x^3 \, dx=b^4 f^a \log ^4(f) \Gamma \left (-4,-\frac {b \log (f)}{x}\right ) \]

[In]

Int[f^(a + b/x)*x^3,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int f^{a+\frac {b}{x}} x^3 \, dx=b^4 f^a \Gamma \left (-4,-\frac {b \log (f)}{x}\right ) \log ^4(f) \]

[In]

Integrate[f^(a + b/x)*x^3,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(98\) vs. \(2(17)=34\).

Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 4.71

method	result	size
risch	\(\frac {f^{a} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x}\right ) b^{4} \ln \left (f \right )^{4}}{24}+\frac {f^{a} f^{\frac {b}{x}} x \,b^{3} \ln \left (f \right )^{3}}{24}+\frac {f^{a} f^{\frac {b}{x}} x^{2} b^{2} \ln \left (f \right )^{2}}{24}+\frac {f^{a} f^{\frac {b}{x}} x^{3} b \ln \left (f \right )}{12}+\frac {f^{a} f^{\frac {b}{x}} x^{4}}{4}\)	\(99\)
meijerg	\(-f^{a} \ln \left (f \right )^{4} b^{4} \left (-\frac {x^{4}}{4 b^{4} \ln \left (f \right )^{4}}-\frac {x^{3}}{3 b^{3} \ln \left (f \right )^{3}}-\frac {x^{2}}{4 b^{2} \ln \left (f \right )^{2}}-\frac {x}{6 b \ln \left (f \right )}-\frac {25}{288}-\frac {\ln \left (x \right )}{24}+\frac {\ln \left (-b \right )}{24}+\frac {\ln \left (\ln \left (f \right )\right )}{24}+\frac {x^{4} \left (\frac {125 b^{4} \ln \left (f \right )^{4}}{x^{4}}+\frac {240 b^{3} \ln \left (f \right )^{3}}{x^{3}}+\frac {360 b^{2} \ln \left (f \right )^{2}}{x^{2}}+\frac {480 b \ln \left (f \right )}{x}+360\right )}{1440 b^{4} \ln \left (f \right )^{4}}-\frac {x^{4} \left (\frac {5 b^{3} \ln \left (f \right )^{3}}{x^{3}}+\frac {5 b^{2} \ln \left (f \right )^{2}}{x^{2}}+\frac {10 b \ln \left (f \right )}{x}+30\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x}}}{120 b^{4} \ln \left (f \right )^{4}}-\frac {\ln \left (-\frac {b \ln \left (f \right )}{x}\right )}{24}-\frac {\operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x}\right )}{24}\right )\)	\(211\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (17) = 34\).

Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.24 \[ \int f^{a+\frac {b}{x}} x^3 \, dx=-\frac {1}{24} \, b^{4} f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x}\right ) \log \left (f\right )^{4} + \frac {1}{24} \, {\left (b^{3} x \log \left (f\right )^{3} + b^{2} x^{2} \log \left (f\right )^{2} + 2 \, b x^{3} \log \left (f\right ) + 6 \, x^{4}\right )} f^{\frac {a x + b}{x}} \]

[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)

Sympy [F]

\[ \int f^{a+\frac {b}{x}} x^3 \, dx=\int f^{a + \frac {b}{x}} x^{3}\, dx \]

[In]

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

[Out]

Integral(f**(a + b/x)*x**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int f^{a+\frac {b}{x}} x^3 \, dx=b^{4} f^{a} \Gamma \left (-4, -\frac {b \log \left (f\right )}{x}\right ) \log \left (f\right )^{4} \]

[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

Giac [F]

\[ \int f^{a+\frac {b}{x}} x^3 \, dx=\int { f^{a + \frac {b}{x}} x^{3} \,d x } \]

[In]

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

[Out]

integrate(f^(a + b/x)*x^3, x)

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 4.14 \[ \int f^{a+\frac {b}{x}} x^3 \, dx=\frac {b^4\,f^a\,{\ln \left (f\right )}^4\,\mathrm {expint}\left (-\frac {b\,\ln \left (f\right )}{x}\right )}{24}+b^4\,f^a\,f^{b/x}\,{\ln \left (f\right )}^4\,\left (\frac {x^2}{24\,b^2\,{\ln \left (f\right )}^2}+\frac {x^3}{12\,b^3\,{\ln \left (f\right )}^3}+\frac {x^4}{4\,b^4\,{\ln \left (f\right )}^4}+\frac {x}{24\,b\,\ln \left (f\right )}\right ) \]

[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)))