3.156 \(\int f^{a+\frac{b}{x^3}} x^{11} \, dx\)

Optimal. Leaf size=24 \[ \frac{1}{3} b^4 f^a \log ^4(f) \text{Gamma}\left (-4,-\frac{b \log (f)}{x^3}\right ) \]

[Out]

(b^4*f^a*Gamma[-4, -((b*Log[f])/x^3)]*Log[f]^4)/3

________________________________________________________________________________________

Rubi [A]  time = 0.0330287, antiderivative size = 24, normalized size of antiderivative = 1., 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} \[ \frac{1}{3} b^4 f^a \log ^4(f) \text{Gamma}\left (-4,-\frac{b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3)*x^11,x]

[Out]

(b^4*f^a*Gamma[-4, -((b*Log[f])/x^3)]*Log[f]^4)/3

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^3}} x^{11} \, dx &=\frac{1}{3} b^4 f^a \Gamma \left (-4,-\frac{b \log (f)}{x^3}\right ) \log ^4(f)\\ \end{align*}

Mathematica [A]  time = 0.0023848, size = 24, normalized size = 1. \[ \frac{1}{3} b^4 f^a \log ^4(f) \text{Gamma}\left (-4,-\frac{b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^3)*x^11,x]

[Out]

(b^4*f^a*Gamma[-4, -((b*Log[f])/x^3)]*Log[f]^4)/3

________________________________________________________________________________________

Maple [B]  time = 0.052, size = 213, normalized size = 8.9 \begin{align*} -{\frac{{f}^{a}{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}{3} \left ( -{\frac{{x}^{12}}{4\,{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}}-{\frac{{x}^{9}}{3\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}}-{\frac{{x}^{6}}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}}-{\frac{{x}^{3}}{6\,b\ln \left ( f \right ) }}-{\frac{25}{288}}-{\frac{\ln \left ( x \right ) }{8}}+{\frac{\ln \left ( -b \right ) }{24}}+{\frac{\ln \left ( \ln \left ( f \right ) \right ) }{24}}+{\frac{{x}^{12}}{1440\,{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}} \left ( 125\,{\frac{{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}{{x}^{12}}}+240\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{x}^{9}}}+360\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{x}^{6}}}+480\,{\frac{b\ln \left ( f \right ) }{{x}^{3}}}+360 \right ) }-{\frac{{x}^{12}}{120\,{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}} \left ( 5\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{x}^{9}}}+5\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{x}^{6}}}+10\,{\frac{b\ln \left ( f \right ) }{{x}^{3}}}+30 \right ){{\rm e}^{{\frac{b\ln \left ( f \right ) }{{x}^{3}}}}}}-{\frac{1}{24}\ln \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) }-{\frac{1}{24}{\it Ei} \left ( 1,-{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^3)*x^11,x)

[Out]

-1/3*f^a*b^4*ln(f)^4*(-1/4*x^12/b^4/ln(f)^4-1/3*x^9/b^3/ln(f)^3-1/4*x^6/b^2/ln(f)^2-1/6*x^3/b/ln(f)-25/288-1/8
*ln(x)+1/24*ln(-b)+1/24*ln(ln(f))+1/1440/b^4/ln(f)^4*x^12*(125*b^4*ln(f)^4/x^12+240*b^3*ln(f)^3/x^9+360*b^2*ln
(f)^2/x^6+480*b*ln(f)/x^3+360)-1/120/b^4/ln(f)^4*x^12*(5*b^3*ln(f)^3/x^9+5*b^2*ln(f)^2/x^6+10*b*ln(f)/x^3+30)*
exp(b*ln(f)/x^3)-1/24*ln(-b*ln(f)/x^3)-1/24*Ei(1,-b*ln(f)/x^3))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)*x^11,x, algorithm="maxima")

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)*x^11,x, algorithm="fricas")

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**3)*x**11,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x^{3}}} x^{11}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)*x^11,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^3)*x^11, x)