3.155 \(\int f^{a+\frac{b}{x^3}} x^{14} \, dx\)

Optimal. Leaf size=24 \[ -\frac{1}{3} b^5 f^a \log ^5(f) \text{Gamma}\left (-5,-\frac{b \log (f)}{x^3}\right ) \]

[Out]

-(b^5*f^a*Gamma[-5, -((b*Log[f])/x^3)]*Log[f]^5)/3

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Rubi [A]  time = 0.0279187, 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} \[ -\frac{1}{3} b^5 f^a \log ^5(f) \text{Gamma}\left (-5,-\frac{b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3)*x^14,x]

[Out]

-(b^5*f^a*Gamma[-5, -((b*Log[f])/x^3)]*Log[f]^5)/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^{14} \, dx &=-\frac{1}{3} b^5 f^a \Gamma \left (-5,-\frac{b \log (f)}{x^3}\right ) \log ^5(f)\\ \end{align*}

Mathematica [A]  time = 0.0023997, size = 24, normalized size = 1. \[ -\frac{1}{3} b^5 f^a \log ^5(f) \text{Gamma}\left (-5,-\frac{b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^3)*x^14,x]

[Out]

-(b^5*f^a*Gamma[-5, -((b*Log[f])/x^3)]*Log[f]^5)/3

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Maple [B]  time = 0.056, size = 249, normalized size = 10.4 \begin{align*}{\frac{{f}^{a}{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}{3} \left ({\frac{{x}^{15}}{5\,{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}}+{\frac{{x}^{12}}{4\,{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}}+{\frac{{x}^{9}}{6\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}}+{\frac{{x}^{6}}{12\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}}+{\frac{{x}^{3}}{24\,b\ln \left ( f \right ) }}+{\frac{137}{7200}}+{\frac{\ln \left ( x \right ) }{40}}-{\frac{\ln \left ( -b \right ) }{120}}-{\frac{\ln \left ( \ln \left ( f \right ) \right ) }{120}}-{\frac{{x}^{15}}{7200\,{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}} \left ( 137\,{\frac{{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}{{x}^{15}}}+300\,{\frac{{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}{{x}^{12}}}+600\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{x}^{9}}}+1200\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{x}^{6}}}+1800\,{\frac{b\ln \left ( f \right ) }{{x}^{3}}}+1440 \right ) }+{\frac{{x}^{15}}{720\,{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}} \left ( 6\,{\frac{{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}{{x}^{12}}}+6\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{x}^{9}}}+12\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{x}^{6}}}+36\,{\frac{b\ln \left ( f \right ) }{{x}^{3}}}+144 \right ){{\rm e}^{{\frac{b\ln \left ( f \right ) }{{x}^{3}}}}}}+{\frac{1}{120}\ln \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) }+{\frac{1}{120}{\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^14,x)

[Out]

1/3*f^a*b^5*ln(f)^5*(1/5*x^15/b^5/ln(f)^5+1/4*x^12/b^4/ln(f)^4+1/6*x^9/b^3/ln(f)^3+1/12*x^6/b^2/ln(f)^2+1/24*x
^3/b/ln(f)+137/7200+1/40*ln(x)-1/120*ln(-b)-1/120*ln(ln(f))-1/7200/b^5/ln(f)^5*x^15*(137*b^5*ln(f)^5/x^15+300*
b^4*ln(f)^4/x^12+600*b^3*ln(f)^3/x^9+1200*b^2*ln(f)^2/x^6+1800*b*ln(f)/x^3+1440)+1/720/b^5/ln(f)^5*x^15*(6*b^4
*ln(f)^4/x^12+6*b^3*ln(f)^3/x^9+12*b^2*ln(f)^2/x^6+36*b*ln(f)/x^3+144)*exp(b*ln(f)/x^3)+1/120*ln(-b*ln(f)/x^3)
+1/120*Ei(1,-b*ln(f)/x^3))

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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^14,x, algorithm="maxima")

[Out]

Exception raised: TypeError

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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^14,x, algorithm="fricas")

[Out]

Exception raised: TypeError

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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**14,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x^{3}}} x^{14}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)*x^14,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^3)*x^14, x)