3.106 \(\int \frac{f^{a+b x^3}}{x^{13}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0209979, 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} \[ -\frac{1}{3} b^4 f^a \log ^4(f) \text{Gamma}\left (-4,-b x^3 \log (f)\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^3)/x^13,x]

[Out]

-(b^4*f^a*Gamma[-4, -(b*x^3*Log[f])]*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 \frac{f^{a+b x^3}}{x^{13}} \, dx &=-\frac{1}{3} b^4 f^a \Gamma \left (-4,-b x^3 \log (f)\right ) \log ^4(f)\\ \end{align*}

Mathematica [A]  time = 0.0023959, size = 24, normalized size = 1. \[ -\frac{1}{3} b^4 f^a \log ^4(f) \text{Gamma}\left (-4,-b x^3 \log (f)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^3)/x^13,x]

[Out]

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

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Maple [B]  time = 0.054, size = 213, normalized size = 8.9 \begin{align*}{\frac{{f}^{a}{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}{3} \left ( -{\frac{1}{4\,{b}^{4}{x}^{12} \left ( \ln \left ( f \right ) \right ) ^{4}}}-{\frac{1}{3\,{b}^{3}{x}^{9} \left ( \ln \left ( f \right ) \right ) ^{3}}}-{\frac{1}{4\,{b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}}}-{\frac{1}{6\,b{x}^{3}\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{125\,{b}^{4}{x}^{12} \left ( \ln \left ( f \right ) \right ) ^{4}+240\,{b}^{3}{x}^{9} \left ( \ln \left ( f \right ) \right ) ^{3}+360\,{b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}+480\,b{x}^{3}\ln \left ( f \right ) +360}{1440\,{b}^{4}{x}^{12} \left ( \ln \left ( f \right ) \right ) ^{4}}}-{\frac{ \left ( 5\,{b}^{3}{x}^{9} \left ( \ln \left ( f \right ) \right ) ^{3}+5\,{b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}+10\,b{x}^{3}\ln \left ( f \right ) +30 \right ){{\rm e}^{b{x}^{3}\ln \left ( f \right ) }}}{120\,{b}^{4}{x}^{12} \left ( \ln \left ( f \right ) \right ) ^{4}}}-{\frac{\ln \left ( -b{x}^{3}\ln \left ( f \right ) \right ) }{24}}-{\frac{{\it Ei} \left ( 1,-b{x}^{3}\ln \left ( f \right ) \right ) }{24}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^3+a)/x^13,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/x^12/ln(f)^4*(125*b^4*x^12*ln(f)^4+240*b^3*x^9*ln(f)^3+360*b^2*x^6
*ln(f)^2+480*b*x^3*ln(f)+360)-1/120/b^4/x^12/ln(f)^4*(5*b^3*x^9*ln(f)^3+5*b^2*x^6*ln(f)^2+10*b*x^3*ln(f)+30)*e
xp(b*x^3*ln(f))-1/24*ln(-b*x^3*ln(f))-1/24*Ei(1,-b*x^3*ln(f)))

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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^(b*x^3+a)/x^13,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^(b*x^3+a)/x^13,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**(b*x**3+a)/x**13,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{b x^{3} + a}}{x^{13}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)/x^13,x, algorithm="giac")

[Out]

integrate(f^(b*x^3 + a)/x^13, x)