3.107 \(\int \frac{f^{a+b x^3}}{x^{16}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0209877, 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} \[ \frac{1}{3} b^5 f^a \log ^5(f) \text{Gamma}\left (-5,-b x^3 \log (f)\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^3)/x^16,x]

[Out]

(b^5*f^a*Gamma[-5, -(b*x^3*Log[f])]*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 \frac{f^{a+b x^3}}{x^{16}} \, dx &=\frac{1}{3} b^5 f^a \Gamma \left (-5,-b x^3 \log (f)\right ) \log ^5(f)\\ \end{align*}

Mathematica [A]  time = 0.0023457, size = 24, normalized size = 1. \[ \frac{1}{3} b^5 f^a \log ^5(f) \text{Gamma}\left (-5,-b x^3 \log (f)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^3)/x^16,x]

[Out]

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

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Maple [B]  time = 0.06, size = 249, normalized size = 10.4 \begin{align*} -{\frac{{f}^{a}{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}{3} \left ({\frac{1}{5\,{b}^{5}{x}^{15} \left ( \ln \left ( f \right ) \right ) ^{5}}}+{\frac{1}{4\,{b}^{4}{x}^{12} \left ( \ln \left ( f \right ) \right ) ^{4}}}+{\frac{1}{6\,{b}^{3}{x}^{9} \left ( \ln \left ( f \right ) \right ) ^{3}}}+{\frac{1}{12\,{b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}}}+{\frac{1}{24\,b{x}^{3}\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{137\,{b}^{5}{x}^{15} \left ( \ln \left ( f \right ) \right ) ^{5}+300\,{b}^{4}{x}^{12} \left ( \ln \left ( f \right ) \right ) ^{4}+600\,{b}^{3}{x}^{9} \left ( \ln \left ( f \right ) \right ) ^{3}+1200\,{b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}+1800\,b{x}^{3}\ln \left ( f \right ) +1440}{7200\,{b}^{5}{x}^{15} \left ( \ln \left ( f \right ) \right ) ^{5}}}+{\frac{ \left ( 6\,{b}^{4}{x}^{12} \left ( \ln \left ( f \right ) \right ) ^{4}+6\,{b}^{3}{x}^{9} \left ( \ln \left ( f \right ) \right ) ^{3}+12\,{b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}+36\,b{x}^{3}\ln \left ( f \right ) +144 \right ){{\rm e}^{b{x}^{3}\ln \left ( f \right ) }}}{720\,{b}^{5}{x}^{15} \left ( \ln \left ( f \right ) \right ) ^{5}}}+{\frac{\ln \left ( -b{x}^{3}\ln \left ( f \right ) \right ) }{120}}+{\frac{{\it Ei} \left ( 1,-b{x}^{3}\ln \left ( f \right ) \right ) }{120}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^3+a)/x^16,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/x^15/ln(f)^5*(137*b^5*x^15*ln(f)^5+300
*b^4*x^12*ln(f)^4+600*b^3*x^9*ln(f)^3+1200*b^2*x^6*ln(f)^2+1800*b*x^3*ln(f)+1440)+1/720/b^5/x^15/ln(f)^5*(6*b^
4*x^12*ln(f)^4+6*b^3*x^9*ln(f)^3+12*b^2*x^6*ln(f)^2+36*b*x^3*ln(f)+144)*exp(b*x^3*ln(f))+1/120*ln(-b*x^3*ln(f)
)+1/120*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^16,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^16,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**16,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^{16}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)/x^16,x, algorithm="giac")

[Out]

integrate(f^(b*x^3 + a)/x^16, x)