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

Optimal. Leaf size=34 \[ -\frac{f^a \left (-b x^3 \log (f)\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-b x^3 \log (f)\right )}{3 x^2} \]

[Out]

-(f^a*Gamma[-2/3, -(b*x^3*Log[f])]*(-(b*x^3*Log[f]))^(2/3))/(3*x^2)

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Rubi [A]  time = 0.0208568, antiderivative size = 34, 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} \[ -\frac{f^a \left (-b x^3 \log (f)\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-b x^3 \log (f)\right )}{3 x^2} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^3)/x^3,x]

[Out]

-(f^a*Gamma[-2/3, -(b*x^3*Log[f])]*(-(b*x^3*Log[f]))^(2/3))/(3*x^2)

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^3} \, dx &=-\frac{f^a \Gamma \left (-\frac{2}{3},-b x^3 \log (f)\right ) \left (-b x^3 \log (f)\right )^{2/3}}{3 x^2}\\ \end{align*}

Mathematica [A]  time = 0.003847, size = 34, normalized size = 1. \[ -\frac{f^a \left (-b x^3 \log (f)\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-b x^3 \log (f)\right )}{3 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^3)/x^3,x]

[Out]

-(f^a*Gamma[-2/3, -(b*x^3*Log[f])]*(-(b*x^3*Log[f]))^(2/3))/(3*x^2)

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Maple [B]  time = 0.023, size = 102, normalized size = 3. \begin{align*} -{\frac{{f}^{a}b}{3} \left ( \ln \left ( f \right ) \right ) ^{{\frac{2}{3}}} \left ({\frac{bx\pi \,\sqrt{3}}{\Gamma \left ({\frac{2}{3}} \right ) }\sqrt [3]{\ln \left ( f \right ) } \left ( -b \right ) ^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{-b{x}^{3}\ln \left ( f \right ) }}}}-{\frac{3\,{{\rm e}^{b{x}^{3}\ln \left ( f \right ) }}}{2\,{x}^{2}} \left ( -b \right ) ^{-{\frac{2}{3}}} \left ( \ln \left ( f \right ) \right ) ^{-{\frac{2}{3}}}}-{\frac{3\,bx}{2}\sqrt [3]{\ln \left ( f \right ) }\Gamma \left ({\frac{1}{3}},-b{x}^{3}\ln \left ( f \right ) \right ) \left ( -b \right ) ^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{-b{x}^{3}\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt [3]{-b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*f^a*b*ln(f)^(2/3)/(-b)^(1/3)*(x/(-b)^(2/3)*ln(f)^(1/3)*b*Pi*3^(1/2)/GAMMA(2/3)/(-b*x^3*ln(f))^(1/3)-3/2/x
^2/(-b)^(2/3)/ln(f)^(2/3)*exp(b*x^3*ln(f))-3/2*x/(-b)^(2/3)*ln(f)^(1/3)*b/(-b*x^3*ln(f))^(1/3)*GAMMA(1/3,-b*x^
3*ln(f)))

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Maxima [A]  time = 1.21619, size = 38, normalized size = 1.12 \begin{align*} -\frac{\left (-b x^{3} \log \left (f\right )\right )^{\frac{2}{3}} f^{a} \Gamma \left (-\frac{2}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-b*x^3*log(f))^(2/3)*f^a*gamma(-2/3, -b*x^3*log(f))/x^2

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Fricas [A]  time = 1.71017, size = 107, normalized size = 3.15 \begin{align*} \frac{\left (-b \log \left (f\right )\right )^{\frac{2}{3}} f^{a} x^{2} \Gamma \left (\frac{1}{3}, -b x^{3} \log \left (f\right )\right ) - f^{b x^{3} + a}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((-b*log(f))^(2/3)*f^a*x^2*gamma(1/3, -b*x^3*log(f)) - f^(b*x^3 + a))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b*x**3)/x**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(b*x^3 + a)/x^3, x)