3.112 \(\int \frac{f^{a+b x^3}}{x^2} \, dx\)

Optimal. Leaf size=34 \[ -\frac{f^a \sqrt [3]{-b x^3 \log (f)} \text{Gamma}\left (-\frac{1}{3},-b x^3 \log (f)\right )}{3 x} \]

[Out]

-(f^a*Gamma[-1/3, -(b*x^3*Log[f])]*(-(b*x^3*Log[f]))^(1/3))/(3*x)

________________________________________________________________________________________

Rubi [A]  time = 0.0216656, 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} \[ -\frac{f^a \sqrt [3]{-b x^3 \log (f)} \text{Gamma}\left (-\frac{1}{3},-b x^3 \log (f)\right )}{3 x} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^3)/x^2,x]

[Out]

-(f^a*Gamma[-1/3, -(b*x^3*Log[f])]*(-(b*x^3*Log[f]))^(1/3))/(3*x)

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^2} \, dx &=-\frac{f^a \Gamma \left (-\frac{1}{3},-b x^3 \log (f)\right ) \sqrt [3]{-b x^3 \log (f)}}{3 x}\\ \end{align*}

Mathematica [A]  time = 0.0037348, size = 34, normalized size = 1. \[ -\frac{f^a \sqrt [3]{-b x^3 \log (f)} \text{Gamma}\left (-\frac{1}{3},-b x^3 \log (f)\right )}{3 x} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^3)/x^2,x]

[Out]

-(f^a*Gamma[-1/3, -(b*x^3*Log[f])]*(-(b*x^3*Log[f]))^(1/3))/(3*x)

________________________________________________________________________________________

Maple [B]  time = 0.022, size = 100, normalized size = 2.9 \begin{align*}{\frac{{f}^{a}}{3}\sqrt [3]{-b}\sqrt [3]{\ln \left ( f \right ) } \left ( 3\,{\frac{{x}^{2} \left ( \ln \left ( f \right ) \right ) ^{2/3}b\Gamma \left ( 2/3 \right ) }{\sqrt [3]{-b} \left ( -b{x}^{3}\ln \left ( f \right ) \right ) ^{2/3}}}-3\,{\frac{{{\rm e}^{b{x}^{3}\ln \left ( f \right ) }}}{x\sqrt [3]{-b}\sqrt [3]{\ln \left ( f \right ) }}}-3\,{\frac{{x}^{2} \left ( \ln \left ( f \right ) \right ) ^{2/3}b\Gamma \left ( 2/3,-b{x}^{3}\ln \left ( f \right ) \right ) }{\sqrt [3]{-b} \left ( -b{x}^{3}\ln \left ( f \right ) \right ) ^{2/3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^3+a)/x^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.23225, size = 38, normalized size = 1.12 \begin{align*} -\frac{\left (-b x^{3} \log \left (f\right )\right )^{\frac{1}{3}} f^{a} \Gamma \left (-\frac{1}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)/x^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.81136, size = 96, normalized size = 2.82 \begin{align*} \frac{\left (-b \log \left (f\right )\right )^{\frac{1}{3}} f^{a} x \Gamma \left (\frac{2}{3}, -b x^{3} \log \left (f\right )\right ) - f^{b x^{3} + a}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)/x^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + b x^{3}}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**3+a)/x**2,x)

[Out]

Integral(f**(a + b*x**3)/x**2, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{b x^{3} + a}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)/x^2,x, algorithm="giac")

[Out]

integrate(f^(b*x^3 + a)/x^2, x)