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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 = 0.00, size = 34, normalized size = 1.00 \[ -\frac {f^a \sqrt [3]{-b x^3 \log (f)} \Gamma \left (-\frac {1}{3},-b x^3 \log (f)\right )}{3 x} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^3)/x^2,x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 38, normalized size = 1.12 \[ \frac {\left (-b \log \relax (f)\right )^{\frac {1}{3}} f^{a} x \Gamma \left (\frac {2}{3}, -b x^{3} \log \relax (f)\right ) - f^{b x^{3} + a}}{x} \]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{b x^{3} + a}}{x^{2}}\,{d x} \]

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)

________________________________________________________________________________________

maple [B]  time = 0.04, size = 100, normalized size = 2.94 \[ \frac {\left (-b \right )^{\frac {1}{3}} \left (-\frac {3 b \,x^{2} \Gamma \left (\frac {2}{3}, -b \,x^{3} \ln \relax (f )\right ) \ln \relax (f )^{\frac {2}{3}}}{\left (-b \right )^{\frac {1}{3}} \left (-b \,x^{3} \ln \relax (f )\right )^{\frac {2}{3}}}+\frac {3 \Gamma \left (\frac {2}{3}\right ) b \,x^{2} \ln \relax (f )^{\frac {2}{3}}}{\left (-b \right )^{\frac {1}{3}} \left (-b \,x^{3} \ln \relax (f )\right )^{\frac {2}{3}}}-\frac {3 \,{\mathrm e}^{b \,x^{3} \ln \relax (f )}}{\left (-b \right )^{\frac {1}{3}} x \ln \relax (f )^{\frac {1}{3}}}\right ) f^{a} \ln \relax (f )^{\frac {1}{3}}}{3} \]

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 = 1.29, size = 28, normalized size = 0.82 \[ -\frac {\left (-b x^{3} \log \relax (f)\right )^{\frac {1}{3}} f^{a} \Gamma \left (-\frac {1}{3}, -b x^{3} \log \relax (f)\right )}{3 \, x} \]

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

________________________________________________________________________________________

mupad [B]  time = 3.47, size = 63, normalized size = 1.85 \[ \frac {f^a\,\Gamma \left (\frac {2}{3},-b\,x^3\,\ln \relax (f)\right )\,{\left (-b\,x^3\,\ln \relax (f)\right )}^{1/3}}{x}-\frac {f^a\,\Gamma \left (\frac {2}{3}\right )\,{\left (-b\,x^3\,\ln \relax (f)\right )}^{1/3}}{x}-\frac {f^a\,f^{b\,x^3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^3)/x^2,x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + b x^{3}}}{x^{2}}\, dx \]

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)

________________________________________________________________________________________