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

Optimal. Leaf size=46 \[ \frac{1}{3} f^a x^{m+1} \left (-\frac{b \log (f)}{x^3}\right )^{\frac{m+1}{3}} \text{Gamma}\left (\frac{1}{3} (-m-1),-\frac{b \log (f)}{x^3}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0250903, antiderivative size = 46, 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} \[ \frac{1}{3} f^a x^{m+1} \left (-\frac{b \log (f)}{x^3}\right )^{\frac{m+1}{3}} \text{Gamma}\left (\frac{1}{3} (-m-1),-\frac{b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3)*x^m,x]

[Out]

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

Mathematica [A]  time = 0.0102192, size = 46, normalized size = 1. \[ \frac{1}{3} f^a x^{m+1} \left (-\frac{b \log (f)}{x^3}\right )^{\frac{m+1}{3}} \text{Gamma}\left (\frac{1}{3} (-m-1),-\frac{b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^3)*x^m,x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.037, size = 169, normalized size = 3.7 \begin{align*} -{\frac{{f}^{a}}{3} \left ( -b \right ) ^{{\frac{1}{3}}+{\frac{m}{3}}} \left ( \ln \left ( f \right ) \right ) ^{{\frac{1}{3}}+{\frac{m}{3}}} \left ( 3\,{\frac{{x}^{-2+m} \left ( -b \right ) ^{-m/3-1/3} \left ( \ln \left ( f \right ) \right ) ^{2/3-m/3}b\Gamma \left ( 2/3-m/3 \right ) }{1+m} \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) ^{-2/3+m/3}}-3\,{\frac{{x}^{1+m} \left ( -b \right ) ^{-m/3-1/3} \left ( \ln \left ( f \right ) \right ) ^{-m/3-1/3}}{1+m}{{\rm e}^{{\frac{b\ln \left ( f \right ) }{{x}^{3}}}}}}-3\,{\frac{{x}^{-2+m} \left ( -b \right ) ^{-m/3-1/3} \left ( \ln \left ( f \right ) \right ) ^{2/3-m/3}b}{1+m} \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) ^{-2/3+m/3}\Gamma \left ( 2/3-m/3,-{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.3426, size = 51, normalized size = 1.11 \begin{align*} \frac{1}{3} \, f^{a} x^{m + 1} \left (-\frac{b \log \left (f\right )}{x^{3}}\right )^{\frac{1}{3} \, m + \frac{1}{3}} \Gamma \left (-\frac{1}{3} \, m - \frac{1}{3}, -\frac{b \log \left (f\right )}{x^{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (f^{\frac{a x^{3} + b}{x^{3}}} x^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(f^((a*x^3 + b)/x^3)*x^m, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b/x**3)*x**m, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x^3)*x^m, x)