3.158 \(\int f^{a+\frac{b}{x^3}} x^5 \, dx\)

Optimal. Leaf size=58 \[ -\frac{1}{6} b^2 f^a \log ^2(f) \text{Ei}\left (\frac{b \log (f)}{x^3}\right )+\frac{1}{6} x^6 f^{a+\frac{b}{x^3}}+\frac{1}{6} b x^3 \log (f) f^{a+\frac{b}{x^3}} \]

[Out]

(f^(a + b/x^3)*x^6)/6 + (b*f^(a + b/x^3)*x^3*Log[f])/6 - (b^2*f^a*ExpIntegralEi[(b*Log[f])/x^3]*Log[f]^2)/6

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Rubi [A]  time = 0.0760661, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2214, 2210} \[ -\frac{1}{6} b^2 f^a \log ^2(f) \text{Ei}\left (\frac{b \log (f)}{x^3}\right )+\frac{1}{6} x^6 f^{a+\frac{b}{x^3}}+\frac{1}{6} b x^3 \log (f) f^{a+\frac{b}{x^3}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3)*x^5,x]

[Out]

(f^(a + b/x^3)*x^6)/6 + (b*f^(a + b/x^3)*x^3*Log[f])/6 - (b^2*f^a*ExpIntegralEi[(b*Log[f])/x^3]*Log[f]^2)/6

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int f^{a+\frac{b}{x^3}} x^5 \, dx &=\frac{1}{6} f^{a+\frac{b}{x^3}} x^6+\frac{1}{2} (b \log (f)) \int f^{a+\frac{b}{x^3}} x^2 \, dx\\ &=\frac{1}{6} f^{a+\frac{b}{x^3}} x^6+\frac{1}{6} b f^{a+\frac{b}{x^3}} x^3 \log (f)+\frac{1}{2} \left (b^2 \log ^2(f)\right ) \int \frac{f^{a+\frac{b}{x^3}}}{x} \, dx\\ &=\frac{1}{6} f^{a+\frac{b}{x^3}} x^6+\frac{1}{6} b f^{a+\frac{b}{x^3}} x^3 \log (f)-\frac{1}{6} b^2 f^a \text{Ei}\left (\frac{b \log (f)}{x^3}\right ) \log ^2(f)\\ \end{align*}

Mathematica [A]  time = 0.0147116, size = 44, normalized size = 0.76 \[ \frac{1}{6} f^a \left (x^3 f^{\frac{b}{x^3}} \left (b \log (f)+x^3\right )-b^2 \log ^2(f) \text{Ei}\left (\frac{b \log (f)}{x^3}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^3)*x^5,x]

[Out]

(f^a*(-(b^2*ExpIntegralEi[(b*Log[f])/x^3]*Log[f]^2) + f^(b/x^3)*x^3*(x^3 + b*Log[f])))/6

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Maple [B]  time = 0.04, size = 141, normalized size = 2.4 \begin{align*} -{\frac{{f}^{a}{b}^{2} \left ( \ln \left ( f \right ) \right ) ^{2}}{3} \left ( -{\frac{{x}^{6}}{2\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}}-{\frac{{x}^{3}}{b\ln \left ( f \right ) }}-{\frac{3}{4}}-{\frac{3\,\ln \left ( x \right ) }{2}}+{\frac{\ln \left ( -b \right ) }{2}}+{\frac{\ln \left ( \ln \left ( f \right ) \right ) }{2}}+{\frac{{x}^{6}}{12\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}} \left ( 9\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{x}^{6}}}+12\,{\frac{b\ln \left ( f \right ) }{{x}^{3}}}+6 \right ) }-{\frac{{x}^{6}}{6\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}} \left ( 3+3\,{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ){{\rm e}^{{\frac{b\ln \left ( f \right ) }{{x}^{3}}}}}}-{\frac{1}{2}\ln \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) }-{\frac{1}{2}{\it Ei} \left ( 1,-{\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^5,x)

[Out]

-1/3*f^a*b^2*ln(f)^2*(-1/2*x^6/b^2/ln(f)^2-x^3/b/ln(f)-3/4-3/2*ln(x)+1/2*ln(-b)+1/2*ln(ln(f))+1/12/b^2/ln(f)^2
*x^6*(9*b^2*ln(f)^2/x^6+12*b*ln(f)/x^3+6)-1/6/b^2/ln(f)^2*x^6*(3+3*b*ln(f)/x^3)*exp(b*ln(f)/x^3)-1/2*ln(-b*ln(
f)/x^3)-1/2*Ei(1,-b*ln(f)/x^3))

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Maxima [A]  time = 1.18913, size = 30, normalized size = 0.52 \begin{align*} \frac{1}{3} \, b^{2} f^{a} \Gamma \left (-2, -\frac{b \log \left (f\right )}{x^{3}}\right ) \log \left (f\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)*x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.77484, size = 117, normalized size = 2.02 \begin{align*} -\frac{1}{6} \, b^{2} f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x^{3}}\right ) \log \left (f\right )^{2} + \frac{1}{6} \,{\left (x^{6} + b x^{3} \log \left (f\right )\right )} f^{\frac{a x^{3} + b}{x^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)*x^5,x, algorithm="fricas")

[Out]

-1/6*b^2*f^a*Ei(b*log(f)/x^3)*log(f)^2 + 1/6*(x^6 + b*x^3*log(f))*f^((a*x^3 + b)/x^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x^{3}}} x^{5}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**3)*x**5,x)

[Out]

Integral(f**(a + b/x**3)*x**5, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x^{3}}} x^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)*x^5,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^3)*x^5, x)