3.131 \(\int f^{a+\frac{b}{x^2}} x^5 \, dx\)

Optimal. Leaf size=81 \[ -\frac{1}{12} b^3 f^a \log ^3(f) \text{Ei}\left (\frac{b \log (f)}{x^2}\right )+\frac{1}{12} b^2 x^2 \log ^2(f) f^{a+\frac{b}{x^2}}+\frac{1}{6} x^6 f^{a+\frac{b}{x^2}}+\frac{1}{12} b x^4 \log (f) f^{a+\frac{b}{x^2}} \]

[Out]

(f^(a + b/x^2)*x^6)/6 + (b*f^(a + b/x^2)*x^4*Log[f])/12 + (b^2*f^(a + b/x^2)*x^2*Log[f]^2)/12 - (b^3*f^a*ExpIn
tegralEi[(b*Log[f])/x^2]*Log[f]^3)/12

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Rubi [A]  time = 0.0892356, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2214, 2210} \[ -\frac{1}{12} b^3 f^a \log ^3(f) \text{Ei}\left (\frac{b \log (f)}{x^2}\right )+\frac{1}{12} b^2 x^2 \log ^2(f) f^{a+\frac{b}{x^2}}+\frac{1}{6} x^6 f^{a+\frac{b}{x^2}}+\frac{1}{12} b x^4 \log (f) f^{a+\frac{b}{x^2}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^2)*x^5,x]

[Out]

(f^(a + b/x^2)*x^6)/6 + (b*f^(a + b/x^2)*x^4*Log[f])/12 + (b^2*f^(a + b/x^2)*x^2*Log[f]^2)/12 - (b^3*f^a*ExpIn
tegralEi[(b*Log[f])/x^2]*Log[f]^3)/12

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^2}} x^5 \, dx &=\frac{1}{6} f^{a+\frac{b}{x^2}} x^6+\frac{1}{3} (b \log (f)) \int f^{a+\frac{b}{x^2}} x^3 \, dx\\ &=\frac{1}{6} f^{a+\frac{b}{x^2}} x^6+\frac{1}{12} b f^{a+\frac{b}{x^2}} x^4 \log (f)+\frac{1}{6} \left (b^2 \log ^2(f)\right ) \int f^{a+\frac{b}{x^2}} x \, dx\\ &=\frac{1}{6} f^{a+\frac{b}{x^2}} x^6+\frac{1}{12} b f^{a+\frac{b}{x^2}} x^4 \log (f)+\frac{1}{12} b^2 f^{a+\frac{b}{x^2}} x^2 \log ^2(f)+\frac{1}{6} \left (b^3 \log ^3(f)\right ) \int \frac{f^{a+\frac{b}{x^2}}}{x} \, dx\\ &=\frac{1}{6} f^{a+\frac{b}{x^2}} x^6+\frac{1}{12} b f^{a+\frac{b}{x^2}} x^4 \log (f)+\frac{1}{12} b^2 f^{a+\frac{b}{x^2}} x^2 \log ^2(f)-\frac{1}{12} b^3 f^a \text{Ei}\left (\frac{b \log (f)}{x^2}\right ) \log ^3(f)\\ \end{align*}

Mathematica [A]  time = 0.0204158, size = 57, normalized size = 0.7 \[ \frac{1}{12} f^a \left (x^2 f^{\frac{b}{x^2}} \left (b^2 \log ^2(f)+b x^2 \log (f)+2 x^4\right )-b^3 \log ^3(f) \text{Ei}\left (\frac{b \log (f)}{x^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^2)*x^5,x]

[Out]

(f^a*(-(b^3*ExpIntegralEi[(b*Log[f])/x^2]*Log[f]^3) + f^(b/x^2)*x^2*(2*x^4 + b*x^2*Log[f] + b^2*Log[f]^2)))/12

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Maple [A]  time = 0.031, size = 79, normalized size = 1. \begin{align*}{\frac{{f}^{a}{x}^{6}}{6}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{{f}^{a}\ln \left ( f \right ) b{x}^{4}}{12}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{x}^{2}}{12}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{12}{\it Ei} \left ( 1,-{\frac{b\ln \left ( f \right ) }{{x}^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^2)*x^5,x)

[Out]

1/6*f^a*x^6*f^(b/x^2)+1/12*f^a*ln(f)*b*x^4*f^(b/x^2)+1/12*f^a*ln(f)^2*b^2*x^2*f^(b/x^2)+1/12*f^a*ln(f)^3*b^3*E
i(1,-b*ln(f)/x^2)

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Maxima [A]  time = 1.27026, size = 30, normalized size = 0.37 \begin{align*} -\frac{1}{2} \, b^{3} f^{a} \Gamma \left (-3, -\frac{b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)*x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.91733, size = 149, normalized size = 1.84 \begin{align*} -\frac{1}{12} \, b^{3} f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right )^{3} + \frac{1}{12} \,{\left (2 \, x^{6} + b x^{4} \log \left (f\right ) + b^{2} x^{2} \log \left (f\right )^{2}\right )} f^{\frac{a x^{2} + b}{x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)*x^5,x, algorithm="fricas")

[Out]

-1/12*b^3*f^a*Ei(b*log(f)/x^2)*log(f)^3 + 1/12*(2*x^6 + b*x^4*log(f) + b^2*x^2*log(f)^2)*f^((a*x^2 + b)/x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x^{2}}} x^{5}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**2)*x**5,x)

[Out]

Integral(f**(a + b/x**2)*x**5, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x^{2}}} x^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)*x^5,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)*x^5, x)