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

Optimal. Leaf size=58 \[ -\frac{1}{4} b^2 f^a \log ^2(f) \text{Ei}\left (\frac{b \log (f)}{x^2}\right )+\frac{1}{4} x^4 f^{a+\frac{b}{x^2}}+\frac{1}{4} b x^2 \log (f) f^{a+\frac{b}{x^2}} \]

[Out]

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

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Rubi [A]  time = 0.062517, 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} \[ -\frac{1}{4} b^2 f^a \log ^2(f) \text{Ei}\left (\frac{b \log (f)}{x^2}\right )+\frac{1}{4} x^4 f^{a+\frac{b}{x^2}}+\frac{1}{4} b x^2 \log (f) f^{a+\frac{b}{x^2}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^2)*x^3,x]

[Out]

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

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

Mathematica [A]  time = 0.0141431, size = 44, normalized size = 0.76 \[ \frac{1}{4} f^a \left (x^2 f^{\frac{b}{x^2}} \left (b \log (f)+x^2\right )-b^2 \log ^2(f) \text{Ei}\left (\frac{b \log (f)}{x^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^2)*x^3,x]

[Out]

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

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Maple [A]  time = 0.026, size = 57, normalized size = 1. \begin{align*}{\frac{{f}^{a}{x}^{4}}{4}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{{f}^{a}\ln \left ( f \right ) b{x}^{2}}{4}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{4}{\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^3,x)

[Out]

1/4*f^a*x^4*f^(b/x^2)+1/4*f^a*ln(f)*b*x^2*f^(b/x^2)+1/4*f^a*ln(f)^2*b^2*Ei(1,-b*ln(f)/x^2)

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Maxima [A]  time = 1.20352, size = 30, normalized size = 0.52 \begin{align*} \frac{1}{2} \, b^{2} f^{a} \Gamma \left (-2, -\frac{b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.77214, size = 117, normalized size = 2.02 \begin{align*} -\frac{1}{4} \, b^{2} f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right )^{2} + \frac{1}{4} \,{\left (x^{4} + b x^{2} \log \left (f\right )\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^3,x, algorithm="fricas")

[Out]

-1/4*b^2*f^a*Ei(b*log(f)/x^2)*log(f)^2 + 1/4*(x^4 + b*x^2*log(f))*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^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b/x**2)*x**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x^2)*x^3, x)