3.144 \(\int f^{a+\frac{b}{x^2}} x^4 \, dx\)

Optimal. Leaf size=96 \[ -\frac{4}{15} \sqrt{\pi } b^{5/2} f^a \log ^{\frac{5}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )+\frac{4}{15} b^2 x \log ^2(f) f^{a+\frac{b}{x^2}}+\frac{1}{5} x^5 f^{a+\frac{b}{x^2}}+\frac{2}{15} b x^3 \log (f) f^{a+\frac{b}{x^2}} \]

[Out]

(f^(a + b/x^2)*x^5)/5 + (2*b*f^(a + b/x^2)*x^3*Log[f])/15 + (4*b^2*f^(a + b/x^2)
*x*Log[f]^2)/15 - (4*b^(5/2)*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x]*Log[f]^
(5/2))/15

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Rubi [A]  time = 0.146236, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{4}{15} \sqrt{\pi } b^{5/2} f^a \log ^{\frac{5}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )+\frac{4}{15} b^2 x \log ^2(f) f^{a+\frac{b}{x^2}}+\frac{1}{5} x^5 f^{a+\frac{b}{x^2}}+\frac{2}{15} b x^3 \log (f) f^{a+\frac{b}{x^2}} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x^2)*x^4,x]

[Out]

(f^(a + b/x^2)*x^5)/5 + (2*b*f^(a + b/x^2)*x^3*Log[f])/15 + (4*b^2*f^(a + b/x^2)
*x*Log[f]^2)/15 - (4*b^(5/2)*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x]*Log[f]^
(5/2))/15

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Rubi in Sympy [A]  time = 12.9026, size = 94, normalized size = 0.98 \[ - \frac{4 \sqrt{\pi } b^{\frac{5}{2}} f^{a} \log{\left (f \right )}^{\frac{5}{2}} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{\log{\left (f \right )}}}{x} \right )}}{15} + \frac{4 b^{2} f^{a + \frac{b}{x^{2}}} x \log{\left (f \right )}^{2}}{15} + \frac{2 b f^{a + \frac{b}{x^{2}}} x^{3} \log{\left (f \right )}}{15} + \frac{f^{a + \frac{b}{x^{2}}} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b/x**2)*x**4,x)

[Out]

-4*sqrt(pi)*b**(5/2)*f**a*log(f)**(5/2)*erfi(sqrt(b)*sqrt(log(f))/x)/15 + 4*b**2
*f**(a + b/x**2)*x*log(f)**2/15 + 2*b*f**(a + b/x**2)*x**3*log(f)/15 + f**(a + b
/x**2)*x**5/5

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Mathematica [A]  time = 0.0443343, size = 74, normalized size = 0.77 \[ \frac{1}{15} f^a \left (x f^{\frac{b}{x^2}} \left (4 b^2 \log ^2(f)+2 b x^2 \log (f)+3 x^4\right )-4 \sqrt{\pi } b^{5/2} \log ^{\frac{5}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x^2)*x^4,x]

[Out]

(f^a*(-4*b^(5/2)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x]*Log[f]^(5/2) + f^(b/x^2
)*x*(3*x^4 + 2*b*x^2*Log[f] + 4*b^2*Log[f]^2)))/15

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Maple [A]  time = 0.032, size = 89, normalized size = 0.9 \[{\frac{{f}^{a}{x}^{5}}{5}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{2\,{f}^{a}\ln \left ( f \right ) b{x}^{3}}{15}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{4\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}x}{15}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{4\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}\sqrt{\pi }}{15}{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x^2)*x^4,x)

[Out]

1/5*f^a*f^(b/x^2)*x^5+2/15*f^a*ln(f)*b*f^(b/x^2)*x^3+4/15*f^a*ln(f)^2*b^2*f^(b/x
^2)*x-4/15*f^a*ln(f)^3*b^3*Pi^(1/2)/(-b*ln(f))^(1/2)*erf((-b*ln(f))^(1/2)/x)

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Maxima [A]  time = 0.927673, size = 113, normalized size = 1.18 \[ -\frac{4 \, \sqrt{\pi } b^{3} f^{a}{\left (\operatorname{erf}\left (\sqrt{-\frac{b \log \left (f\right )}{x^{2}}}\right ) - 1\right )} \log \left (f\right )^{3}}{15 \, x \sqrt{-\frac{b \log \left (f\right )}{x^{2}}}} + \frac{1}{15} \,{\left (3 \, f^{a} x^{5} + 2 \, b f^{a} x^{3} \log \left (f\right ) + 4 \, b^{2} f^{a} x \log \left (f\right )^{2}\right )} f^{\frac{b}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)*x^4,x, algorithm="maxima")

[Out]

-4/15*sqrt(pi)*b^3*f^a*(erf(sqrt(-b*log(f)/x^2)) - 1)*log(f)^3/(x*sqrt(-b*log(f)
/x^2)) + 1/15*(3*f^a*x^5 + 2*b*f^a*x^3*log(f) + 4*b^2*f^a*x*log(f)^2)*f^(b/x^2)

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Fricas [A]  time = 0.319589, size = 112, normalized size = 1.17 \[ -\frac{4 \, \sqrt{\pi } b^{3} f^{a} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) \log \left (f\right )^{3} -{\left (3 \, x^{5} + 2 \, b x^{3} \log \left (f\right ) + 4 \, b^{2} x \log \left (f\right )^{2}\right )} \sqrt{-b \log \left (f\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{15 \, \sqrt{-b \log \left (f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)*x^4,x, algorithm="fricas")

[Out]

-1/15*(4*sqrt(pi)*b^3*f^a*erf(sqrt(-b*log(f))/x)*log(f)^3 - (3*x^5 + 2*b*x^3*log
(f) + 4*b^2*x*log(f)^2)*sqrt(-b*log(f))*f^((a*x^2 + b)/x^2))/sqrt(-b*log(f))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{a + \frac{b}{x^{2}}} x^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b/x**2)*x**4,x)

[Out]

Integral(f**(a + b/x**2)*x**4, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{a + \frac{b}{x^{2}}} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)*x^4,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)*x^4, x)