3.191 \(\int f^{a+b x^n} x^{-1+\frac{n}{2}} \, dx\)

Optimal. Leaf size=43 \[ \frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (f)} x^{n/2}\right )}{\sqrt{b} n \sqrt{\log (f)}} \]

[Out]

(f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x^(n/2)*Sqrt[Log[f]]])/(Sqrt[b]*n*Sqrt[Log[f]])

_______________________________________________________________________________________

Rubi [A]  time = 0.0639646, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (f)} x^{n/2}\right )}{\sqrt{b} n \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^n)*x^(-1 + n/2),x]

[Out]

(f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x^(n/2)*Sqrt[Log[f]]])/(Sqrt[b]*n*Sqrt[Log[f]])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.5116, size = 39, normalized size = 0.91 \[ \frac{\sqrt{\pi } f^{a} \operatorname{erfi}{\left (\sqrt{b} x^{\frac{n}{2}} \sqrt{\log{\left (f \right )}} \right )}}{\sqrt{b} n \sqrt{\log{\left (f \right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b*x**n)*x**(-1+1/2*n),x)

[Out]

sqrt(pi)*f**a*erfi(sqrt(b)*x**(n/2)*sqrt(log(f)))/(sqrt(b)*n*sqrt(log(f)))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0332997, size = 46, normalized size = 1.07 \[ \frac{\sqrt{\pi } f^a x^{n/2} \left (\text{Erf}\left (\sqrt{-b \log (f) x^n}\right )-1\right )}{n \sqrt{-b \log (f) x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^n)*x^(-1 + n/2),x]

[Out]

(f^a*Sqrt[Pi]*x^(n/2)*(-1 + Erf[Sqrt[-(b*x^n*Log[f])]]))/(n*Sqrt[-(b*x^n*Log[f])
])

_______________________________________________________________________________________

Maple [A]  time = 0.068, size = 32, normalized size = 0.7 \[{\frac{{f}^{a}\sqrt{\pi }}{n}{\it Erf} \left ( \sqrt{-b\ln \left ( f \right ) }{x}^{{\frac{n}{2}}} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b*x^n)*x^(-1+1/2*n),x)

[Out]

1/n*f^a*Pi^(1/2)/(-b*ln(f))^(1/2)*erf((-b*ln(f))^(1/2)*x^(1/2*n))

_______________________________________________________________________________________

Maxima [A]  time = 1.04757, size = 51, normalized size = 1.19 \[ \frac{\sqrt{\pi } f^{a} x^{\frac{1}{2} \, n}{\left (\operatorname{erf}\left (\sqrt{-b x^{n} \log \left (f\right )}\right ) - 1\right )}}{\sqrt{-b x^{n} \log \left (f\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^n + a)*x^(1/2*n - 1),x, algorithm="maxima")

[Out]

sqrt(pi)*f^a*x^(1/2*n)*(erf(sqrt(-b*x^n*log(f))) - 1)/(sqrt(-b*x^n*log(f))*n)

_______________________________________________________________________________________

Fricas [A]  time = 0.281294, size = 46, normalized size = 1.07 \[ \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x x^{\frac{1}{2} \, n - 1}\right )}{\sqrt{-b \log \left (f\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^n + a)*x^(1/2*n - 1),x, algorithm="fricas")

[Out]

sqrt(pi)*f^a*erf(sqrt(-b*log(f))*x*x^(1/2*n - 1))/(sqrt(-b*log(f))*n)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b*x**n)*x**(-1+1/2*n),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{b x^{n} + a} x^{\frac{1}{2} \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^n + a)*x^(1/2*n - 1),x, algorithm="giac")

[Out]

integrate(f^(b*x^n + a)*x^(1/2*n - 1), x)