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]])

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Rubi [A]  time = 0.0363298, 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, Rules used = {2211, 2204} \[ \frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (f)} x^{n/2}\right )}{\sqrt{b} n \sqrt{\log (f)}} \]

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]])

Rule 2211

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/(d*(m + 1))
, Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(m + 1
)]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin{align*} \int f^{a+b x^n} x^{-1+\frac{n}{2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int f^{a+b x^2} \, dx,x,x^{n/2}\right )}{n}\\ &=\frac{f^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} x^{n/2} \sqrt{\log (f)}\right )}{\sqrt{b} n \sqrt{\log (f)}}\\ \end{align*}

Mathematica [A]  time = 0.008519, size = 43, normalized size = 1. \[ \frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (f)} x^{n/2}\right )}{\sqrt{b} n \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

Integrate[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]])

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Maple [A]  time = 0.05, size = 32, normalized size = 0.7 \begin{align*}{\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 ) }}}} \end{align*}

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))

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Maxima [A]  time = 1.14225, size = 51, normalized size = 1.19 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]  time = 1.52508, size = 109, normalized size = 2.53 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x x^{\frac{1}{2} \, n - 1}\right )}{b n \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{b x^{n} + a} x^{\frac{1}{2} \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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