3.34 \(\int \frac{F^{a+b x}}{\sqrt{x}} \, dx\)

Optimal. Leaf size=38 \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{\sqrt{b} \sqrt{\log (F)}} \]

[Out]

(F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(Sqrt[b]*Sqrt[Log[F]])

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Rubi [A]  time = 0.025179, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2180, 2204} \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{\sqrt{b} \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*x)/Sqrt[x],x]

[Out]

(F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(Sqrt[b]*Sqrt[Log[F]])

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

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

Mathematica [A]  time = 0.0063854, size = 30, normalized size = 0.79 \[ -\frac{\sqrt{x} F^a \text{Gamma}\left (\frac{1}{2},-b x \log (F)\right )}{\sqrt{-b x \log (F)}} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*x)/Sqrt[x],x]

[Out]

-((F^a*Sqrt[x]*Gamma[1/2, -(b*x*Log[F])])/Sqrt[-(b*x*Log[F])])

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Maple [A]  time = 0.008, size = 27, normalized size = 0.7 \begin{align*}{{F}^{a}\sqrt{\pi }{\it erfi} \left ( \sqrt{b}\sqrt{x}\sqrt{\ln \left ( F \right ) } \right ){\frac{1}{\sqrt{b}}}{\frac{1}{\sqrt{\ln \left ( F \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(b*x+a)/x^(1/2),x)

[Out]

F^a*erfi(b^(1/2)*x^(1/2)*ln(F)^(1/2))*Pi^(1/2)/b^(1/2)/ln(F)^(1/2)

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Maxima [A]  time = 1.15337, size = 39, normalized size = 1.03 \begin{align*} \frac{\sqrt{\pi } F^{a} \sqrt{x}{\left (\operatorname{erf}\left (\sqrt{-b x \log \left (F\right )}\right ) - 1\right )}}{\sqrt{-b x \log \left (F\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(b*x+a)/x^(1/2),x, algorithm="maxima")

[Out]

sqrt(pi)*F^a*sqrt(x)*(erf(sqrt(-b*x*log(F))) - 1)/sqrt(-b*x*log(F))

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Fricas [A]  time = 1.5287, size = 96, 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 )} \sqrt{x}\right )}{b \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(pi)*sqrt(-b*log(F))*F^a*erf(sqrt(-b*log(F))*sqrt(x))/(b*log(F))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(b*x+a)/x**(1/2),x)

[Out]

Integral(F**(a + b*x)/sqrt(x), x)

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Giac [A]  time = 1.34626, size = 38, normalized size = 1. \begin{align*} -\frac{\sqrt{\pi } F^{a} \operatorname{erf}\left (-\sqrt{-b \log \left (F\right )} \sqrt{x}\right )}{\sqrt{-b \log \left (F\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(pi)*F^a*erf(-sqrt(-b*log(F))*sqrt(x))/sqrt(-b*log(F))