3.43 \(\int \frac{F^{c (a+b x)}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=72 \[ \frac{\sqrt{\pi } F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{\sqrt{b} \sqrt{c} \sqrt{e} \sqrt{\log (F)}} \]

[Out]

(F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]])/(Sqrt[b]*Sqrt[c]*Sqr
t[e]*Sqrt[Log[F]])

________________________________________________________________________________________

Rubi [A]  time = 0.0454783, antiderivative size = 72, 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 = {2180, 2204} \[ \frac{\sqrt{\pi } F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{\sqrt{b} \sqrt{c} \sqrt{e} \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))/Sqrt[d + e*x],x]

[Out]

(F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]])/(Sqrt[b]*Sqrt[c]*Sqr
t[e]*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^{c (a+b x)}}{\sqrt{d+e x}} \, dx &=\frac{2 \operatorname{Subst}\left (\int F^{c \left (a-\frac{b d}{e}\right )+\frac{b c x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{e}\\ &=\frac{F^{c \left (a-\frac{b d}{e}\right )} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{d+e x} \sqrt{\log (F)}}{\sqrt{e}}\right )}{\sqrt{b} \sqrt{c} \sqrt{e} \sqrt{\log (F)}}\\ \end{align*}

Mathematica [A]  time = 0.0279241, size = 63, normalized size = 0.88 \[ -\frac{\sqrt{d+e x} F^{c \left (a-\frac{b d}{e}\right )} \text{Gamma}\left (\frac{1}{2},-\frac{b c \log (F) (d+e x)}{e}\right )}{e \sqrt{-\frac{b c \log (F) (d+e x)}{e}}} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(c*(a + b*x))/Sqrt[d + e*x],x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.022, size = 0, normalized size = 0. \begin{align*} \int{{F}^{c \left ( bx+a \right ) }{\frac{1}{\sqrt{ex+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (b x + a\right )} c}}{\sqrt{e x + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(F^((b*x + a)*c)/sqrt(e*x + d), x)

________________________________________________________________________________________

Fricas [A]  time = 2.04574, size = 142, normalized size = 1.97 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-\frac{b c \log \left (F\right )}{e}} \operatorname{erf}\left (\sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}\right )}{F^{\frac{b c d - a c e}{e}} b c \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(pi)*sqrt(-b*c*log(F)/e)*erf(sqrt(e*x + d)*sqrt(-b*c*log(F)/e))/(F^((b*c*d - a*c*e)/e)*b*c*log(F))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{c \left (a + b x\right )}}{\sqrt{d + e x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(F**(c*(a + b*x))/sqrt(d + e*x), x)

________________________________________________________________________________________

Giac [A]  time = 1.25441, size = 78, normalized size = 1.08 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b c e \log \left (F\right )} \sqrt{x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d \log \left (F\right ) - a c e \log \left (F\right )\right )} e^{\left (-1\right )}\right )}}{\sqrt{-b c e \log \left (F\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(pi)*erf(-sqrt(-b*c*e*log(F))*sqrt(x*e + d)*e^(-1))*e^(-(b*c*d*log(F) - a*c*e*log(F))*e^(-1))/sqrt(-b*c*e
*log(F))