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

Optimal. Leaf size=105 \[ \frac{\sqrt{d+e x} F^{c (a+b x)}}{b c \log (F)}-\frac{\sqrt{\pi } \sqrt{e} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{2 b^{3/2} c^{3/2} \log ^{\frac{3}{2}}(F)} \]

[Out]

-(Sqrt[e]*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]])/(2*b^(3/2)*
c^(3/2)*Log[F]^(3/2)) + (F^(c*(a + b*x))*Sqrt[d + e*x])/(b*c*Log[F])

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Rubi [A]  time = 0.0772365, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2176, 2180, 2204} \[ \frac{\sqrt{d+e x} F^{c (a+b x)}}{b c \log (F)}-\frac{\sqrt{\pi } \sqrt{e} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{2 b^{3/2} c^{3/2} \log ^{\frac{3}{2}}(F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))*Sqrt[d + e*x],x]

[Out]

-(Sqrt[e]*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]])/(2*b^(3/2)*
c^(3/2)*Log[F]^(3/2)) + (F^(c*(a + b*x))*Sqrt[d + e*x])/(b*c*Log[F])

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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 F^{c (a+b x)} \sqrt{d+e x} \, dx &=\frac{F^{c (a+b x)} \sqrt{d+e x}}{b c \log (F)}-\frac{e \int \frac{F^{c (a+b x)}}{\sqrt{d+e x}} \, dx}{2 b c \log (F)}\\ &=\frac{F^{c (a+b x)} \sqrt{d+e x}}{b c \log (F)}-\frac{\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 )}{b c \log (F)}\\ &=-\frac{\sqrt{e} 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 )}{2 b^{3/2} c^{3/2} \log ^{\frac{3}{2}}(F)}+\frac{F^{c (a+b x)} \sqrt{d+e x}}{b c \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.0380728, size = 63, normalized size = 0.6 \[ -\frac{(d+e x)^{3/2} F^{c \left (a-\frac{b d}{e}\right )} \text{Gamma}\left (\frac{3}{2},-\frac{b c \log (F) (d+e x)}{e}\right )}{e \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((F^(c*(a - (b*d)/e))*(d + e*x)^(3/2)*Gamma[3/2, -((b*c*(d + e*x)*Log[F])/e)])/(e*(-((b*c*(d + e*x)*Log[F])/e
))^(3/2)))

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Maple [F]  time = 0.019, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) }\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)

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

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Fricas [A]  time = 2.08921, size = 220, normalized size = 2.1 \begin{align*} \frac{2 \, \sqrt{e x + d} F^{b c x + a c} b c \log \left (F\right ) + \frac{\sqrt{\pi } \sqrt{-\frac{b c \log \left (F\right )}{e}} 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}}}}{2 \, b^{2} c^{2} \log \left (F\right )^{2}} \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]

1/2*(2*sqrt(e*x + d)*F^(b*c*x + a*c)*b*c*log(F) + sqrt(pi)*sqrt(-b*c*log(F)/e)*e*erf(sqrt(e*x + d)*sqrt(-b*c*l
og(F)/e))/F^((b*c*d - a*c*e)/e))/(b^2*c^2*log(F)^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int 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)

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Giac [A]  time = 1.30169, size = 170, normalized size = 1.62 \begin{align*} \frac{1}{2} \,{\left (\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 )} + 2\right )}}{\sqrt{-b c e \log \left (F\right )} b c \log \left (F\right )} + \frac{2 \, \sqrt{x e + d} e^{\left ({\left ({\left (x e + d\right )} b c \log \left (F\right ) - b c d \log \left (F\right ) + a c e \log \left (F\right )\right )} e^{\left (-1\right )} + 1\right )}}{b c \log \left (F\right )}\right )} e^{\left (-1\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]

1/2*(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) + 2)/(sq
rt(-b*c*e*log(F))*b*c*log(F)) + 2*sqrt(x*e + d)*e^(((x*e + d)*b*c*log(F) - b*c*d*log(F) + a*c*e*log(F))*e^(-1)
+ 1)/(b*c*log(F)))*e^(-1)