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

Optimal. Leaf size=71 \[ \frac{2 a (c+d x)^{3/2}}{3 d}-\frac{1}{2} \sqrt{\pi } b \sqrt{d} e^{-\frac{c}{d}} \text{Erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )+b e^x \sqrt{c+d x} \]

[Out]

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

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Rubi [A]  time = 0.0881561, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2183, 2176, 2180, 2204} \[ \frac{2 a (c+d x)^{3/2}}{3 d}-\frac{1}{2} \sqrt{\pi } b \sqrt{d} e^{-\frac{c}{d}} \text{Erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )+b e^x \sqrt{c+d x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*E^x)*Sqrt[c + d*x],x]

[Out]

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

Rule 2183

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> In
t[ExpandIntegrand[(c + d*x)^m, (a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n},
x] && IGtQ[p, 0]

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 \left (a+b e^x\right ) \sqrt{c+d x} \, dx &=\int \left (a \sqrt{c+d x}+b e^x \sqrt{c+d x}\right ) \, dx\\ &=\frac{2 a (c+d x)^{3/2}}{3 d}+b \int e^x \sqrt{c+d x} \, dx\\ &=b e^x \sqrt{c+d x}+\frac{2 a (c+d x)^{3/2}}{3 d}-\frac{1}{2} (b d) \int \frac{e^x}{\sqrt{c+d x}} \, dx\\ &=b e^x \sqrt{c+d x}+\frac{2 a (c+d x)^{3/2}}{3 d}-b \operatorname{Subst}\left (\int e^{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x}\right )\\ &=b e^x \sqrt{c+d x}+\frac{2 a (c+d x)^{3/2}}{3 d}-\frac{1}{2} b \sqrt{d} e^{-\frac{c}{d}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0940573, size = 71, normalized size = 1. \[ \frac{2 a (c+d x)^{3/2}}{3 d}-\frac{1}{2} \sqrt{\pi } b \sqrt{d} e^{-\frac{c}{d}} \text{Erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )+b e^x \sqrt{c+d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 77, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{d} \left ( 1/3\,a \left ( dx+c \right ) ^{3/2}+{b \left ( 1/2\,\sqrt{dx+c}{{\rm e}^{{\frac{dx+c}{d}}}}d-1/4\,{d\sqrt{\pi }{\it Erf} \left ( \sqrt{-{d}^{-1}}\sqrt{dx+c} \right ){\frac{1}{\sqrt{-{d}^{-1}}}}} \right ) \left ({{\rm e}^{{\frac{c}{d}}}} \right ) ^{-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*(1/3*a*(d*x+c)^(3/2)+b/exp(c/d)*(1/2*(d*x+c)^(1/2)*exp((d*x+c)/d)*d-1/4*d*Pi^(1/2)/(-1/d)^(1/2)*erf((-1/d)
^(1/2)*(d*x+c)^(1/2))))

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Maxima [A]  time = 1.19239, size = 111, normalized size = 1.56 \begin{align*} \frac{4 \,{\left (d x + c\right )}^{\frac{3}{2}} a - 3 \,{\left (\frac{\sqrt{\pi } d \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{c}{d}\right )}}{\sqrt{-\frac{1}{d}}} - 2 \, \sqrt{d x + c} d e^{\left (\frac{d x + c}{d} - \frac{c}{d}\right )}\right )} b}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(4*(d*x + c)^(3/2)*a - 3*(sqrt(pi)*d*erf(sqrt(d*x + c)*sqrt(-1/d))*e^(-c/d)/sqrt(-1/d) - 2*sqrt(d*x + c)*d
*e^((d*x + c)/d - c/d))*b)/d

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Fricas [A]  time = 1.636, size = 167, normalized size = 2.35 \begin{align*} \frac{3 \, \sqrt{\pi } b d^{2} \sqrt{-\frac{1}{d}} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{c}{d}\right )} + 2 \,{\left (2 \, a d x + 3 \, b d e^{x} + 2 \, a c\right )} \sqrt{d x + c}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*sqrt(pi)*b*d^2*sqrt(-1/d)*erf(sqrt(d*x + c)*sqrt(-1/d))*e^(-c/d) + 2*(2*a*d*x + 3*b*d*e^x + 2*a*c)*sqrt
(d*x + c))/d

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Sympy [A]  time = 1.89973, size = 85, normalized size = 1.2 \begin{align*} \frac{2 a \left (c + d x\right )^{\frac{3}{2}}}{3 d} + b \sqrt{d} \sqrt{c + d x} \sqrt{\frac{1}{d}} e^{- \frac{c}{d}} e^{\frac{c}{d} + x} - \frac{\sqrt{\pi } b \sqrt{d} e^{- \frac{c}{d}} \operatorname{erfi}{\left (\frac{\sqrt{c + d x}}{d \sqrt{\frac{1}{d}}} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*(c + d*x)**(3/2)/(3*d) + b*sqrt(d)*sqrt(c + d*x)*sqrt(1/d)*exp(-c/d)*exp(c/d + x) - sqrt(pi)*b*sqrt(d)*exp
(-c/d)*erfi(sqrt(c + d*x)/(d*sqrt(1/d)))/2

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Giac [A]  time = 1.3116, size = 93, normalized size = 1.31 \begin{align*} \frac{4 \,{\left (d x + c\right )}^{\frac{3}{2}} a + 3 \,{\left (\frac{\sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{d x + c} \sqrt{-d}}{d}\right ) e^{\left (-\frac{c}{d}\right )}}{\sqrt{-d}} + 2 \, \sqrt{d x + c} d e^{x}\right )} b}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(4*(d*x + c)^(3/2)*a + 3*(sqrt(pi)*d^2*erf(-sqrt(d*x + c)*sqrt(-d)/d)*e^(-c/d)/sqrt(-d) + 2*sqrt(d*x + c)*
d*e^x)*b)/d