3.65 \(\int (a+b e^x)^2 \sqrt{c+d x} \, dx\)

Optimal. Leaf size=145 \[ \frac{2 a^2 (c+d x)^{3/2}}{3 d}-\sqrt{\pi } a b \sqrt{d} e^{-\frac{c}{d}} \text{Erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )+2 a b e^x \sqrt{c+d x}-\frac{1}{4} \sqrt{\frac{\pi }{2}} b^2 \sqrt{d} e^{-\frac{2 c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{c+d x}}{\sqrt{d}}\right )+\frac{1}{2} b^2 e^{2 x} \sqrt{c+d x} \]

[Out]

2*a*b*E^x*Sqrt[c + d*x] + (b^2*E^(2*x)*Sqrt[c + d*x])/2 + (2*a^2*(c + d*x)^(3/2))/(3*d) - (a*b*Sqrt[d]*Sqrt[Pi
]*Erfi[Sqrt[c + d*x]/Sqrt[d]])/E^(c/d) - (b^2*Sqrt[d]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c + d*x])/Sqrt[d]])/(4*E^(
(2*c)/d))

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Rubi [A]  time = 0.184614, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2183, 2176, 2180, 2204} \[ \frac{2 a^2 (c+d x)^{3/2}}{3 d}-\sqrt{\pi } a b \sqrt{d} e^{-\frac{c}{d}} \text{Erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )+2 a b e^x \sqrt{c+d x}-\frac{1}{4} \sqrt{\frac{\pi }{2}} b^2 \sqrt{d} e^{-\frac{2 c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{c+d x}}{\sqrt{d}}\right )+\frac{1}{2} b^2 e^{2 x} \sqrt{c+d x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*E^x)^2*Sqrt[c + d*x],x]

[Out]

2*a*b*E^x*Sqrt[c + d*x] + (b^2*E^(2*x)*Sqrt[c + d*x])/2 + (2*a^2*(c + d*x)^(3/2))/(3*d) - (a*b*Sqrt[d]*Sqrt[Pi
]*Erfi[Sqrt[c + d*x]/Sqrt[d]])/E^(c/d) - (b^2*Sqrt[d]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c + d*x])/Sqrt[d]])/(4*E^(
(2*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 )^2 \sqrt{c+d x} \, dx &=\int \left (a^2 \sqrt{c+d x}+2 a b e^x \sqrt{c+d x}+b^2 e^{2 x} \sqrt{c+d x}\right ) \, dx\\ &=\frac{2 a^2 (c+d x)^{3/2}}{3 d}+(2 a b) \int e^x \sqrt{c+d x} \, dx+b^2 \int e^{2 x} \sqrt{c+d x} \, dx\\ &=2 a b e^x \sqrt{c+d x}+\frac{1}{2} b^2 e^{2 x} \sqrt{c+d x}+\frac{2 a^2 (c+d x)^{3/2}}{3 d}-(a b d) \int \frac{e^x}{\sqrt{c+d x}} \, dx-\frac{1}{4} \left (b^2 d\right ) \int \frac{e^{2 x}}{\sqrt{c+d x}} \, dx\\ &=2 a b e^x \sqrt{c+d x}+\frac{1}{2} b^2 e^{2 x} \sqrt{c+d x}+\frac{2 a^2 (c+d x)^{3/2}}{3 d}-(2 a b) \operatorname{Subst}\left (\int e^{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x}\right )-\frac{1}{2} b^2 \operatorname{Subst}\left (\int e^{-\frac{2 c}{d}+\frac{2 x^2}{d}} \, dx,x,\sqrt{c+d x}\right )\\ &=2 a b e^x \sqrt{c+d x}+\frac{1}{2} b^2 e^{2 x} \sqrt{c+d x}+\frac{2 a^2 (c+d x)^{3/2}}{3 d}-a b \sqrt{d} e^{-\frac{c}{d}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )-\frac{1}{4} b^2 \sqrt{d} e^{-\frac{2 c}{d}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{c+d x}}{\sqrt{d}}\right )\\ \end{align*}

Mathematica [A]  time = 0.310615, size = 134, normalized size = 0.92 \[ \frac{4 \sqrt{c+d x} \left (4 a^2 (c+d x)+12 a b d e^x+3 b^2 d e^{2 x}\right )-3 \sqrt{2 \pi } b^2 d^{3/2} e^{-\frac{2 c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{c+d x}}{\sqrt{d}}\right )}{24 d}-\sqrt{\pi } a b \sqrt{d} e^{-\frac{c}{d}} \text{Erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*E^x)^2*Sqrt[c + d*x],x]

[Out]

-((a*b*Sqrt[d]*Sqrt[Pi]*Erfi[Sqrt[c + d*x]/Sqrt[d]])/E^(c/d)) + (4*Sqrt[c + d*x]*(12*a*b*d*E^x + 3*b^2*d*E^(2*
x) + 4*a^2*(c + d*x)) - (3*b^2*d^(3/2)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[c + d*x])/Sqrt[d]])/E^((2*c)/d))/(24*d)

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Maple [A]  time = 0.006, size = 144, normalized size = 1. \begin{align*} 2\,{\frac{1}{d} \left ( 1/3\,{a}^{2} \left ( dx+c \right ) ^{3/2}+{{b}^{2} \left ( 1/4\,d\sqrt{dx+c}{{\rm e}^{2\,{\frac{dx+c}{d}}}}-1/8\,{d\sqrt{\pi }{\it Erf} \left ( \sqrt{-2\,{d}^{-1}}\sqrt{dx+c} \right ){\frac{1}{\sqrt{-2\,{d}^{-1}}}}} \right ) \left ({{\rm e}^{{\frac{c}{d}}}} \right ) ^{-2}}+2\,{ab \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))^2*(d*x+c)^(1/2),x)

[Out]

2/d*(1/3*a^2*(d*x+c)^(3/2)+b^2/exp(c/d)^2*(1/4*d*(d*x+c)^(1/2)*exp(2*(d*x+c)/d)-1/8*d*Pi^(1/2)/(-2/d)^(1/2)*er
f((-2/d)^(1/2)*(d*x+c)^(1/2)))+2*a*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 = 1.83941, size = 216, normalized size = 1.49 \begin{align*} \frac{16 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} - 24 \,{\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 )} a b - 3 \,{\left (\frac{\sqrt{2} \sqrt{\pi } d \operatorname{erf}\left (\sqrt{2} \sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{2 \, c}{d}\right )}}{\sqrt{-\frac{1}{d}}} - 4 \, \sqrt{d x + c} d e^{\left (\frac{2 \,{\left (d x + c\right )}}{d} - \frac{2 \, c}{d}\right )}\right )} b^{2}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(16*(d*x + c)^(3/2)*a^2 - 24*(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))*a*b - 3*(sqrt(2)*sqrt(pi)*d*erf(sqrt(2)*sqrt(d*x + c)*sqrt(-1/d))*e^(-2*c/d)/sqrt
(-1/d) - 4*sqrt(d*x + c)*d*e^(2*(d*x + c)/d - 2*c/d))*b^2)/d

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Fricas [A]  time = 1.6394, size = 327, normalized size = 2.26 \begin{align*} \frac{3 \, \sqrt{2} \sqrt{\pi } b^{2} d^{2} \sqrt{-\frac{1}{d}} \operatorname{erf}\left (\sqrt{2} \sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{2 \, c}{d}\right )} + 24 \, \sqrt{\pi } a 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 )} + 4 \,{\left (4 \, a^{2} d x + 3 \, b^{2} d e^{\left (2 \, x\right )} + 12 \, a b d e^{x} + 4 \, a^{2} c\right )} \sqrt{d x + c}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*sqrt(2)*sqrt(pi)*b^2*d^2*sqrt(-1/d)*erf(sqrt(2)*sqrt(d*x + c)*sqrt(-1/d))*e^(-2*c/d) + 24*sqrt(pi)*a*b
*d^2*sqrt(-1/d)*erf(sqrt(d*x + c)*sqrt(-1/d))*e^(-c/d) + 4*(4*a^2*d*x + 3*b^2*d*e^(2*x) + 12*a*b*d*e^x + 4*a^2
*c)*sqrt(d*x + c))/d

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Sympy [A]  time = 2.87097, size = 184, normalized size = 1.27 \begin{align*} \frac{2 a^{2} \left (c + d x\right )^{\frac{3}{2}}}{3 d} + 2 a b \sqrt{d} \sqrt{c + d x} \sqrt{\frac{1}{d}} e^{- \frac{c}{d}} e^{\frac{c}{d} + x} - \sqrt{\pi } a b \sqrt{d} e^{- \frac{c}{d}} \operatorname{erfi}{\left (\frac{\sqrt{c + d x}}{d \sqrt{\frac{1}{d}}} \right )} + \frac{b^{2} \sqrt{d} \sqrt{c + d x} \sqrt{\frac{1}{d}} e^{- \frac{2 c}{d}} e^{\frac{2 c}{d} + 2 x}}{2} - \frac{\sqrt{2} \sqrt{\pi } b^{2} \sqrt{d} e^{- \frac{2 c}{d}} \operatorname{erfi}{\left (\frac{\sqrt{2} \sqrt{c + d x}}{d \sqrt{\frac{1}{d}}} \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*(c + d*x)**(3/2)/(3*d) + 2*a*b*sqrt(d)*sqrt(c + d*x)*sqrt(1/d)*exp(-c/d)*exp(c/d + x) - sqrt(pi)*a*b*sq
rt(d)*exp(-c/d)*erfi(sqrt(c + d*x)/(d*sqrt(1/d))) + b**2*sqrt(d)*sqrt(c + d*x)*sqrt(1/d)*exp(-2*c/d)*exp(2*c/d
+ 2*x)/2 - sqrt(2)*sqrt(pi)*b**2*sqrt(d)*exp(-2*c/d)*erfi(sqrt(2)*sqrt(c + d*x)/(d*sqrt(1/d)))/8

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Giac [A]  time = 1.25575, size = 182, normalized size = 1.26 \begin{align*} \frac{16 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} + 24 \,{\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 )} a b + 3 \,{\left (\frac{\sqrt{2} \sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{d x + c} \sqrt{-d}}{d}\right ) e^{\left (-\frac{2 \, c}{d}\right )}}{\sqrt{-d}} + 4 \, \sqrt{d x + c} d e^{\left (2 \, x\right )}\right )} b^{2}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(16*(d*x + c)^(3/2)*a^2 + 24*(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)*a*b + 3*(sqrt(2)*sqrt(pi)*d^2*erf(-sqrt(2)*sqrt(d*x + c)*sqrt(-d)/d)*e^(-2*c/d)/sqrt(-d) + 4*sqrt(
d*x + c)*d*e^(2*x))*b^2)/d