3.66 \(\int (a+b e^x)^3 \sqrt{c+d x} \, dx\)

Optimal. Leaf size=224 \[ -\frac{3}{2} \sqrt{\pi } a^2 b \sqrt{d} e^{-\frac{c}{d}} \text{Erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )+3 a^2 b e^x \sqrt{c+d x}+\frac{2 a^3 (c+d x)^{3/2}}{3 d}-\frac{3}{4} \sqrt{\frac{\pi }{2}} a b^2 \sqrt{d} e^{-\frac{2 c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{c+d x}}{\sqrt{d}}\right )+\frac{3}{2} a b^2 e^{2 x} \sqrt{c+d x}-\frac{1}{6} \sqrt{\frac{\pi }{3}} b^3 \sqrt{d} e^{-\frac{3 c}{d}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{c+d x}}{\sqrt{d}}\right )+\frac{1}{3} b^3 e^{3 x} \sqrt{c+d x} \]

[Out]

3*a^2*b*E^x*Sqrt[c + d*x] + (3*a*b^2*E^(2*x)*Sqrt[c + d*x])/2 + (b^3*E^(3*x)*Sqrt[c + d*x])/3 + (2*a^3*(c + d*
x)^(3/2))/(3*d) - (3*a^2*b*Sqrt[d]*Sqrt[Pi]*Erfi[Sqrt[c + d*x]/Sqrt[d]])/(2*E^(c/d)) - (3*a*b^2*Sqrt[d]*Sqrt[P
i/2]*Erfi[(Sqrt[2]*Sqrt[c + d*x])/Sqrt[d]])/(4*E^((2*c)/d)) - (b^3*Sqrt[d]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[c + d
*x])/Sqrt[d]])/(6*E^((3*c)/d))

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Rubi [A]  time = 0.263767, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2183, 2176, 2180, 2204} \[ -\frac{3}{2} \sqrt{\pi } a^2 b \sqrt{d} e^{-\frac{c}{d}} \text{Erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )+3 a^2 b e^x \sqrt{c+d x}+\frac{2 a^3 (c+d x)^{3/2}}{3 d}-\frac{3}{4} \sqrt{\frac{\pi }{2}} a b^2 \sqrt{d} e^{-\frac{2 c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{c+d x}}{\sqrt{d}}\right )+\frac{3}{2} a b^2 e^{2 x} \sqrt{c+d x}-\frac{1}{6} \sqrt{\frac{\pi }{3}} b^3 \sqrt{d} e^{-\frac{3 c}{d}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{c+d x}}{\sqrt{d}}\right )+\frac{1}{3} b^3 e^{3 x} \sqrt{c+d x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*E^x)^3*Sqrt[c + d*x],x]

[Out]

3*a^2*b*E^x*Sqrt[c + d*x] + (3*a*b^2*E^(2*x)*Sqrt[c + d*x])/2 + (b^3*E^(3*x)*Sqrt[c + d*x])/3 + (2*a^3*(c + d*
x)^(3/2))/(3*d) - (3*a^2*b*Sqrt[d]*Sqrt[Pi]*Erfi[Sqrt[c + d*x]/Sqrt[d]])/(2*E^(c/d)) - (3*a*b^2*Sqrt[d]*Sqrt[P
i/2]*Erfi[(Sqrt[2]*Sqrt[c + d*x])/Sqrt[d]])/(4*E^((2*c)/d)) - (b^3*Sqrt[d]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[c + d
*x])/Sqrt[d]])/(6*E^((3*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 )^3 \sqrt{c+d x} \, dx &=\int \left (a^3 \sqrt{c+d x}+3 a^2 b e^x \sqrt{c+d x}+3 a b^2 e^{2 x} \sqrt{c+d x}+b^3 e^{3 x} \sqrt{c+d x}\right ) \, dx\\ &=\frac{2 a^3 (c+d x)^{3/2}}{3 d}+\left (3 a^2 b\right ) \int e^x \sqrt{c+d x} \, dx+\left (3 a b^2\right ) \int e^{2 x} \sqrt{c+d x} \, dx+b^3 \int e^{3 x} \sqrt{c+d x} \, dx\\ &=3 a^2 b e^x \sqrt{c+d x}+\frac{3}{2} a b^2 e^{2 x} \sqrt{c+d x}+\frac{1}{3} b^3 e^{3 x} \sqrt{c+d x}+\frac{2 a^3 (c+d x)^{3/2}}{3 d}-\frac{1}{2} \left (3 a^2 b d\right ) \int \frac{e^x}{\sqrt{c+d x}} \, dx-\frac{1}{4} \left (3 a b^2 d\right ) \int \frac{e^{2 x}}{\sqrt{c+d x}} \, dx-\frac{1}{6} \left (b^3 d\right ) \int \frac{e^{3 x}}{\sqrt{c+d x}} \, dx\\ &=3 a^2 b e^x \sqrt{c+d x}+\frac{3}{2} a b^2 e^{2 x} \sqrt{c+d x}+\frac{1}{3} b^3 e^{3 x} \sqrt{c+d x}+\frac{2 a^3 (c+d x)^{3/2}}{3 d}-\left (3 a^2 b\right ) \operatorname{Subst}\left (\int e^{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x}\right )-\frac{1}{2} \left (3 a b^2\right ) \operatorname{Subst}\left (\int e^{-\frac{2 c}{d}+\frac{2 x^2}{d}} \, dx,x,\sqrt{c+d x}\right )-\frac{1}{3} b^3 \operatorname{Subst}\left (\int e^{-\frac{3 c}{d}+\frac{3 x^2}{d}} \, dx,x,\sqrt{c+d x}\right )\\ &=3 a^2 b e^x \sqrt{c+d x}+\frac{3}{2} a b^2 e^{2 x} \sqrt{c+d x}+\frac{1}{3} b^3 e^{3 x} \sqrt{c+d x}+\frac{2 a^3 (c+d x)^{3/2}}{3 d}-\frac{3}{2} a^2 b \sqrt{d} e^{-\frac{c}{d}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )-\frac{3}{4} a 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 )-\frac{1}{6} b^3 \sqrt{d} e^{-\frac{3 c}{d}} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\frac{\sqrt{3} \sqrt{c+d x}}{\sqrt{d}}\right )\\ \end{align*}

Mathematica [A]  time = 0.669476, size = 196, normalized size = 0.88 \[ -\frac{-12 \sqrt{c+d x} \left (18 a^2 b d e^x+4 a^3 (c+d x)+9 a b^2 d e^{2 x}+2 b^3 d e^{3 x}\right )+108 \sqrt{\pi } a^2 b d^{3/2} e^{-\frac{c}{d}} \text{Erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )+27 \sqrt{2 \pi } a b^2 d^{3/2} e^{-\frac{2 c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{c+d x}}{\sqrt{d}}\right )+4 \sqrt{3 \pi } b^3 d^{3/2} e^{-\frac{3 c}{d}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{c+d x}}{\sqrt{d}}\right )}{72 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*E^x)^3*Sqrt[c + d*x],x]

[Out]

-(-12*Sqrt[c + d*x]*(18*a^2*b*d*E^x + 9*a*b^2*d*E^(2*x) + 2*b^3*d*E^(3*x) + 4*a^3*(c + d*x)) + (108*a^2*b*d^(3
/2)*Sqrt[Pi]*Erfi[Sqrt[c + d*x]/Sqrt[d]])/E^(c/d) + (27*a*b^2*d^(3/2)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[c + d*x])/
Sqrt[d]])/E^((2*c)/d) + (4*b^3*d^(3/2)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[c + d*x])/Sqrt[d]])/E^((3*c)/d))/(72*d)

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Maple [A]  time = 0.005, size = 211, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{d} \left ( 1/3\, \left ( dx+c \right ) ^{3/2}{a}^{3}+{{b}^{3} \left ( 1/6\,d\sqrt{dx+c}{{\rm e}^{3\,{\frac{dx+c}{d}}}}-1/12\,{d\sqrt{\pi }{\it Erf} \left ( \sqrt{-3\,{d}^{-1}}\sqrt{dx+c} \right ){\frac{1}{\sqrt{-3\,{d}^{-1}}}}} \right ) \left ({{\rm e}^{{\frac{c}{d}}}} \right ) ^{-3}}+3\,{a{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}}+3\,{{a}^{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))^3*(d*x+c)^(1/2),x)

[Out]

2/d*(1/3*(d*x+c)^(3/2)*a^3+b^3/exp(c/d)^3*(1/6*d*(d*x+c)^(1/2)*exp(3*(d*x+c)/d)-1/12*d*Pi^(1/2)/(-3/d)^(1/2)*e
rf((-3/d)^(1/2)*(d*x+c)^(1/2)))+3*a*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)*erf((-2/d)^(1/2)*(d*x+c)^(1/2)))+3*a^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 = 1.97191, size = 321, normalized size = 1.43 \begin{align*} \frac{48 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} - 108 \,{\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^{2} b - 27 \,{\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 )} a b^{2} - 4 \,{\left (\frac{\sqrt{3} \sqrt{\pi } d \operatorname{erf}\left (\sqrt{3} \sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{3 \, c}{d}\right )}}{\sqrt{-\frac{1}{d}}} - 6 \, \sqrt{d x + c} d e^{\left (\frac{3 \,{\left (d x + c\right )}}{d} - \frac{3 \, c}{d}\right )}\right )} b^{3}}{72 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(x))^3*(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

1/72*(48*(d*x + c)^(3/2)*a^3 - 108*(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^2*b - 27*(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))*a*b^2 - 4*(sqrt(3)*sqrt(pi)*d*erf(sqrt(3)*sqrt(d*x +
c)*sqrt(-1/d))*e^(-3*c/d)/sqrt(-1/d) - 6*sqrt(d*x + c)*d*e^(3*(d*x + c)/d - 3*c/d))*b^3)/d

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Fricas [A]  time = 1.56272, size = 486, normalized size = 2.17 \begin{align*} \frac{27 \, \sqrt{2} \sqrt{\pi } a 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 )} + 4 \, \sqrt{3} \sqrt{\pi } b^{3} d^{2} \sqrt{-\frac{1}{d}} \operatorname{erf}\left (\sqrt{3} \sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{3 \, c}{d}\right )} + 108 \, \sqrt{\pi } a^{2} 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 )} + 12 \,{\left (4 \, a^{3} d x + 2 \, b^{3} d e^{\left (3 \, x\right )} + 9 \, a b^{2} d e^{\left (2 \, x\right )} + 18 \, a^{2} b d e^{x} + 4 \, a^{3} c\right )} \sqrt{d x + c}}{72 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(x))^3*(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

1/72*(27*sqrt(2)*sqrt(pi)*a*b^2*d^2*sqrt(-1/d)*erf(sqrt(2)*sqrt(d*x + c)*sqrt(-1/d))*e^(-2*c/d) + 4*sqrt(3)*sq
rt(pi)*b^3*d^2*sqrt(-1/d)*erf(sqrt(3)*sqrt(d*x + c)*sqrt(-1/d))*e^(-3*c/d) + 108*sqrt(pi)*a^2*b*d^2*sqrt(-1/d)
*erf(sqrt(d*x + c)*sqrt(-1/d))*e^(-c/d) + 12*(4*a^3*d*x + 2*b^3*d*e^(3*x) + 9*a*b^2*d*e^(2*x) + 18*a^2*b*d*e^x
+ 4*a^3*c)*sqrt(d*x + c))/d

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Sympy [A]  time = 3.91588, size = 291, normalized size = 1.3 \begin{align*} \frac{2 a^{3} \left (c + d x\right )^{\frac{3}{2}}}{3 d} + 3 a^{2} b \sqrt{d} \sqrt{c + d x} \sqrt{\frac{1}{d}} e^{- \frac{c}{d}} e^{\frac{c}{d} + x} - \frac{3 \sqrt{\pi } a^{2} b \sqrt{d} e^{- \frac{c}{d}} \operatorname{erfi}{\left (\frac{\sqrt{c + d x}}{d \sqrt{\frac{1}{d}}} \right )}}{2} + \frac{3 a 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{3 \sqrt{2} \sqrt{\pi } a 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} + \frac{b^{3} \sqrt{d} \sqrt{c + d x} \sqrt{\frac{1}{d}} e^{- \frac{3 c}{d}} e^{\frac{3 c}{d} + 3 x}}{3} - \frac{\sqrt{3} \sqrt{\pi } b^{3} \sqrt{d} e^{- \frac{3 c}{d}} \operatorname{erfi}{\left (\frac{\sqrt{3} \sqrt{c + d x}}{d \sqrt{\frac{1}{d}}} \right )}}{18} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(x))**3*(d*x+c)**(1/2),x)

[Out]

2*a**3*(c + d*x)**(3/2)/(3*d) + 3*a**2*b*sqrt(d)*sqrt(c + d*x)*sqrt(1/d)*exp(-c/d)*exp(c/d + x) - 3*sqrt(pi)*a
**2*b*sqrt(d)*exp(-c/d)*erfi(sqrt(c + d*x)/(d*sqrt(1/d)))/2 + 3*a*b**2*sqrt(d)*sqrt(c + d*x)*sqrt(1/d)*exp(-2*
c/d)*exp(2*c/d + 2*x)/2 - 3*sqrt(2)*sqrt(pi)*a*b**2*sqrt(d)*exp(-2*c/d)*erfi(sqrt(2)*sqrt(c + d*x)/(d*sqrt(1/d
)))/8 + b**3*sqrt(d)*sqrt(c + d*x)*sqrt(1/d)*exp(-3*c/d)*exp(3*c/d + 3*x)/3 - sqrt(3)*sqrt(pi)*b**3*sqrt(d)*ex
p(-3*c/d)*erfi(sqrt(3)*sqrt(c + d*x)/(d*sqrt(1/d)))/18

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Giac [A]  time = 1.23147, size = 271, normalized size = 1.21 \begin{align*} \frac{48 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} + 108 \,{\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^{2} b + 27 \,{\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 )} a b^{2} + 4 \,{\left (\frac{\sqrt{3} \sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{3} \sqrt{d x + c} \sqrt{-d}}{d}\right ) e^{\left (-\frac{3 \, c}{d}\right )}}{\sqrt{-d}} + 6 \, \sqrt{d x + c} d e^{\left (3 \, x\right )}\right )} b^{3}}{72 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(x))^3*(d*x+c)^(1/2),x, algorithm="giac")

[Out]

1/72*(48*(d*x + c)^(3/2)*a^3 + 108*(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^2*b + 27*(sqrt(2)*sqrt(pi)*d^2*erf(-sqrt(2)*sqrt(d*x + c)*sqrt(-d)/d)*e^(-2*c/d)/sqrt(-d) + 4*s
qrt(d*x + c)*d*e^(2*x))*a*b^2 + 4*(sqrt(3)*sqrt(pi)*d^2*erf(-sqrt(3)*sqrt(d*x + c)*sqrt(-d)/d)*e^(-3*c/d)/sqrt
(-d) + 6*sqrt(d*x + c)*d*e^(3*x))*b^3)/d