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

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

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

_______________________________________________________________________________________

Rubi [A]  time = 0.449331, 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 \[ \frac{2 a^3 (c+d x)^{3/2}}{3 d}-\frac{3}{2} \sqrt{\pi } a^2 b \sqrt{d} e^{-\frac{c}{d}} \text{Erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )+3 a^2 b e^x \sqrt{c+d x}-\frac{3}{4} \sqrt{\frac{\pi }{2}} a b^2 \sqrt{d} e^{-\frac{2 c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{c+d x}}{\sqrt{d}}\right )+\frac{3}{2} a b^2 e^{2 x} \sqrt{c+d x}-\frac{1}{6} \sqrt{\frac{\pi }{3}} b^3 \sqrt{d} e^{-\frac{3 c}{d}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{c+d x}}{\sqrt{d}}\right )+\frac{1}{3} b^3 e^{3 x} \sqrt{c+d x} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 37.1849, size = 212, normalized size = 0.95 \[ \frac{2 a^{3} \left (c + d x\right )^{\frac{3}{2}}}{3 d} - \frac{3 \sqrt{\pi } a^{2} b \sqrt{d} e^{- \frac{c}{d}} \operatorname{erfi}{\left (\frac{\sqrt{c + d x}}{\sqrt{d}} \right )}}{2} + 3 a^{2} b \sqrt{c + d x} e^{x} - \frac{3 \sqrt{2} \sqrt{\pi } a b^{2} \sqrt{d} e^{- \frac{2 c}{d}} \operatorname{erfi}{\left (\frac{\sqrt{2} \sqrt{c + d x}}{\sqrt{d}} \right )}}{8} + \frac{3 a b^{2} \sqrt{c + d x} e^{2 x}}{2} - \frac{\sqrt{3} \sqrt{\pi } b^{3} \sqrt{d} e^{- \frac{3 c}{d}} \operatorname{erfi}{\left (\frac{\sqrt{3} \sqrt{c + d x}}{\sqrt{d}} \right )}}{18} + \frac{b^{3} \sqrt{c + d x} e^{3 x}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*exp(x))**3*(d*x+c)**(1/2),x)

[Out]

2*a**3*(c + d*x)**(3/2)/(3*d) - 3*sqrt(pi)*a**2*b*sqrt(d)*exp(-c/d)*erfi(sqrt(c
+ d*x)/sqrt(d))/2 + 3*a**2*b*sqrt(c + d*x)*exp(x) - 3*sqrt(2)*sqrt(pi)*a*b**2*sq
rt(d)*exp(-2*c/d)*erfi(sqrt(2)*sqrt(c + d*x)/sqrt(d))/8 + 3*a*b**2*sqrt(c + d*x)
*exp(2*x)/2 - sqrt(3)*sqrt(pi)*b**3*sqrt(d)*exp(-3*c/d)*erfi(sqrt(3)*sqrt(c + d*
x)/sqrt(d))/18 + b**3*sqrt(c + d*x)*exp(3*x)/3

_______________________________________________________________________________________

Mathematica [A]  time = 1.08626, size = 285, normalized size = 1.27 \[ -\frac{\sqrt{-\frac{c+d x}{d}} \left (-24 a^3 d \left (-\frac{c+d x}{d}\right )^{3/2}+54 a^2 b d e^{-\frac{c}{d}} \left (-\sqrt{\pi } \text{Erf}\left (\sqrt{-\frac{c+d x}{d}}\right )+2 e^{\frac{c}{d}+x} \sqrt{-\frac{c+d x}{d}}+\sqrt{\pi }\right )+27 \sqrt{2} a b^2 d e^{-\frac{2 c}{d}} \left (\sqrt{2} e^{2 \left (\frac{c}{d}+x\right )} \sqrt{-\frac{c+d x}{d}}-\frac{1}{2} \sqrt{\pi } \left (\text{Erf}\left (\sqrt{2} \sqrt{-\frac{c+d x}{d}}\right )-1\right )\right )+2 \sqrt{3} b^3 d e^{-\frac{3 c}{d}} \left (-\sqrt{\pi } \text{Erf}\left (\sqrt{3} \sqrt{-\frac{c+d x}{d}}\right )+2 \sqrt{3} e^{3 \left (\frac{c}{d}+x\right )} \sqrt{-\frac{c+d x}{d}}+\sqrt{\pi }\right )\right )}{36 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[-((c + d*x)/d)]*(-24*a^3*d*(-((c + d*x)/d))^(3/2) + (54*a^2*b*d*(Sqrt[Pi]
+ 2*E^(c/d + x)*Sqrt[-((c + d*x)/d)] - Sqrt[Pi]*Erf[Sqrt[-((c + d*x)/d)]]))/E^(
c/d) + (27*Sqrt[2]*a*b^2*d*(Sqrt[2]*E^(2*(c/d + x))*Sqrt[-((c + d*x)/d)] - (Sqrt
[Pi]*(-1 + Erf[Sqrt[2]*Sqrt[-((c + d*x)/d)]]))/2))/E^((2*c)/d) + (2*Sqrt[3]*b^3*
d*(Sqrt[Pi] + 2*Sqrt[3]*E^(3*(c/d + x))*Sqrt[-((c + d*x)/d)] - Sqrt[Pi]*Erf[Sqrt
[3]*Sqrt[-((c + d*x)/d)]]))/E^((3*c)/d)))/(36*Sqrt[c + d*x])

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 211, normalized size = 0.9 \[ 2\,{\frac{1}{d} \left ( 1/3\, \left ( dx+c \right ) ^{3/2}{a}^{3}+{{b}^{3} \left ( 1/6\,d\sqrt{dx+c}{{\rm e}^{3\,{\frac{dx+c}{d}}}}-1/12\,{d\sqrt{\pi }{\it Erf} \left ( \sqrt{-3\,{d}^{-1}}\sqrt{dx+c} \right ){\frac{1}{\sqrt{-3\,{d}^{-1}}}}} \right ) \left ({{\rm e}^{{\frac{c}{d}}}} \right ) ^{-3}}+3\,{a{b}^{2} \left ( 1/4\,d\sqrt{dx+c}{{\rm e}^{2\,{\frac{dx+c}{d}}}}-1/8\,{d\sqrt{\pi }{\it Erf} \left ( \sqrt{-2\,{d}^{-1}}\sqrt{dx+c} \right ){\frac{1}{\sqrt{-2\,{d}^{-1}}}}} \right ) \left ({{\rm e}^{{\frac{c}{d}}}} \right ) ^{-2}}+3\,{{a}^{2}b \left ( 1/2\,\sqrt{dx+c}{{\rm e}^{{\frac{dx+c}{d}}}}d-1/4\,{d\sqrt{\pi }{\it Erf} \left ( \sqrt{-{d}^{-1}}\sqrt{dx+c} \right ){\frac{1}{\sqrt{-{d}^{-1}}}}} \right ) \left ({{\rm e}^{{\frac{c}{d}}}} \right ) ^{-1}} \right ) } \]

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*(d*x+c))-
1/12*d*Pi^(1/2)/(-3/d)^(1/2)*erf((-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*(d*x+c))-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(1/d*(d*x+c))*d-1/4*d
*Pi^(1/2)/(-1/d)^(1/2)*erf((-1/d)^(1/2)*(d*x+c)^(1/2))))

_______________________________________________________________________________________

Maxima [A]  time = 0.878151, size = 321, normalized size = 1.43 \[ \frac{48 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} - 108 \,{\left (\frac{\sqrt{\pi } d \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{c}{d}\right )}}{\sqrt{-\frac{1}{d}}} - 2 \, \sqrt{d x + c} d e^{\left (\frac{d x + c}{d} - \frac{c}{d}\right )}\right )} a^{2} b - 27 \,{\left (\frac{\sqrt{2} \sqrt{\pi } d \operatorname{erf}\left (\sqrt{2} \sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{2 \, c}{d}\right )}}{\sqrt{-\frac{1}{d}}} - 4 \, \sqrt{d x + c} d e^{\left (\frac{2 \,{\left (d x + c\right )}}{d} - \frac{2 \, c}{d}\right )}\right )} a b^{2} - 4 \,{\left (\frac{\sqrt{3} \sqrt{\pi } d \operatorname{erf}\left (\sqrt{3} \sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{3 \, c}{d}\right )}}{\sqrt{-\frac{1}{d}}} - 6 \, \sqrt{d x + c} d e^{\left (\frac{3 \,{\left (d x + c\right )}}{d} - \frac{3 \, c}{d}\right )}\right )} b^{3}}{72 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)*(b*e^x + a)^3,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

_______________________________________________________________________________________

Fricas [A]  time = 0.257096, size = 267, normalized size = 1.19 \[ -\frac{\sqrt{3} \sqrt{2}{\left (18 \, \sqrt{3} \sqrt{2} \sqrt{\pi } a^{2} b \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{c}{d}\right )} + 9 \, \sqrt{3} \sqrt{\pi } a b^{2} \operatorname{erf}\left (\sqrt{2} \sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{2 \, c}{d}\right )} + 2 \, \sqrt{2} \sqrt{\pi } b^{3} \operatorname{erf}\left (\sqrt{3} \sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{3 \, c}{d}\right )} - \frac{2 \, \sqrt{3} \sqrt{2}{\left (4 \, a^{3} d x + 2 \, b^{3} d e^{\left (3 \, x\right )} + 9 \, a b^{2} d e^{\left (2 \, x\right )} + 18 \, a^{2} b d e^{x} + 4 \, a^{3} c\right )} \sqrt{d x + c} \sqrt{-\frac{1}{d}}}{d}\right )}}{72 \, \sqrt{-\frac{1}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)*(b*e^x + a)^3,x, algorithm="fricas")

[Out]

-1/72*sqrt(3)*sqrt(2)*(18*sqrt(3)*sqrt(2)*sqrt(pi)*a^2*b*erf(sqrt(d*x + c)*sqrt(
-1/d))*e^(-c/d) + 9*sqrt(3)*sqrt(pi)*a*b^2*erf(sqrt(2)*sqrt(d*x + c)*sqrt(-1/d))
*e^(-2*c/d) + 2*sqrt(2)*sqrt(pi)*b^3*erf(sqrt(3)*sqrt(d*x + c)*sqrt(-1/d))*e^(-3
*c/d) - 2*sqrt(3)*sqrt(2)*(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)*sqrt(-1/d)/d)/sqrt(-1/d)

_______________________________________________________________________________________

Sympy [A]  time = 9.35451, size = 265, normalized size = 1.18 \[ \frac{2 a^{3} \left (c + d x\right )^{\frac{3}{2}}}{3 d} + 3 a^{2} b \sqrt{c + d x} e^{- \frac{c}{d}} e^{\frac{c}{d} + x} + \frac{3 i \sqrt{\pi } a^{2} b e^{- \frac{c}{d}} \operatorname{erf}{\left (i \sqrt{c + d x} \sqrt{\frac{1}{d}} \right )}}{2 \sqrt{\frac{1}{d}}} + \frac{3 a b^{2} \sqrt{c + d x} e^{- \frac{2 c}{d}} e^{\frac{2 c}{d} + 2 x}}{2} + \frac{3 \sqrt{2} i \sqrt{\pi } a b^{2} e^{- \frac{2 c}{d}} \operatorname{erf}{\left (\sqrt{2} i \sqrt{c + d x} \sqrt{\frac{1}{d}} \right )}}{8 \sqrt{\frac{1}{d}}} + \frac{b^{3} \sqrt{c + d x} e^{- \frac{3 c}{d}} e^{\frac{3 c}{d} + 3 x}}{3} + \frac{\sqrt{3} i \sqrt{\pi } b^{3} e^{- \frac{3 c}{d}} \operatorname{erf}{\left (\sqrt{3} i \sqrt{c + d x} \sqrt{\frac{1}{d}} \right )}}{18 \sqrt{\frac{1}{d}}} \]

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(c + d*x)*exp(-c/d)*exp(c/d + x) +
3*I*sqrt(pi)*a**2*b*exp(-c/d)*erf(I*sqrt(c + d*x)*sqrt(1/d))/(2*sqrt(1/d)) + 3*a
*b**2*sqrt(c + d*x)*exp(-2*c/d)*exp(2*c/d + 2*x)/2 + 3*sqrt(2)*I*sqrt(pi)*a*b**2
*exp(-2*c/d)*erf(sqrt(2)*I*sqrt(c + d*x)*sqrt(1/d))/(8*sqrt(1/d)) + b**3*sqrt(c
+ d*x)*exp(-3*c/d)*exp(3*c/d + 3*x)/3 + sqrt(3)*I*sqrt(pi)*b**3*exp(-3*c/d)*erf(
sqrt(3)*I*sqrt(c + d*x)*sqrt(1/d))/(18*sqrt(1/d))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.239575, size = 271, normalized size = 1.21 \[ \frac{48 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} + 108 \,{\left (\frac{\sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{d x + c} \sqrt{-d}}{d}\right ) e^{\left (-\frac{c}{d}\right )}}{\sqrt{-d}} + 2 \, \sqrt{d x + c} d e^{x}\right )} a^{2} b + 27 \,{\left (\frac{\sqrt{2} \sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{d x + c} \sqrt{-d}}{d}\right ) e^{\left (-\frac{2 \, c}{d}\right )}}{\sqrt{-d}} + 4 \, \sqrt{d x + c} d e^{\left (2 \, x\right )}\right )} a b^{2} + 4 \,{\left (\frac{\sqrt{3} \sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{3} \sqrt{d x + c} \sqrt{-d}}{d}\right ) e^{\left (-\frac{3 \, c}{d}\right )}}{\sqrt{-d}} + 6 \, \sqrt{d x + c} d e^{\left (3 \, x\right )}\right )} b^{3}}{72 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)*(b*e^x + a)^3,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*sqrt(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