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

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))

_______________________________________________________________________________________

Rubi [A]  time = 0.332069, 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 \[ \frac{2 a^2 (c+d x)^{3/2}}{3 d}-\sqrt{\pi } a b \sqrt{d} e^{-\frac{c}{d}} \text{Erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )+2 a b e^x \sqrt{c+d x}-\frac{1}{4} \sqrt{\frac{\pi }{2}} b^2 \sqrt{d} e^{-\frac{2 c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{c+d x}}{\sqrt{d}}\right )+\frac{1}{2} b^2 e^{2 x} \sqrt{c+d x} \]

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))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 25.8784, size = 133, normalized size = 0.92 \[ \frac{2 a^{2} \left (c + d x\right )^{\frac{3}{2}}}{3 d} - \sqrt{\pi } a b \sqrt{d} e^{- \frac{c}{d}} \operatorname{erfi}{\left (\frac{\sqrt{c + d x}}{\sqrt{d}} \right )} + 2 a b \sqrt{c + d x} e^{x} - \frac{\sqrt{2} \sqrt{\pi } b^{2} \sqrt{d} e^{- \frac{2 c}{d}} \operatorname{erfi}{\left (\frac{\sqrt{2} \sqrt{c + d x}}{\sqrt{d}} \right )}}{8} + \frac{b^{2} \sqrt{c + d x} e^{2 x}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.766045, size = 197, normalized size = 1.36 \[ -\frac{\sqrt{-\frac{c+d x}{d}} \left (-8 a^2 d \left (-\frac{c+d x}{d}\right )^{3/2}+12 a b d e^{-\frac{c}{d}} \left (-\sqrt{\pi } \text{Erf}\left (\sqrt{-\frac{c+d x}{d}}\right )+2 e^{\frac{c}{d}+x} \sqrt{-\frac{c+d x}{d}}+\sqrt{\pi }\right )+3 \sqrt{2} b^2 d e^{-\frac{2 c}{d}} \left (\sqrt{2} e^{2 \left (\frac{c}{d}+x\right )} \sqrt{-\frac{c+d x}{d}}-\frac{1}{2} \sqrt{\pi } \left (\text{Erf}\left (\sqrt{2} \sqrt{-\frac{c+d x}{d}}\right )-1\right )\right )\right )}{12 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[-((c + d*x)/d)]*(-8*a^2*d*(-((c + d*x)/d))^(3/2) + (12*a*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
) + (3*Sqrt[2]*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)))/(12*Sqrt[c + d*x])

_______________________________________________________________________________________

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

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*(d*x+c)^(3/2)*a^2+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)))+2*a*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 = 0.853262, size = 216, normalized size = 1.49 \[ \frac{16 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} - 24 \,{\left (\frac{\sqrt{\pi } d \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{c}{d}\right )}}{\sqrt{-\frac{1}{d}}} - 2 \, \sqrt{d x + c} d e^{\left (\frac{d x + c}{d} - \frac{c}{d}\right )}\right )} a b - 3 \,{\left (\frac{\sqrt{2} \sqrt{\pi } d \operatorname{erf}\left (\sqrt{2} \sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{2 \, c}{d}\right )}}{\sqrt{-\frac{1}{d}}} - 4 \, \sqrt{d x + c} d e^{\left (\frac{2 \,{\left (d x + c\right )}}{d} - \frac{2 \, c}{d}\right )}\right )} b^{2}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

Fricas [A]  time = 0.255003, size = 182, normalized size = 1.26 \[ \frac{1}{24} \, \sqrt{2}{\left (12 \, \sqrt{2} \sqrt{\pi } a b d \sqrt{-\frac{1}{d}} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{c}{d}\right )} + 3 \, \sqrt{\pi } b^{2} d \sqrt{-\frac{1}{d}} \operatorname{erf}\left (\sqrt{2} \sqrt{d x + c} \sqrt{-\frac{1}{d}}\right ) e^{\left (-\frac{2 \, c}{d}\right )} + \frac{2 \, \sqrt{2}{\left (4 \, a^{2} d x + 3 \, b^{2} d e^{\left (2 \, x\right )} + 12 \, a b d e^{x} + 4 \, a^{2} c\right )} \sqrt{d x + c}}{d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)*(b*e^x + a)^2,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 6.62214, size = 167, normalized size = 1.15 \[ \frac{2 a^{2} \left (c + d x\right )^{\frac{3}{2}}}{3 d} + 2 a b \sqrt{c + d x} e^{- \frac{c}{d}} e^{\frac{c}{d} + x} + \frac{i \sqrt{\pi } a b e^{- \frac{c}{d}} \operatorname{erf}{\left (i \sqrt{c + d x} \sqrt{\frac{1}{d}} \right )}}{\sqrt{\frac{1}{d}}} + \frac{b^{2} \sqrt{c + d x} e^{- \frac{2 c}{d}} e^{\frac{2 c}{d} + 2 x}}{2} + \frac{\sqrt{2} i \sqrt{\pi } b^{2} e^{- \frac{2 c}{d}} \operatorname{erf}{\left (\sqrt{2} i \sqrt{c + d x} \sqrt{\frac{1}{d}} \right )}}{8 \sqrt{\frac{1}{d}}} \]

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(c + d*x)*exp(-c/d)*exp(c/d + x) + I*s
qrt(pi)*a*b*exp(-c/d)*erf(I*sqrt(c + d*x)*sqrt(1/d))/sqrt(1/d) + b**2*sqrt(c + d
*x)*exp(-2*c/d)*exp(2*c/d + 2*x)/2 + sqrt(2)*I*sqrt(pi)*b**2*exp(-2*c/d)*erf(sqr
t(2)*I*sqrt(c + d*x)*sqrt(1/d))/(8*sqrt(1/d))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.264448, size = 182, normalized size = 1.26 \[ \frac{16 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} + 24 \,{\left (\frac{\sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{d x + c} \sqrt{-d}}{d}\right ) e^{\left (-\frac{c}{d}\right )}}{\sqrt{-d}} + 2 \, \sqrt{d x + c} d e^{x}\right )} a b + 3 \,{\left (\frac{\sqrt{2} \sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{d x + c} \sqrt{-d}}{d}\right ) e^{\left (-\frac{2 \, c}{d}\right )}}{\sqrt{-d}} + 4 \, \sqrt{d x + c} d e^{\left (2 \, x\right )}\right )} b^{2}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)*(b*e^x + a)^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(-sqr
t(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