\(\int (a+b e^x)^2 \sqrt {c+d x} \, dx\) [65]

Optimal result

Integrand size = 19, antiderivative size = 145 \[ \int \left (a+b e^x\right )^2 \sqrt {c+d x} \, dx=2 a b e^x \sqrt {c+d x}+\frac {1}{2} b^2 e^{2 x} \sqrt {c+d x}+\frac {2 a^2 (c+d x)^{3/2}}{3 d}-a b \sqrt {d} e^{-\frac {c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )-\frac {1}{4} b^2 \sqrt {d} e^{-\frac {2 c}{d}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {c+d x}}{\sqrt {d}}\right ) \] Output:

2*a*b*exp(x)*(d*x+c)^(1/2)+1/2*b^2*exp(2*x)*(d*x+c)^(1/2)+2/3*a^2*(d*x+c)^ 
(3/2)/d-a*b*d^(1/2)*Pi^(1/2)*erfi((d*x+c)^(1/2)/d^(1/2))/exp(c/d)-1/8*b^2* 
d^(1/2)*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*(d*x+c)^(1/2)/d^(1/2))/exp(2*c/d)
 

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.92 \[ \int \left (a+b e^x\right )^2 \sqrt {c+d x} \, dx=-a b \sqrt {d} e^{-\frac {c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )+\frac {4 \sqrt {c+d x} \left (12 a b d e^x+3 b^2 d e^{2 x}+4 a^2 (c+d x)\right )-3 b^2 d^{3/2} e^{-\frac {2 c}{d}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {c+d x}}{\sqrt {d}}\right )}{24 d} \] Input:

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

Output:

-((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)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2614, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2614

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 a^2 (c+d x)^{3/2}}{3 d}-\sqrt {\pi } a b \sqrt {d} e^{-\frac {c}{d}} \text {erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )+2 a b e^x \sqrt {c+d x}-\frac {1}{4} \sqrt {\frac {\pi }{2}} b^2 \sqrt {d} e^{-\frac {2 c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {c+d x}}{\sqrt {d}}\right )+\frac {1}{2} b^2 e^{2 x} \sqrt {c+d x}\)

Input:

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

Output:

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))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2614

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + 
 (d_.)*(x_))^(m_.), x_Symbol] :> Int[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]
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.99

method	result	size
derivativedivides	\(\frac {\frac {2 a^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+2 b^{2} {\mathrm e}^{-\frac {2 c}{d}} \left (\frac {d \sqrt {d x +c}\, {\mathrm e}^{\frac {2 d x +2 c}{d}}}{4}-\frac {d \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {2}{d}}\, \sqrt {d x +c}\right )}{8 \sqrt {-\frac {2}{d}}}\right )+4 a b \,{\mathrm e}^{-\frac {c}{d}} \left (\frac {\sqrt {d x +c}\, {\mathrm e}^{\frac {d x +c}{d}} d}{2}-\frac {d \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {1}{d}}\, \sqrt {d x +c}\right )}{4 \sqrt {-\frac {1}{d}}}\right )}{d}\)	\(144\)
default	\(\frac {\frac {2 a^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+2 b^{2} {\mathrm e}^{-\frac {2 c}{d}} \left (\frac {d \sqrt {d x +c}\, {\mathrm e}^{\frac {2 d x +2 c}{d}}}{4}-\frac {d \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {2}{d}}\, \sqrt {d x +c}\right )}{8 \sqrt {-\frac {2}{d}}}\right )+4 a b \,{\mathrm e}^{-\frac {c}{d}} \left (\frac {\sqrt {d x +c}\, {\mathrm e}^{\frac {d x +c}{d}} d}{2}-\frac {d \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {1}{d}}\, \sqrt {d x +c}\right )}{4 \sqrt {-\frac {1}{d}}}\right )}{d}\)	\(144\)
parts	\(\frac {2 a^{2} \left (d x +c \right )^{\frac {3}{2}}}{3 d}+\frac {2 b^{2} {\mathrm e}^{-\frac {2 c}{d}} \left (\frac {d \sqrt {d x +c}\, {\mathrm e}^{\frac {2 d x +2 c}{d}}}{4}-\frac {d \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {2}{d}}\, \sqrt {d x +c}\right )}{8 \sqrt {-\frac {2}{d}}}\right )}{d}+\frac {4 a b \,{\mathrm e}^{-\frac {c}{d}} \left (\frac {\sqrt {d x +c}\, {\mathrm e}^{\frac {d x +c}{d}} d}{2}-\frac {d \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {1}{d}}\, \sqrt {d x +c}\right )}{4 \sqrt {-\frac {1}{d}}}\right )}{d}\)	\(149\)

Input:

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

Output:

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*(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/e 
xp(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))))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.92 \[ \int \left (a+b e^x\right )^2 \sqrt {c+d x} \, dx=\frac {3 \, \sqrt {2} \sqrt {\pi } b^{2} d^{2} \sqrt {-\frac {1}{d}} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {2 \, c}{d}\right )} + 24 \, \sqrt {\pi } a b d^{2} \sqrt {-\frac {1}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {c}{d}\right )} + 4 \, {\left (4 \, a^{2} d x + 3 \, b^{2} d e^{\left (2 \, x\right )} + 12 \, a b d e^{x} + 4 \, a^{2} c\right )} \sqrt {d x + c}}{24 \, d} \] Input:

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

Output:

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
 

Sympy [F]

\[ \int \left (a+b e^x\right )^2 \sqrt {c+d x} \, dx=\int \left (a + b e^{x}\right )^{2} \sqrt {c + d x}\, dx \] Input:

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

Output:

Integral((a + b*exp(x))**2*sqrt(c + d*x), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.10 \[ \int \left (a+b e^x\right )^2 \sqrt {c+d x} \, dx=\frac {16 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} - 24 \, {\left (\frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {c}{d}\right )}}{\sqrt {-\frac {1}{d}}} - 2 \, \sqrt {d x + c} d e^{\left (\frac {d x + c}{d} - \frac {c}{d}\right )}\right )} a b - 3 \, {\left (\frac {\sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {2 \, c}{d}\right )}}{\sqrt {-\frac {1}{d}}} - 4 \, \sqrt {d x + c} d e^{\left (\frac {2 \, {\left (d x + c\right )}}{d} - \frac {2 \, c}{d}\right )}\right )} b^{2}}{24 \, d} \] Input:

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

Output:

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
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (110) = 220\).

Time = 0.11 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.72 \[ \int \left (a+b e^x\right )^2 \sqrt {c+d x} \, dx=-\frac {\frac {12 \, \sqrt {2} \sqrt {\pi } b^{2} c d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {2 \, c}{d}\right )}}{\sqrt {-d}} + \frac {48 \, \sqrt {\pi } a b c d \operatorname {erf}\left (-\frac {\sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {c}{d}\right )}}{\sqrt {-d}} - 48 \, \sqrt {d x + c} a^{2} c - 16 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{2} - 24 \, {\left (\frac {\sqrt {\pi } {\left (2 \, c + d\right )} d \operatorname {erf}\left (-\frac {\sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {c}{d}\right )}}{\sqrt {-d}} + 2 \, \sqrt {d x + c} d e^{x}\right )} a b - 3 \, {\left (\frac {\sqrt {2} \sqrt {\pi } {\left (4 \, c + d\right )} d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {2 \, c}{d}\right )}}{\sqrt {-d}} + 4 \, \sqrt {d x + c} d e^{\left (2 \, x\right )}\right )} b^{2}}{24 \, d} \] Input:

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

Output:

-1/24*(12*sqrt(2)*sqrt(pi)*b^2*c*d*erf(-sqrt(2)*sqrt(d*x + c)*sqrt(-d)/d)* 
e^(-2*c/d)/sqrt(-d) + 48*sqrt(pi)*a*b*c*d*erf(-sqrt(d*x + c)*sqrt(-d)/d)*e 
^(-c/d)/sqrt(-d) - 48*sqrt(d*x + c)*a^2*c - 16*((d*x + c)^(3/2) - 3*sqrt(d 
*x + c)*c)*a^2 - 24*(sqrt(pi)*(2*c + d)*d*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)*(4*c + 
 d)*d*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
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b e^x\right )^2 \sqrt {c+d x} \, dx=\int {\left (a+b\,{\mathrm {e}}^x\right )}^2\,\sqrt {c+d\,x} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \left (a+b e^x\right )^2 \sqrt {c+d x} \, dx=\frac {12 \sqrt {\pi }\, \sqrt {d}\, \mathrm {erf}\left (\frac {\sqrt {d x +c}\, i}{\sqrt {d}}\right ) a b d i +6 e^{\frac {2 d x +c}{d}} \sqrt {d x +c}\, b^{2} d +24 e^{\frac {d x +c}{d}} \sqrt {d x +c}\, a b d +8 e^{\frac {c}{d}} \sqrt {d x +c}\, a^{2} c +8 e^{\frac {c}{d}} \sqrt {d x +c}\, a^{2} d x -3 e^{\frac {c}{d}} \left (\int \frac {e^{2 x} \sqrt {d x +c}}{d x +c}d x \right ) b^{2} d^{2}}{12 e^{\frac {c}{d}} d} \] Input:

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

Output:

(12*sqrt(pi)*sqrt(d)*erf((sqrt(c + d*x)*i)/sqrt(d))*a*b*d*i + 6*e**((c + 2 
*d*x)/d)*sqrt(c + d*x)*b**2*d + 24*e**((c + d*x)/d)*sqrt(c + d*x)*a*b*d + 
8*e**(c/d)*sqrt(c + d*x)*a**2*c + 8*e**(c/d)*sqrt(c + d*x)*a**2*d*x - 3*e* 
*(c/d)*int((e**(2*x)*sqrt(c + d*x))/(c + d*x),x)*b**2*d**2)/(12*e**(c/d)*d 
)