\(\int (a+b e^x)^3 \sqrt {c+d x} \, dx\) [66]

Optimal result

Integrand size = 19, antiderivative size = 224 \[ \int \left (a+b e^x\right )^3 \sqrt {c+d x} \, dx=3 a^2 b e^x \sqrt {c+d x}+\frac {3}{2} a b^2 e^{2 x} \sqrt {c+d x}+\frac {1}{3} b^3 e^{3 x} \sqrt {c+d x}+\frac {2 a^3 (c+d x)^{3/2}}{3 d}-\frac {3}{2} a^2 b \sqrt {d} e^{-\frac {c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )-\frac {3}{4} a b^2 \sqrt {d} e^{-\frac {2 c}{d}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {c+d x}}{\sqrt {d}}\right )-\frac {1}{6} b^3 \sqrt {d} e^{-\frac {3 c}{d}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {c+d x}}{\sqrt {d}}\right ) \] Output:

3*a^2*b*exp(x)*(d*x+c)^(1/2)+3/2*a*b^2*exp(2*x)*(d*x+c)^(1/2)+1/3*b^3*exp( 
3*x)*(d*x+c)^(1/2)+2/3*a^3*(d*x+c)^(3/2)/d-3/2*a^2*b*d^(1/2)*Pi^(1/2)*erfi 
((d*x+c)^(1/2)/d^(1/2))/exp(c/d)-3/8*a*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)-1/18*b^3*d^(1/2)*3^(1/2)*Pi^(1/2) 
*erfi(3^(1/2)*(d*x+c)^(1/2)/d^(1/2))/exp(3*c/d)
 

Mathematica [A] (verified)

Time = 2.54 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.88 \[ \int \left (a+b e^x\right )^3 \sqrt {c+d x} \, dx=-\frac {-12 \sqrt {c+d x} \left (18 a^2 b d e^x+9 a b^2 d e^{2 x}+2 b^3 d e^{3 x}+4 a^3 (c+d x)\right )+108 a^2 b d^{3/2} e^{-\frac {c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )+27 a b^2 d^{3/2} e^{-\frac {2 c}{d}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {c+d x}}{\sqrt {d}}\right )+4 b^3 d^{3/2} e^{-\frac {3 c}{d}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {c+d x}}{\sqrt {d}}\right )}{72 d} \] Input:

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

Output:

-1/72*(-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]/S 
qrt[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))/d
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 224, 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.

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

\(\Big \downarrow \) 2614

\(\displaystyle \int \left (a^3 \sqrt {c+d x}+3 a^2 b e^x \sqrt {c+d x}+3 a b^2 e^{2 x} \sqrt {c+d x}+b^3 e^{3 x} \sqrt {c+d x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 a^3 (c+d x)^{3/2}}{3 d}-\frac {3}{2} \sqrt {\pi } a^2 b \sqrt {d} e^{-\frac {c}{d}} \text {erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )+3 a^2 b e^x \sqrt {c+d x}-\frac {3}{4} \sqrt {\frac {\pi }{2}} a b^2 \sqrt {d} e^{-\frac {2 c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {c+d x}}{\sqrt {d}}\right )+\frac {3}{2} a b^2 e^{2 x} \sqrt {c+d x}-\frac {1}{6} \sqrt {\frac {\pi }{3}} b^3 \sqrt {d} e^{-\frac {3 c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {c+d x}}{\sqrt {d}}\right )+\frac {1}{3} b^3 e^{3 x} \sqrt {c+d x}\)

Input:

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

Output:

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]*Sqr 
t[Pi]*Erfi[Sqrt[c + d*x]/Sqrt[d]])/(2*E^(c/d)) - (3*a*b^2*Sqrt[d]*Sqrt[Pi/ 
2]*Erfi[(Sqrt[2]*Sqrt[c + d*x])/Sqrt[d]])/(4*E^((2*c)/d)) - (b^3*Sqrt[d]*S 
qrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[c + d*x])/Sqrt[d]])/(6*E^((3*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 = 0.14 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(\frac {\frac {2 a^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}+2 b^{3} {\mathrm e}^{-\frac {3 c}{d}} \left (\frac {d \sqrt {d x +c}\, {\mathrm e}^{\frac {3 d x +3 c}{d}}}{6}-\frac {d \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {3}{d}}\, \sqrt {d x +c}\right )}{12 \sqrt {-\frac {3}{d}}}\right )+6 a \,b^{2} {\mathrm e}^{-\frac {2 c}{d}} \left (\frac {d \sqrt {d x +c}\, {\mathrm e}^{\frac {2 d x +2 c}{d}}}{4}-\frac {d \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {2}{d}}\, \sqrt {d x +c}\right )}{8 \sqrt {-\frac {2}{d}}}\right )+6 a^{2} b \,{\mathrm e}^{-\frac {c}{d}} \left (\frac {\sqrt {d x +c}\, {\mathrm e}^{\frac {d x +c}{d}} d}{2}-\frac {d \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {1}{d}}\, \sqrt {d x +c}\right )}{4 \sqrt {-\frac {1}{d}}}\right )}{d}\)	\(211\)
default	\(\frac {\frac {2 a^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}+2 b^{3} {\mathrm e}^{-\frac {3 c}{d}} \left (\frac {d \sqrt {d x +c}\, {\mathrm e}^{\frac {3 d x +3 c}{d}}}{6}-\frac {d \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {3}{d}}\, \sqrt {d x +c}\right )}{12 \sqrt {-\frac {3}{d}}}\right )+6 a \,b^{2} {\mathrm e}^{-\frac {2 c}{d}} \left (\frac {d \sqrt {d x +c}\, {\mathrm e}^{\frac {2 d x +2 c}{d}}}{4}-\frac {d \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {2}{d}}\, \sqrt {d x +c}\right )}{8 \sqrt {-\frac {2}{d}}}\right )+6 a^{2} b \,{\mathrm e}^{-\frac {c}{d}} \left (\frac {\sqrt {d x +c}\, {\mathrm e}^{\frac {d x +c}{d}} d}{2}-\frac {d \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {1}{d}}\, \sqrt {d x +c}\right )}{4 \sqrt {-\frac {1}{d}}}\right )}{d}\)	\(211\)
parts	\(\frac {2 a^{3} \left (d x +c \right )^{\frac {3}{2}}}{3 d}+\frac {2 b^{3} {\mathrm e}^{-\frac {3 c}{d}} \left (\frac {d \sqrt {d x +c}\, {\mathrm e}^{\frac {3 d x +3 c}{d}}}{6}-\frac {d \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {3}{d}}\, \sqrt {d x +c}\right )}{12 \sqrt {-\frac {3}{d}}}\right )}{d}+\frac {6 a \,b^{2} {\mathrm e}^{-\frac {2 c}{d}} \left (\frac {d \sqrt {d x +c}\, {\mathrm e}^{\frac {2 d x +2 c}{d}}}{4}-\frac {d \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {2}{d}}\, \sqrt {d x +c}\right )}{8 \sqrt {-\frac {2}{d}}}\right )}{d}+\frac {6 a^{2} b \,{\mathrm e}^{-\frac {c}{d}} \left (\frac {\sqrt {d x +c}\, {\mathrm e}^{\frac {d x +c}{d}} d}{2}-\frac {d \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {1}{d}}\, \sqrt {d x +c}\right )}{4 \sqrt {-\frac {1}{d}}}\right )}{d}\)	\(219\)

Input:

int((a+b*exp(x))^3*(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/d*(1/3*a^3*(d*x+c)^(3/2)+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))))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.88 \[ \int \left (a+b e^x\right )^3 \sqrt {c+d x} \, dx=\frac {27 \, \sqrt {2} \sqrt {\pi } a b^{2} d^{2} \sqrt {-\frac {1}{d}} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {2 \, c}{d}\right )} + 4 \, \sqrt {3} \sqrt {\pi } b^{3} d^{2} \sqrt {-\frac {1}{d}} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {3 \, c}{d}\right )} + 108 \, \sqrt {\pi } a^{2} b d^{2} \sqrt {-\frac {1}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {c}{d}\right )} + 12 \, {\left (4 \, a^{3} d x + 2 \, b^{3} d e^{\left (3 \, x\right )} + 9 \, a b^{2} d e^{\left (2 \, x\right )} + 18 \, a^{2} b d e^{x} + 4 \, a^{3} c\right )} \sqrt {d x + c}}{72 \, d} \] Input:

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

Output:

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

Sympy [F]

\[ \int \left (a+b e^x\right )^3 \sqrt {c+d x} \, dx=\int \left (a + b e^{x}\right )^{3} \sqrt {c + d x}\, dx \] Input:

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

Output:

Integral((a + b*exp(x))**3*sqrt(c + d*x), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.06 \[ \int \left (a+b e^x\right )^3 \sqrt {c+d x} \, dx=\frac {48 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} - 108 \, {\left (\frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {c}{d}\right )}}{\sqrt {-\frac {1}{d}}} - 2 \, \sqrt {d x + c} d e^{\left (\frac {d x + c}{d} - \frac {c}{d}\right )}\right )} a^{2} b - 27 \, {\left (\frac {\sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {2 \, c}{d}\right )}}{\sqrt {-\frac {1}{d}}} - 4 \, \sqrt {d x + c} d e^{\left (\frac {2 \, {\left (d x + c\right )}}{d} - \frac {2 \, c}{d}\right )}\right )} a b^{2} - 4 \, {\left (\frac {\sqrt {3} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {3 \, c}{d}\right )}}{\sqrt {-\frac {1}{d}}} - 6 \, \sqrt {d x + c} d e^{\left (\frac {3 \, {\left (d x + c\right )}}{d} - \frac {3 \, c}{d}\right )}\right )} b^{3}}{72 \, d} \] Input:

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

Output:

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)/sq 
rt(-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
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (168) = 336\).

Time = 0.15 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.64 \[ \int \left (a+b e^x\right )^3 \sqrt {c+d x} \, dx=-\frac {\frac {108 \, \sqrt {2} \sqrt {\pi } a b^{2} c d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {2 \, c}{d}\right )}}{\sqrt {-d}} + \frac {24 \, \sqrt {3} \sqrt {\pi } b^{3} c d \operatorname {erf}\left (-\frac {\sqrt {3} \sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {3 \, c}{d}\right )}}{\sqrt {-d}} + \frac {216 \, \sqrt {\pi } a^{2} b c d \operatorname {erf}\left (-\frac {\sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {c}{d}\right )}}{\sqrt {-d}} - 144 \, \sqrt {d x + c} a^{3} c - 48 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{3} - 108 \, {\left (\frac {\sqrt {\pi } {\left (2 \, c + d\right )} d \operatorname {erf}\left (-\frac {\sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {c}{d}\right )}}{\sqrt {-d}} + 2 \, \sqrt {d x + c} d e^{x}\right )} a^{2} b - 27 \, {\left (\frac {\sqrt {2} \sqrt {\pi } {\left (4 \, c + d\right )} d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {2 \, c}{d}\right )}}{\sqrt {-d}} + 4 \, \sqrt {d x + c} d e^{\left (2 \, x\right )}\right )} a b^{2} - 4 \, {\left (\frac {\sqrt {3} \sqrt {\pi } {\left (6 \, c + d\right )} d \operatorname {erf}\left (-\frac {\sqrt {3} \sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {3 \, c}{d}\right )}}{\sqrt {-d}} + 6 \, \sqrt {d x + c} d e^{\left (3 \, x\right )}\right )} b^{3}}{72 \, d} \] Input:

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

Output:

-1/72*(108*sqrt(2)*sqrt(pi)*a*b^2*c*d*erf(-sqrt(2)*sqrt(d*x + c)*sqrt(-d)/ 
d)*e^(-2*c/d)/sqrt(-d) + 24*sqrt(3)*sqrt(pi)*b^3*c*d*erf(-sqrt(3)*sqrt(d*x 
 + c)*sqrt(-d)/d)*e^(-3*c/d)/sqrt(-d) + 216*sqrt(pi)*a^2*b*c*d*erf(-sqrt(d 
*x + c)*sqrt(-d)/d)*e^(-c/d)/sqrt(-d) - 144*sqrt(d*x + c)*a^3*c - 48*((d*x 
 + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^3 - 108*(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^2*b - 2 
7*(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))*a*b^2 - 4*(sqrt(3)*sqrt(pi)* 
(6*c + d)*d*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
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b e^x\right )^3 \sqrt {c+d x} \, dx=\int {\left (a+b\,{\mathrm {e}}^x\right )}^3\,\sqrt {c+d\,x} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \left (a+b e^x\right )^3 \sqrt {c+d x} \, dx=\frac {18 \sqrt {\pi }\, \sqrt {d}\, \mathrm {erf}\left (\frac {\sqrt {d x +c}\, i}{\sqrt {d}}\right ) a^{2} b d i +4 e^{\frac {3 d x +c}{d}} \sqrt {d x +c}\, b^{3} d +18 e^{\frac {2 d x +c}{d}} \sqrt {d x +c}\, a \,b^{2} d +36 e^{\frac {d x +c}{d}} \sqrt {d x +c}\, a^{2} b d +8 e^{\frac {c}{d}} \sqrt {d x +c}\, a^{3} c +8 e^{\frac {c}{d}} \sqrt {d x +c}\, a^{3} d x -2 e^{\frac {c}{d}} \left (\int \frac {e^{3 x} \sqrt {d x +c}}{d x +c}d x \right ) b^{3} d^{2}-9 e^{\frac {c}{d}} \left (\int \frac {e^{2 x} \sqrt {d x +c}}{d x +c}d x \right ) a \,b^{2} d^{2}}{12 e^{\frac {c}{d}} d} \] Input:

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

Output:

(18*sqrt(pi)*sqrt(d)*erf((sqrt(c + d*x)*i)/sqrt(d))*a**2*b*d*i + 4*e**((c 
+ 3*d*x)/d)*sqrt(c + d*x)*b**3*d + 18*e**((c + 2*d*x)/d)*sqrt(c + d*x)*a*b 
**2*d + 36*e**((c + d*x)/d)*sqrt(c + d*x)*a**2*b*d + 8*e**(c/d)*sqrt(c + d 
*x)*a**3*c + 8*e**(c/d)*sqrt(c + d*x)*a**3*d*x - 2*e**(c/d)*int((e**(3*x)* 
sqrt(c + d*x))/(c + d*x),x)*b**3*d**2 - 9*e**(c/d)*int((e**(2*x)*sqrt(c + 
d*x))/(c + d*x),x)*a*b**2*d**2)/(12*e**(c/d)*d)