\(\int (d+e x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2} \, dx\) [288]

Optimal result

Integrand size = 39, antiderivative size = 240 \[ \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {2 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 c^4 d^4 (d+e x)^{5/2}}+\frac {6 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c^4 d^4 (d+e x)^{7/2}}+\frac {2 e^2 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{3 c^4 d^4 (d+e x)^{9/2}}+\frac {2 e^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{11/2}}{11 c^4 d^4 (d+e x)^{11/2}} \] Output:

2/5*(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^4/d^4/(e*x+ 
d)^(5/2)+6/7*e*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^ 
4/d^4/(e*x+d)^(7/2)+2/3*e^2*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^ 
2)^(9/2)/c^4/d^4/(e*x+d)^(9/2)+2/11*e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^ 
(11/2)/c^4/d^4/(e*x+d)^(11/2)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.55 \[ \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {2 ((a e+c d x) (d+e x))^{5/2} \left (-16 a^3 e^6+8 a^2 c d e^4 (11 d+5 e x)-2 a c^2 d^2 e^2 \left (99 d^2+110 d e x+35 e^2 x^2\right )+c^3 d^3 \left (231 d^3+495 d^2 e x+385 d e^2 x^2+105 e^3 x^3\right )\right )}{1155 c^4 d^4 (d+e x)^{5/2}} \] Input:

Integrate[(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]
 

Output:

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-16*a^3*e^6 + 8*a^2*c*d*e^4*(11*d + 5* 
e*x) - 2*a*c^2*d^2*e^2*(99*d^2 + 110*d*e*x + 35*e^2*x^2) + c^3*d^3*(231*d^ 
3 + 495*d^2*e*x + 385*d*e^2*x^2 + 105*e^3*x^3)))/(1155*c^4*d^4*(d + e*x)^( 
5/2))
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1128, 1128, 1128, 1122}

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.

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

\(\Big \downarrow \) 1128

\(\displaystyle \frac {6 \left (d^2-\frac {a e^2}{c}\right ) \int \sqrt {d+e x} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}dx}{11 d}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {6 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{\sqrt {d+e x}}dx}{9 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d \sqrt {d+e x}}\right )}{11 d}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {6 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^{3/2}}dx}{7 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}}\right )}{9 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d \sqrt {d+e x}}\right )}{11 d}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d}\)

\(\Big \downarrow \) 1122

\(\displaystyle \frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d}+\frac {6 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d \sqrt {d+e x}}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 c d^2 (d+e x)^{5/2}}\right )}{9 d}\right )}{11 d}\)

Input:

Int[(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]
 

Output:

(2*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(11*c*d) + 
 (6*(d^2 - (a*e^2)/c)*((2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/( 
9*c*d*Sqrt[d + e*x]) + (4*(d^2 - (a*e^2)/c)*((4*(d^2 - (a*e^2)/c)*(a*d*e + 
 (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(35*c*d^2*(d + e*x)^(5/2)) + (2*(a* 
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*c*d*(d + e*x)^(3/2))))/(9*d 
)))/(11*d)
 

Defintions of rubi rules used

rule 1122

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), 
 x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && 
EqQ[m + p, 0]
 

rule 1128

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1)))   Int[(d 
 + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, 
 x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[Simplify[m + p], 0]
 

Maple [A] (verified)

Time = 1.24 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.67

method	result	size
default	\(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (c d x +a e \right )^{2} \left (-105 c^{3} d^{3} e^{3} x^{3}+70 x^{2} a \,c^{2} d^{2} e^{4}-385 c^{3} d^{4} e^{2} x^{2}-40 x \,a^{2} c d \,e^{5}+220 x a \,c^{2} d^{3} e^{3}-495 c^{3} d^{5} e x +16 e^{6} a^{3}-88 d^{2} e^{4} a^{2} c +198 d^{4} e^{2} a \,c^{2}-231 d^{6} c^{3}\right )}{1155 \sqrt {e x +d}\, d^{4} c^{4}}\)	\(160\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-105 c^{3} d^{3} e^{3} x^{3}+70 x^{2} a \,c^{2} d^{2} e^{4}-385 c^{3} d^{4} e^{2} x^{2}-40 x \,a^{2} c d \,e^{5}+220 x a \,c^{2} d^{3} e^{3}-495 c^{3} d^{5} e x +16 e^{6} a^{3}-88 d^{2} e^{4} a^{2} c +198 d^{4} e^{2} a \,c^{2}-231 d^{6} c^{3}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{1155 d^{4} c^{4} \left (e x +d \right )^{\frac {3}{2}}}\)	\(168\)
orering	\(-\frac {2 \left (-105 c^{3} d^{3} e^{3} x^{3}+70 x^{2} a \,c^{2} d^{2} e^{4}-385 c^{3} d^{4} e^{2} x^{2}-40 x \,a^{2} c d \,e^{5}+220 x a \,c^{2} d^{3} e^{3}-495 c^{3} d^{5} e x +16 e^{6} a^{3}-88 d^{2} e^{4} a^{2} c +198 d^{4} e^{2} a \,c^{2}-231 d^{6} c^{3}\right ) \left (c d x +a e \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}{1155 d^{4} c^{4} \left (e x +d \right )^{\frac {3}{2}}}\)	\(169\)

Input:

int((e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RETURN 
VERBOSE)
 

Output:

-2/1155/(e*x+d)^(1/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(c*d*x+a*e)^2*(-105*c^3* 
d^3*e^3*x^3+70*a*c^2*d^2*e^4*x^2-385*c^3*d^4*e^2*x^2-40*a^2*c*d*e^5*x+220* 
a*c^2*d^3*e^3*x-495*c^3*d^5*e*x+16*a^3*e^6-88*a^2*c*d^2*e^4+198*a*c^2*d^4* 
e^2-231*c^3*d^6)/d^4/c^4
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.22 \[ \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {2 \, {\left (105 \, c^{5} d^{5} e^{3} x^{5} + 231 \, a^{2} c^{3} d^{6} e^{2} - 198 \, a^{3} c^{2} d^{4} e^{4} + 88 \, a^{4} c d^{2} e^{6} - 16 \, a^{5} e^{8} + 35 \, {\left (11 \, c^{5} d^{6} e^{2} + 4 \, a c^{4} d^{4} e^{4}\right )} x^{4} + 5 \, {\left (99 \, c^{5} d^{7} e + 110 \, a c^{4} d^{5} e^{3} + a^{2} c^{3} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (77 \, c^{5} d^{8} + 264 \, a c^{4} d^{6} e^{2} + 11 \, a^{2} c^{3} d^{4} e^{4} - 2 \, a^{3} c^{2} d^{2} e^{6}\right )} x^{2} + {\left (462 \, a c^{4} d^{7} e + 99 \, a^{2} c^{3} d^{5} e^{3} - 44 \, a^{3} c^{2} d^{3} e^{5} + 8 \, a^{4} c d e^{7}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{1155 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \] Input:

integrate((e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit 
hm="fricas")
 

Output:

2/1155*(105*c^5*d^5*e^3*x^5 + 231*a^2*c^3*d^6*e^2 - 198*a^3*c^2*d^4*e^4 + 
88*a^4*c*d^2*e^6 - 16*a^5*e^8 + 35*(11*c^5*d^6*e^2 + 4*a*c^4*d^4*e^4)*x^4 
+ 5*(99*c^5*d^7*e + 110*a*c^4*d^5*e^3 + a^2*c^3*d^3*e^5)*x^3 + 3*(77*c^5*d 
^8 + 264*a*c^4*d^6*e^2 + 11*a^2*c^3*d^4*e^4 - 2*a^3*c^2*d^2*e^6)*x^2 + (46 
2*a*c^4*d^7*e + 99*a^2*c^3*d^5*e^3 - 44*a^3*c^2*d^3*e^5 + 8*a^4*c*d*e^7)*x 
)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^4*d^4*e*x + 
 c^4*d^5)
 

Sympy [F]

\[ \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\int \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}\, dx \] Input:

integrate((e*x+d)**(3/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
 

Output:

Integral(((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)**(3/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.14 \[ \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {2 \, {\left (105 \, c^{5} d^{5} e^{3} x^{5} + 231 \, a^{2} c^{3} d^{6} e^{2} - 198 \, a^{3} c^{2} d^{4} e^{4} + 88 \, a^{4} c d^{2} e^{6} - 16 \, a^{5} e^{8} + 35 \, {\left (11 \, c^{5} d^{6} e^{2} + 4 \, a c^{4} d^{4} e^{4}\right )} x^{4} + 5 \, {\left (99 \, c^{5} d^{7} e + 110 \, a c^{4} d^{5} e^{3} + a^{2} c^{3} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (77 \, c^{5} d^{8} + 264 \, a c^{4} d^{6} e^{2} + 11 \, a^{2} c^{3} d^{4} e^{4} - 2 \, a^{3} c^{2} d^{2} e^{6}\right )} x^{2} + {\left (462 \, a c^{4} d^{7} e + 99 \, a^{2} c^{3} d^{5} e^{3} - 44 \, a^{3} c^{2} d^{3} e^{5} + 8 \, a^{4} c d e^{7}\right )} x\right )} \sqrt {c d x + a e} {\left (e x + d\right )}}{1155 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \] Input:

integrate((e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit 
hm="maxima")
 

Output:

2/1155*(105*c^5*d^5*e^3*x^5 + 231*a^2*c^3*d^6*e^2 - 198*a^3*c^2*d^4*e^4 + 
88*a^4*c*d^2*e^6 - 16*a^5*e^8 + 35*(11*c^5*d^6*e^2 + 4*a*c^4*d^4*e^4)*x^4 
+ 5*(99*c^5*d^7*e + 110*a*c^4*d^5*e^3 + a^2*c^3*d^3*e^5)*x^3 + 3*(77*c^5*d 
^8 + 264*a*c^4*d^6*e^2 + 11*a^2*c^3*d^4*e^4 - 2*a^3*c^2*d^2*e^6)*x^2 + (46 
2*a*c^4*d^7*e + 99*a^2*c^3*d^5*e^3 - 44*a^3*c^2*d^3*e^5 + 8*a^4*c*d*e^7)*x 
)*sqrt(c*d*x + a*e)*(e*x + d)/(c^4*d^4*e*x + c^4*d^5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (216) = 432\).

Time = 0.19 (sec) , antiderivative size = 784, normalized size of antiderivative = 3.27 \[ \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit 
hm="giac")
 

Output:

2/3465*(3465*sqrt(c*d*x + a*e)*a^2*d^3*e^2 - 2310*(3*sqrt(c*d*x + a*e)*a*e 
 - (c*d*x + a*e)^(3/2))*a*d^3*e - 3465*(3*sqrt(c*d*x + a*e)*a*e - (c*d*x + 
 a*e)^(3/2))*a^2*d*e^3/c + 231*(15*sqrt(c*d*x + a*e)*a^2*e^2 - 10*(c*d*x + 
 a*e)^(3/2)*a*e + 3*(c*d*x + a*e)^(5/2))*d^3 + 1386*(15*sqrt(c*d*x + a*e)* 
a^2*e^2 - 10*(c*d*x + a*e)^(3/2)*a*e + 3*(c*d*x + a*e)^(5/2))*a*d*e^2/c + 
693*(15*sqrt(c*d*x + a*e)*a^2*e^2 - 10*(c*d*x + a*e)^(3/2)*a*e + 3*(c*d*x 
+ a*e)^(5/2))*a^2*e^4/(c^2*d) - 297*(35*sqrt(c*d*x + a*e)*a^3*e^3 - 35*(c* 
d*x + a*e)^(3/2)*a^2*e^2 + 21*(c*d*x + a*e)^(5/2)*a*e - 5*(c*d*x + a*e)^(7 
/2))*d*e/c - 594*(35*sqrt(c*d*x + a*e)*a^3*e^3 - 35*(c*d*x + a*e)^(3/2)*a^ 
2*e^2 + 21*(c*d*x + a*e)^(5/2)*a*e - 5*(c*d*x + a*e)^(7/2))*a*e^3/(c^2*d) 
- 99*(35*sqrt(c*d*x + a*e)*a^3*e^3 - 35*(c*d*x + a*e)^(3/2)*a^2*e^2 + 21*( 
c*d*x + a*e)^(5/2)*a*e - 5*(c*d*x + a*e)^(7/2))*a^2*e^5/(c^3*d^3) + 33*(31 
5*sqrt(c*d*x + a*e)*a^4*e^4 - 420*(c*d*x + a*e)^(3/2)*a^3*e^3 + 378*(c*d*x 
 + a*e)^(5/2)*a^2*e^2 - 180*(c*d*x + a*e)^(7/2)*a*e + 35*(c*d*x + a*e)^(9/ 
2))*e^2/(c^2*d) + 22*(315*sqrt(c*d*x + a*e)*a^4*e^4 - 420*(c*d*x + a*e)^(3 
/2)*a^3*e^3 + 378*(c*d*x + a*e)^(5/2)*a^2*e^2 - 180*(c*d*x + a*e)^(7/2)*a* 
e + 35*(c*d*x + a*e)^(9/2))*a*e^4/(c^3*d^3) - 5*(693*sqrt(c*d*x + a*e)*a^5 
*e^5 - 1155*(c*d*x + a*e)^(3/2)*a^4*e^4 + 1386*(c*d*x + a*e)^(5/2)*a^3*e^3 
 - 990*(c*d*x + a*e)^(7/2)*a^2*e^2 + 385*(c*d*x + a*e)^(9/2)*a*e - 63*(c*d 
*x + a*e)^(11/2))*e^3/(c^3*d^3))/(c*d)
 

Mupad [B] (verification not implemented)

Time = 5.56 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.33 \[ \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,e\,x^4\,\left (11\,c\,d^2+4\,a\,e^2\right )\,\sqrt {d+e\,x}}{33}+\frac {2\,c\,d\,e^2\,x^5\,\sqrt {d+e\,x}}{11}-\frac {\sqrt {d+e\,x}\,\left (32\,a^5\,e^8-176\,a^4\,c\,d^2\,e^6+396\,a^3\,c^2\,d^4\,e^4-462\,a^2\,c^3\,d^6\,e^2\right )}{1155\,c^4\,d^4\,e}+\frac {2\,x^3\,\sqrt {d+e\,x}\,\left (a^2\,e^4+110\,a\,c\,d^2\,e^2+99\,c^2\,d^4\right )}{231\,c\,d}+\frac {x^2\,\sqrt {d+e\,x}\,\left (-12\,a^3\,c^2\,d^2\,e^6+66\,a^2\,c^3\,d^4\,e^4+1584\,a\,c^4\,d^6\,e^2+462\,c^5\,d^8\right )}{1155\,c^4\,d^4\,e}+\frac {2\,a\,x\,\sqrt {d+e\,x}\,\left (8\,a^3\,e^6-44\,a^2\,c\,d^2\,e^4+99\,a\,c^2\,d^4\,e^2+462\,c^3\,d^6\right )}{1155\,c^3\,d^3}\right )}{x+\frac {d}{e}} \] Input:

int((d + e*x)^(3/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2),x)
 

Output:

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*e*x^4*(4*a*e^2 + 11*c*d 
^2)*(d + e*x)^(1/2))/33 + (2*c*d*e^2*x^5*(d + e*x)^(1/2))/11 - ((d + e*x)^ 
(1/2)*(32*a^5*e^8 - 176*a^4*c*d^2*e^6 - 462*a^2*c^3*d^6*e^2 + 396*a^3*c^2* 
d^4*e^4))/(1155*c^4*d^4*e) + (2*x^3*(d + e*x)^(1/2)*(a^2*e^4 + 99*c^2*d^4 
+ 110*a*c*d^2*e^2))/(231*c*d) + (x^2*(d + e*x)^(1/2)*(462*c^5*d^8 + 1584*a 
*c^4*d^6*e^2 + 66*a^2*c^3*d^4*e^4 - 12*a^3*c^2*d^2*e^6))/(1155*c^4*d^4*e) 
+ (2*a*x*(d + e*x)^(1/2)*(8*a^3*e^6 + 462*c^3*d^6 + 99*a*c^2*d^4*e^2 - 44* 
a^2*c*d^2*e^4))/(1155*c^3*d^3)))/(x + d/e)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.10 \[ \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {2 \sqrt {c d x +a e}\, \left (105 c^{5} d^{5} e^{3} x^{5}+140 a \,c^{4} d^{4} e^{4} x^{4}+385 c^{5} d^{6} e^{2} x^{4}+5 a^{2} c^{3} d^{3} e^{5} x^{3}+550 a \,c^{4} d^{5} e^{3} x^{3}+495 c^{5} d^{7} e \,x^{3}-6 a^{3} c^{2} d^{2} e^{6} x^{2}+33 a^{2} c^{3} d^{4} e^{4} x^{2}+792 a \,c^{4} d^{6} e^{2} x^{2}+231 c^{5} d^{8} x^{2}+8 a^{4} c d \,e^{7} x -44 a^{3} c^{2} d^{3} e^{5} x +99 a^{2} c^{3} d^{5} e^{3} x +462 a \,c^{4} d^{7} e x -16 a^{5} e^{8}+88 a^{4} c \,d^{2} e^{6}-198 a^{3} c^{2} d^{4} e^{4}+231 a^{2} c^{3} d^{6} e^{2}\right )}{1155 c^{4} d^{4}} \] Input:

int((e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
 

Output:

(2*sqrt(a*e + c*d*x)*( - 16*a**5*e**8 + 88*a**4*c*d**2*e**6 + 8*a**4*c*d*e 
**7*x - 198*a**3*c**2*d**4*e**4 - 44*a**3*c**2*d**3*e**5*x - 6*a**3*c**2*d 
**2*e**6*x**2 + 231*a**2*c**3*d**6*e**2 + 99*a**2*c**3*d**5*e**3*x + 33*a* 
*2*c**3*d**4*e**4*x**2 + 5*a**2*c**3*d**3*e**5*x**3 + 462*a*c**4*d**7*e*x 
+ 792*a*c**4*d**6*e**2*x**2 + 550*a*c**4*d**5*e**3*x**3 + 140*a*c**4*d**4* 
e**4*x**4 + 231*c**5*d**8*x**2 + 495*c**5*d**7*e*x**3 + 385*c**5*d**6*e**2 
*x**4 + 105*c**5*d**5*e**3*x**5))/(1155*c**4*d**4)