\(\int (d+e x)^{5/2} \sqrt {a d e+(c d^2+a e^2) x+c d e x^2} \, dx\) [279]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.55 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 ((a e+c d x) (d+e x))^{3/2} \left (-16 a^3 e^6+24 a^2 c d e^4 (3 d+e x)-6 a c^2 d^2 e^2 \left (21 d^2+18 d e x+5 e^2 x^2\right )+c^3 d^3 \left (105 d^3+189 d^2 e x+135 d e^2 x^2+35 e^3 x^3\right )\right )}{315 c^4 d^4 (d+e x)^{3/2}} \] Input:

Integrate[(d + e*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
 

Output:

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-16*a^3*e^6 + 24*a^2*c*d*e^4*(3*d + e* 
x) - 6*a*c^2*d^2*e^2*(21*d^2 + 18*d*e*x + 5*e^2*x^2) + c^3*d^3*(105*d^3 + 
189*d^2*e*x + 135*d*e^2*x^2 + 35*e^3*x^3)))/(315*c^4*d^4*(d + e*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.69 (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.

\(\displaystyle \int (d+e x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \, dx\)

\(\Big \downarrow \) 1128

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

\(\Big \downarrow \) 1128

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

\(\Big \downarrow \) 1128

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

\(\Big \downarrow \) 1122

\(\displaystyle \frac {2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d}+\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d \sqrt {d+e x}}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 c d^2 (d+e x)^{3/2}}\right )}{7 d}\right )}{3 d}\)

Input:

Int[(d + e*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
 

Output:

(2*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(9*c*d) 
+ (2*(d^2 - (a*e^2)/c)*((2*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d* 
e*x^2)^(3/2))/(7*c*d) + (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)^(3/2))/(15*c*d^2*(d + e*x)^(3/2)) + (2*(a 
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*c*d*Sqrt[d + e*x])))/(7*d) 
))/(3*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 = 1.18 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.66

method	result	size
default	\(-\frac {2 \left (c d x +a e \right ) \left (-35 c^{3} d^{3} e^{3} x^{3}+30 x^{2} a \,c^{2} d^{2} e^{4}-135 c^{3} d^{4} e^{2} x^{2}-24 x \,a^{2} c d \,e^{5}+108 x a \,c^{2} d^{3} e^{3}-189 c^{3} d^{5} e x +16 e^{6} a^{3}-72 d^{2} e^{4} a^{2} c +126 d^{4} e^{2} a \,c^{2}-105 d^{6} c^{3}\right ) \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}}{315 d^{4} c^{4} \sqrt {e x +d}}\)	\(158\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-35 c^{3} d^{3} e^{3} x^{3}+30 x^{2} a \,c^{2} d^{2} e^{4}-135 c^{3} d^{4} e^{2} x^{2}-24 x \,a^{2} c d \,e^{5}+108 x a \,c^{2} d^{3} e^{3}-189 c^{3} d^{5} e x +16 e^{6} a^{3}-72 d^{2} e^{4} a^{2} c +126 d^{4} e^{2} a \,c^{2}-105 d^{6} c^{3}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{315 d^{4} c^{4} \sqrt {e x +d}}\)	\(168\)
orering	\(-\frac {2 \left (-35 c^{3} d^{3} e^{3} x^{3}+30 x^{2} a \,c^{2} d^{2} e^{4}-135 c^{3} d^{4} e^{2} x^{2}-24 x \,a^{2} c d \,e^{5}+108 x a \,c^{2} d^{3} e^{3}-189 c^{3} d^{5} e x +16 e^{6} a^{3}-72 d^{2} e^{4} a^{2} c +126 d^{4} e^{2} a \,c^{2}-105 d^{6} c^{3}\right ) \left (c d x +a e \right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{315 d^{4} c^{4} \sqrt {e x +d}}\)	\(169\)

Input:

int((e*x+d)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x,method=_RETURN 
VERBOSE)
 

Output:

-2/315*(c*d*x+a*e)*(-35*c^3*d^3*e^3*x^3+30*a*c^2*d^2*e^4*x^2-135*c^3*d^4*e 
^2*x^2-24*a^2*c*d*e^5*x+108*a*c^2*d^3*e^3*x-189*c^3*d^5*e*x+16*a^3*e^6-72* 
a^2*c*d^2*e^4+126*a*c^2*d^4*e^2-105*c^3*d^6)*((e*x+d)*(c*d*x+a*e))^(1/2)/d 
^4/c^4/(e*x+d)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.96 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (35 \, c^{4} d^{4} e^{3} x^{4} + 105 \, a c^{3} d^{6} e - 126 \, a^{2} c^{2} d^{4} e^{3} + 72 \, a^{3} c d^{2} e^{5} - 16 \, a^{4} e^{7} + 5 \, {\left (27 \, c^{4} d^{5} e^{2} + a c^{3} d^{3} e^{4}\right )} x^{3} + 3 \, {\left (63 \, c^{4} d^{6} e + 9 \, a c^{3} d^{4} e^{3} - 2 \, a^{2} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (105 \, c^{4} d^{7} + 63 \, a c^{3} d^{5} e^{2} - 36 \, a^{2} c^{2} d^{3} e^{4} + 8 \, a^{3} c d e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{315 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \] Input:

integrate((e*x+d)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorit 
hm="fricas")
 

Output:

2/315*(35*c^4*d^4*e^3*x^4 + 105*a*c^3*d^6*e - 126*a^2*c^2*d^4*e^3 + 72*a^3 
*c*d^2*e^5 - 16*a^4*e^7 + 5*(27*c^4*d^5*e^2 + a*c^3*d^3*e^4)*x^3 + 3*(63*c 
^4*d^6*e + 9*a*c^3*d^4*e^3 - 2*a^2*c^2*d^2*e^5)*x^2 + (105*c^4*d^7 + 63*a* 
c^3*d^5*e^2 - 36*a^2*c^2*d^3*e^4 + 8*a^3*c*d*e^6)*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(-1)]

Timed out. \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**(5/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.88 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (35 \, c^{4} d^{4} e^{3} x^{4} + 105 \, a c^{3} d^{6} e - 126 \, a^{2} c^{2} d^{4} e^{3} + 72 \, a^{3} c d^{2} e^{5} - 16 \, a^{4} e^{7} + 5 \, {\left (27 \, c^{4} d^{5} e^{2} + a c^{3} d^{3} e^{4}\right )} x^{3} + 3 \, {\left (63 \, c^{4} d^{6} e + 9 \, a c^{3} d^{4} e^{3} - 2 \, a^{2} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (105 \, c^{4} d^{7} + 63 \, a c^{3} d^{5} e^{2} - 36 \, a^{2} c^{2} d^{3} e^{4} + 8 \, a^{3} c d e^{6}\right )} x\right )} \sqrt {c d x + a e} {\left (e x + d\right )}}{315 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \] Input:

integrate((e*x+d)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorit 
hm="maxima")
 

Output:

2/315*(35*c^4*d^4*e^3*x^4 + 105*a*c^3*d^6*e - 126*a^2*c^2*d^4*e^3 + 72*a^3 
*c*d^2*e^5 - 16*a^4*e^7 + 5*(27*c^4*d^5*e^2 + a*c^3*d^3*e^4)*x^3 + 3*(63*c 
^4*d^6*e + 9*a*c^3*d^4*e^3 - 2*a^2*c^2*d^2*e^5)*x^2 + (105*c^4*d^7 + 63*a* 
c^3*d^5*e^2 - 36*a^2*c^2*d^3*e^4 + 8*a^3*c*d*e^6)*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. 444 vs. \(2 (216) = 432\).

Time = 0.19 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.85 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (315 \, \sqrt {c d x + a e} a d^{3} e - 105 \, {\left (3 \, \sqrt {c d x + a e} a e - {\left (c d x + a e\right )}^{\frac {3}{2}}\right )} d^{3} - \frac {315 \, {\left (3 \, \sqrt {c d x + a e} a e - {\left (c d x + a e\right )}^{\frac {3}{2}}\right )} a d e^{2}}{c} + \frac {63 \, {\left (15 \, \sqrt {c d x + a e} a^{2} e^{2} - 10 \, {\left (c d x + a e\right )}^{\frac {3}{2}} a e + 3 \, {\left (c d x + a e\right )}^{\frac {5}{2}}\right )} d e}{c} + \frac {63 \, {\left (15 \, \sqrt {c d x + a e} a^{2} e^{2} - 10 \, {\left (c d x + a e\right )}^{\frac {3}{2}} a e + 3 \, {\left (c d x + a e\right )}^{\frac {5}{2}}\right )} a e^{3}}{c^{2} d} - \frac {27 \, {\left (35 \, \sqrt {c d x + a e} a^{3} e^{3} - 35 \, {\left (c d x + a e\right )}^{\frac {3}{2}} a^{2} e^{2} + 21 \, {\left (c d x + a e\right )}^{\frac {5}{2}} a e - 5 \, {\left (c d x + a e\right )}^{\frac {7}{2}}\right )} e^{2}}{c^{2} d} - \frac {9 \, {\left (35 \, \sqrt {c d x + a e} a^{3} e^{3} - 35 \, {\left (c d x + a e\right )}^{\frac {3}{2}} a^{2} e^{2} + 21 \, {\left (c d x + a e\right )}^{\frac {5}{2}} a e - 5 \, {\left (c d x + a e\right )}^{\frac {7}{2}}\right )} a e^{4}}{c^{3} d^{3}} + \frac {{\left (315 \, \sqrt {c d x + a e} a^{4} e^{4} - 420 \, {\left (c d x + a e\right )}^{\frac {3}{2}} a^{3} e^{3} + 378 \, {\left (c d x + a e\right )}^{\frac {5}{2}} a^{2} e^{2} - 180 \, {\left (c d x + a e\right )}^{\frac {7}{2}} a e + 35 \, {\left (c d x + a e\right )}^{\frac {9}{2}}\right )} e^{3}}{c^{3} d^{3}}\right )}}{315 \, c d} \] Input:

integrate((e*x+d)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorit 
hm="giac")
 

Output:

2/315*(315*sqrt(c*d*x + a*e)*a*d^3*e - 105*(3*sqrt(c*d*x + a*e)*a*e - (c*d 
*x + a*e)^(3/2))*d^3 - 315*(3*sqrt(c*d*x + a*e)*a*e - (c*d*x + a*e)^(3/2)) 
*a*d*e^2/c + 63*(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*e/c + 63*(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*e^3/(c^2*d) - 27*(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))*e^2/(c^2*d) - 9*(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^4/(c^3*d^3) + (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))*e^3/(c^3*d^3))/(c*d)
 

Mupad [B] (verification not implemented)

Time = 5.59 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.07 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,e^2\,x^4\,\sqrt {d+e\,x}}{9}-\frac {\sqrt {d+e\,x}\,\left (32\,a^4\,e^7-144\,a^3\,c\,d^2\,e^5+252\,a^2\,c^2\,d^4\,e^3-210\,a\,c^3\,d^6\,e\right )}{315\,c^4\,d^4\,e}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (-2\,a^2\,e^4+9\,a\,c\,d^2\,e^2+63\,c^2\,d^4\right )}{105\,c^2\,d^2}+\frac {x\,\sqrt {d+e\,x}\,\left (16\,a^3\,c\,d\,e^6-72\,a^2\,c^2\,d^3\,e^4+126\,a\,c^3\,d^5\,e^2+210\,c^4\,d^7\right )}{315\,c^4\,d^4\,e}+\frac {2\,e\,x^3\,\left (27\,c\,d^2+a\,e^2\right )\,\sqrt {d+e\,x}}{63\,c\,d}\right )}{x+\frac {d}{e}} \] Input:

int((d + e*x)^(5/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2),x)
 

Output:

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*e^2*x^4*(d + e*x)^(1/2) 
)/9 - ((d + e*x)^(1/2)*(32*a^4*e^7 - 144*a^3*c*d^2*e^5 + 252*a^2*c^2*d^4*e 
^3 - 210*a*c^3*d^6*e))/(315*c^4*d^4*e) + (2*x^2*(d + e*x)^(1/2)*(63*c^2*d^ 
4 - 2*a^2*e^4 + 9*a*c*d^2*e^2))/(105*c^2*d^2) + (x*(d + e*x)^(1/2)*(210*c^ 
4*d^7 + 126*a*c^3*d^5*e^2 - 72*a^2*c^2*d^3*e^4 + 16*a^3*c*d*e^6))/(315*c^4 
*d^4*e) + (2*e*x^3*(a*e^2 + 27*c*d^2)*(d + e*x)^(1/2))/(63*c*d)))/(x + d/e 
)
 

Reduce [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.82 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \sqrt {c d x +a e}\, \left (35 c^{4} d^{4} e^{3} x^{4}+5 a \,c^{3} d^{3} e^{4} x^{3}+135 c^{4} d^{5} e^{2} x^{3}-6 a^{2} c^{2} d^{2} e^{5} x^{2}+27 a \,c^{3} d^{4} e^{3} x^{2}+189 c^{4} d^{6} e \,x^{2}+8 a^{3} c d \,e^{6} x -36 a^{2} c^{2} d^{3} e^{4} x +63 a \,c^{3} d^{5} e^{2} x +105 c^{4} d^{7} x -16 a^{4} e^{7}+72 a^{3} c \,d^{2} e^{5}-126 a^{2} c^{2} d^{4} e^{3}+105 a \,c^{3} d^{6} e \right )}{315 c^{4} d^{4}} \] Input:

int((e*x+d)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
 

Output:

(2*sqrt(a*e + c*d*x)*( - 16*a**4*e**7 + 72*a**3*c*d**2*e**5 + 8*a**3*c*d*e 
**6*x - 126*a**2*c**2*d**4*e**3 - 36*a**2*c**2*d**3*e**4*x - 6*a**2*c**2*d 
**2*e**5*x**2 + 105*a*c**3*d**6*e + 63*a*c**3*d**5*e**2*x + 27*a*c**3*d**4 
*e**3*x**2 + 5*a*c**3*d**3*e**4*x**3 + 105*c**4*d**7*x + 189*c**4*d**6*e*x 
**2 + 135*c**4*d**5*e**2*x**3 + 35*c**4*d**4*e**3*x**4))/(315*c**4*d**4)