\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{10}} \, dx\) [233]

Optimal result

Integrand size = 37, antiderivative size = 231 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 \left (c d^2-a e^2\right )^3 (d+e x)^8}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 \left (c d^2-a e^2\right )^4 (d+e x)^7} \] Output:

2/13*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e^2+c*d^2)/(e*x+d)^10+12/ 
143*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e^2+c*d^2)^2/(e*x+d)^9 
+16/429*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e^2+c*d^2)^3/( 
e*x+d)^8+32/3003*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e^2+c 
*d^2)^4/(e*x+d)^7
 

Mathematica [A] (verified)

Time = 1.88 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.64 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} \left (-231 a^3 e^6+63 a^2 c d e^4 (13 d+2 e x)-7 a c^2 d^2 e^2 \left (143 d^2+52 d e x+8 e^2 x^2\right )+c^3 d^3 \left (429 d^3+286 d^2 e x+104 d e^2 x^2+16 e^3 x^3\right )\right )}{3003 \left (c d^2-a e^2\right )^4 (d+e x)^7} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^10,x]
 

Output:

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-231*a^3*e^6 + 63*a^2*c* 
d*e^4*(13*d + 2*e*x) - 7*a*c^2*d^2*e^2*(143*d^2 + 52*d*e*x + 8*e^2*x^2) + 
c^3*d^3*(429*d^3 + 286*d^2*e*x + 104*d*e^2*x^2 + 16*e^3*x^3)))/(3003*(c*d^ 
2 - a*e^2)^4*(d + e*x)^7)
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {1129, 1129, 1129, 1123}

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.

\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{(d+e x)^{10}} \, dx\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {6 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^9}dx}{13 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{10} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {6 c d \left (\frac {4 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^8}dx}{11 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^9 \left (c d^2-a e^2\right )}\right )}{13 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{10} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {6 c d \left (\frac {4 c d \left (\frac {2 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^7}dx}{9 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^8 \left (c d^2-a e^2\right )}\right )}{11 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^9 \left (c d^2-a e^2\right )}\right )}{13 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{10} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1123

\(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{10} \left (c d^2-a e^2\right )}+\frac {6 c d \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^9 \left (c d^2-a e^2\right )}+\frac {4 c d \left (\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^8 \left (c d^2-a e^2\right )}\right )}{11 \left (c d^2-a e^2\right )}\right )}{13 \left (c d^2-a e^2\right )}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^10,x]
 

Output:

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(13*(c*d^2 - a*e^2)*(d + 
 e*x)^10) + (6*c*d*((2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(11* 
(c*d^2 - a*e^2)*(d + e*x)^9) + (4*c*d*((2*(a*d*e + (c*d^2 + a*e^2)*x + c*d 
*e*x^2)^(7/2))/(9*(c*d^2 - a*e^2)*(d + e*x)^8) + (4*c*d*(a*d*e + (c*d^2 + 
a*e^2)*x + c*d*e*x^2)^(7/2))/(63*(c*d^2 - a*e^2)^2*(d + e*x)^7)))/(11*(c*d 
^2 - a*e^2))))/(13*(c*d^2 - a*e^2))
 

Defintions of rubi rules used

rule 1123

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

rule 1129

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

Maple [A] (verified)

Time = 16.93 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.94

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-16 c^{3} d^{3} e^{3} x^{3}+56 x^{2} a \,c^{2} d^{2} e^{4}-104 c^{3} d^{4} e^{2} x^{2}-126 x \,a^{2} c d \,e^{5}+364 x a \,c^{2} d^{3} e^{3}-286 c^{3} d^{5} e x +231 e^{6} a^{3}-819 d^{2} e^{4} a^{2} c +1001 d^{4} e^{2} a \,c^{2}-429 d^{6} c^{3}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{3003 \left (e x +d \right )^{9} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}\)	\(217\)
orering	\(-\frac {2 \left (-16 c^{3} d^{3} e^{3} x^{3}+56 x^{2} a \,c^{2} d^{2} e^{4}-104 c^{3} d^{4} e^{2} x^{2}-126 x \,a^{2} c d \,e^{5}+364 x a \,c^{2} d^{3} e^{3}-286 c^{3} d^{5} e x +231 e^{6} a^{3}-819 d^{2} e^{4} a^{2} c +1001 d^{4} e^{2} a \,c^{2}-429 d^{6} c^{3}\right ) \left (c d x +a e \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}{3003 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{9}}\)	\(218\)
default	\(\frac {-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{13 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{10}}-\frac {6 d e c \left (-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{11 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{9}}-\frac {4 d e c \left (-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{8}}+\frac {4 d e c \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{7}}\right )}{11 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{13 \left (a \,e^{2}-c \,d^{2}\right )}}{e^{10}}\)	\(293\)
trager	\(-\frac {2 \left (-16 c^{6} d^{6} e^{3} x^{6}+8 a \,c^{5} d^{5} e^{4} x^{5}-104 c^{6} d^{7} e^{2} x^{5}-6 a^{2} c^{4} d^{4} e^{5} x^{4}+52 a \,c^{5} d^{6} e^{3} x^{4}-286 c^{6} d^{8} e \,x^{4}+5 a^{3} c^{3} d^{3} e^{6} x^{3}-39 a^{2} c^{4} d^{5} e^{4} x^{3}+143 a \,c^{5} d^{7} e^{2} x^{3}-429 c^{6} d^{9} x^{3}+371 a^{4} c^{2} d^{2} e^{7} x^{2}-1469 a^{3} c^{3} d^{4} e^{5} x^{2}+2145 a^{2} c^{4} d^{6} e^{3} x^{2}-1287 a \,c^{5} d^{8} e \,x^{2}+567 a^{5} c d \,e^{8} x -2093 a^{4} c^{2} d^{3} e^{6} x +2717 a^{3} c^{3} d^{5} e^{4} x -1287 a^{2} c^{4} d^{7} e^{2} x +231 a^{6} e^{9}-819 a^{5} c \,d^{2} e^{7}+1001 a^{4} c^{2} d^{4} e^{5}-429 a^{3} c^{3} d^{6} e^{3}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{3003 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{7}}\)	\(407\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/(e*x+d)^10,x,method=_RETURNVER 
BOSE)
 

Output:

-2/3003*(c*d*x+a*e)*(-16*c^3*d^3*e^3*x^3+56*a*c^2*d^2*e^4*x^2-104*c^3*d^4* 
e^2*x^2-126*a^2*c*d*e^5*x+364*a*c^2*d^3*e^3*x-286*c^3*d^5*e*x+231*a^3*e^6- 
819*a^2*c*d^2*e^4+1001*a*c^2*d^4*e^2-429*c^3*d^6)*(c*d*e*x^2+a*e^2*x+c*d^2 
*x+a*d*e)^(5/2)/(e*x+d)^9/(a^4*e^8-4*a^3*c*d^2*e^6+6*a^2*c^2*d^4*e^4-4*a*c 
^3*d^6*e^2+c^4*d^8)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (215) = 430\).

Time = 40.70 (sec) , antiderivative size = 823, normalized size of antiderivative = 3.56 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{10}} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^10,x, algorithm= 
"fricas")
 

Output:

2/3003*(16*c^6*d^6*e^3*x^6 + 429*a^3*c^3*d^6*e^3 - 1001*a^4*c^2*d^4*e^5 + 
819*a^5*c*d^2*e^7 - 231*a^6*e^9 + 8*(13*c^6*d^7*e^2 - a*c^5*d^5*e^4)*x^5 + 
 2*(143*c^6*d^8*e - 26*a*c^5*d^6*e^3 + 3*a^2*c^4*d^4*e^5)*x^4 + (429*c^6*d 
^9 - 143*a*c^5*d^7*e^2 + 39*a^2*c^4*d^5*e^4 - 5*a^3*c^3*d^3*e^6)*x^3 + (12 
87*a*c^5*d^8*e - 2145*a^2*c^4*d^6*e^3 + 1469*a^3*c^3*d^4*e^5 - 371*a^4*c^2 
*d^2*e^7)*x^2 + (1287*a^2*c^4*d^7*e^2 - 2717*a^3*c^3*d^5*e^4 + 2093*a^4*c^ 
2*d^3*e^6 - 567*a^5*c*d*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x 
)/(c^4*d^15 - 4*a*c^3*d^13*e^2 + 6*a^2*c^2*d^11*e^4 - 4*a^3*c*d^9*e^6 + a^ 
4*d^7*e^8 + (c^4*d^8*e^7 - 4*a*c^3*d^6*e^9 + 6*a^2*c^2*d^4*e^11 - 4*a^3*c* 
d^2*e^13 + a^4*e^15)*x^7 + 7*(c^4*d^9*e^6 - 4*a*c^3*d^7*e^8 + 6*a^2*c^2*d^ 
5*e^10 - 4*a^3*c*d^3*e^12 + a^4*d*e^14)*x^6 + 21*(c^4*d^10*e^5 - 4*a*c^3*d 
^8*e^7 + 6*a^2*c^2*d^6*e^9 - 4*a^3*c*d^4*e^11 + a^4*d^2*e^13)*x^5 + 35*(c^ 
4*d^11*e^4 - 4*a*c^3*d^9*e^6 + 6*a^2*c^2*d^7*e^8 - 4*a^3*c*d^5*e^10 + a^4* 
d^3*e^12)*x^4 + 35*(c^4*d^12*e^3 - 4*a*c^3*d^10*e^5 + 6*a^2*c^2*d^8*e^7 - 
4*a^3*c*d^6*e^9 + a^4*d^4*e^11)*x^3 + 21*(c^4*d^13*e^2 - 4*a*c^3*d^11*e^4 
+ 6*a^2*c^2*d^9*e^6 - 4*a^3*c*d^7*e^8 + a^4*d^5*e^10)*x^2 + 7*(c^4*d^14*e 
- 4*a*c^3*d^12*e^3 + 6*a^2*c^2*d^10*e^5 - 4*a^3*c*d^8*e^7 + a^4*d^6*e^9)*x 
)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**10,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^10,x, algorithm= 
"maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e*(a*e^2-c*d^2)>0)', see `assume 
?` for mor
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^10,x, algorithm= 
"giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{1,[0,0,7]%%%},[14]%%%}+%%%{%%{[%%%{-14,[0,1,6]%%%},0]: 
[1,0,%%%{
 

Mupad [B] (verification not implemented)

Time = 10.94 (sec) , antiderivative size = 5069, normalized size of antiderivative = 21.94 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\text {Too large to display} \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^10,x)
 

Output:

(((d*((8*c^5*d^6)/(143*e*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e)) - (4*c^4 
*d^4*(21*a*e^2 - 17*c*d^2))/(143*e*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e) 
)))/e + (4*c^3*d^3*(110*a^2*e^4 + 53*c^2*d^4 - 157*a*c*d^2*e^2))/(429*e^2* 
(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d 
*e*x^2)^(1/2))/(d + e*x)^4 - (((2*a^3*e^4)/(13*a*e^3 - 13*c*d^2*e) - (d*(( 
d*((2*c^3*d^4)/(13*a*e^3 - 13*c*d^2*e) - (6*a*c^2*d^2*e^2)/(13*a*e^3 - 13* 
c*d^2*e)))/e + (6*a^2*c*d*e^3)/(13*a*e^3 - 13*c*d^2*e)))/e)*(x*(a*e^2 + c* 
d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^7 + (((d*((16*c^6*d^7)/(1287*e* 
(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e)) - (8*c^5*d^5*(33*a*e^2 - 29*c*d^2 
))/(1287*e*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e))))/e + (8*c^4*d^4*(112* 
a^2*e^4 + 81*c^2*d^4 - 191*a*c*d^2*e^2))/(1287*e^2*(a*e^2 - c*d^2)^3*(5*a* 
e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x 
)^3 + (((d*((32*c^7*d^8)/(9009*e*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)) 
- (16*c^6*d^6*(43*a*e^2 - 39*c*d^2))/(9009*e*(a*e^2 - c*d^2)^4*(3*a*e^3 - 
3*c*d^2*e))))/e + (16*c^5*d^5*(1089*a^2*e^4 + 884*c^2*d^4 - 1963*a*c*d^2*e 
^2))/(45045*e^2*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^ 
2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 + (((38*c^4*d^5 + 94*a*c^3*d^3* 
e^2)/(429*e^2*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)) - (4*c^4*d^5)/(13*e^2 
*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d* 
e*x^2)^(1/2))/(d + e*x)^4 + (((348*c^5*d^6 - 292*a*c^4*d^4*e^2)/(1001*e...
 

Reduce [B] (verification not implemented)

Time = 1.21 (sec) , antiderivative size = 1372, normalized size of antiderivative = 5.94 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{10}} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^10,x)
 

Output:

(2*( - 231*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**6*e**13 + 819*sqrt(d + e*x)* 
sqrt(a*e + c*d*x)*a**5*c*d**2*e**11 - 567*sqrt(d + e*x)*sqrt(a*e + c*d*x)* 
a**5*c*d*e**12*x - 1001*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**2*d**4*e** 
9 + 2093*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**2*d**3*e**10*x - 371*sqrt 
(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**2*d**2*e**11*x**2 + 429*sqrt(d + e*x)* 
sqrt(a*e + c*d*x)*a**3*c**3*d**6*e**7 - 2717*sqrt(d + e*x)*sqrt(a*e + c*d* 
x)*a**3*c**3*d**5*e**8*x + 1469*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3* 
d**4*e**9*x**2 - 5*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**3*e**10*x* 
*3 + 1287*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**4*d**7*e**6*x - 2145*sqr 
t(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**4*d**6*e**7*x**2 + 39*sqrt(d + e*x)*s 
qrt(a*e + c*d*x)*a**2*c**4*d**5*e**8*x**3 + 6*sqrt(d + e*x)*sqrt(a*e + c*d 
*x)*a**2*c**4*d**4*e**9*x**4 + 1287*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5 
*d**8*e**5*x**2 - 143*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d**7*e**6*x** 
3 - 52*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d**6*e**7*x**4 - 8*sqrt(d + 
e*x)*sqrt(a*e + c*d*x)*a*c**5*d**5*e**8*x**5 + 429*sqrt(d + e*x)*sqrt(a*e 
+ c*d*x)*c**6*d**9*e**4*x**3 + 286*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d* 
*8*e**5*x**4 + 104*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**7*e**6*x**5 + 1 
6*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**6*e**7*x**6 - 16*sqrt(e)*sqrt(d) 
*sqrt(c)*c**6*d**13 - 112*sqrt(e)*sqrt(d)*sqrt(c)*c**6*d**12*e*x - 336*sqr 
t(e)*sqrt(d)*sqrt(c)*c**6*d**11*e**2*x**2 - 560*sqrt(e)*sqrt(d)*sqrt(c)...