\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{7/2}}{(d+e x)^{12}} \, dx\) [247]

Optimal result

Integrand size = 37, antiderivative size = 231 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{12}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{15 \left (c d^2-a e^2\right ) (d+e x)^{12}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{65 \left (c d^2-a e^2\right )^2 (d+e x)^{11}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{715 \left (c d^2-a e^2\right )^3 (d+e x)^{10}}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{6435 \left (c d^2-a e^2\right )^4 (d+e x)^9} \] Output:

2/15*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(9/2)/(-a*e^2+c*d^2)/(e*x+d)^12+4/6 
5*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(9/2)/(-a*e^2+c*d^2)^2/(e*x+d)^11+ 
16/715*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(9/2)/(-a*e^2+c*d^2)^3/(e 
*x+d)^10+32/6435*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(9/2)/(-a*e^2+c 
*d^2)^4/(e*x+d)^9
 

Mathematica [A] (verified)

Time = 2.42 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.64 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{12}} \, dx=\frac {2 (a e+c d x)^4 \sqrt {(a e+c d x) (d+e x)} \left (-429 a^3 e^6+99 a^2 c d e^4 (15 d+2 e x)-9 a c^2 d^2 e^2 \left (195 d^2+60 d e x+8 e^2 x^2\right )+c^3 d^3 \left (715 d^3+390 d^2 e x+120 d e^2 x^2+16 e^3 x^3\right )\right )}{6435 \left (c d^2-a e^2\right )^4 (d+e x)^8} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x)^12,x]
 

Output:

(2*(a*e + c*d*x)^4*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-429*a^3*e^6 + 99*a^2*c* 
d*e^4*(15*d + 2*e*x) - 9*a*c^2*d^2*e^2*(195*d^2 + 60*d*e*x + 8*e^2*x^2) + 
c^3*d^3*(715*d^3 + 390*d^2*e*x + 120*d*e^2*x^2 + 16*e^3*x^3)))/(6435*(c*d^ 
2 - a*e^2)^4*(d + e*x)^8)
 

Rubi [A] (verified)

Time = 0.70 (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.

\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{(d+e x)^{12}} \, dx\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {2 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{7/2}}{(d+e x)^{11}}dx}{5 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{15 (d+e x)^{12} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {2 c d \left (\frac {4 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{7/2}}{(d+e x)^{10}}dx}{13 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{13 (d+e x)^{11} \left (c d^2-a e^2\right )}\right )}{5 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{15 (d+e x)^{12} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {2 c d \left (\frac {4 c d \left (\frac {2 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{7/2}}{(d+e x)^9}dx}{11 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{11 (d+e x)^{10} \left (c d^2-a e^2\right )}\right )}{13 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{13 (d+e x)^{11} \left (c d^2-a e^2\right )}\right )}{5 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{15 (d+e x)^{12} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1123

\(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{15 (d+e x)^{12} \left (c d^2-a e^2\right )}+\frac {2 c d \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{13 (d+e x)^{11} \left (c d^2-a e^2\right )}+\frac {4 c d \left (\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{99 (d+e x)^9 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{11 (d+e x)^{10} \left (c d^2-a e^2\right )}\right )}{13 \left (c d^2-a e^2\right )}\right )}{5 \left (c d^2-a e^2\right )}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x)^12,x]
 

Output:

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(9/2))/(15*(c*d^2 - a*e^2)*(d + 
 e*x)^12) + (2*c*d*((2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(9/2))/(13* 
(c*d^2 - a*e^2)*(d + e*x)^11) + (4*c*d*((2*(a*d*e + (c*d^2 + a*e^2)*x + c* 
d*e*x^2)^(9/2))/(11*(c*d^2 - a*e^2)*(d + e*x)^10) + (4*c*d*(a*d*e + (c*d^2 
 + a*e^2)*x + c*d*e*x^2)^(9/2))/(99*(c*d^2 - a*e^2)^2*(d + e*x)^9)))/(13*( 
c*d^2 - a*e^2))))/(5*(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 = 18.61 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.94

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-16 c^{3} d^{3} e^{3} x^{3}+72 x^{2} a \,c^{2} d^{2} e^{4}-120 c^{3} d^{4} e^{2} x^{2}-198 x \,a^{2} c d \,e^{5}+540 x a \,c^{2} d^{3} e^{3}-390 c^{3} d^{5} e x +429 e^{6} a^{3}-1485 d^{2} e^{4} a^{2} c +1755 d^{4} e^{2} a \,c^{2}-715 d^{6} c^{3}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {7}{2}}}{6435 \left (e x +d \right )^{11} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}\)	\(217\)
orering	\(-\frac {2 \left (-16 c^{3} d^{3} e^{3} x^{3}+72 x^{2} a \,c^{2} d^{2} e^{4}-120 c^{3} d^{4} e^{2} x^{2}-198 x \,a^{2} c d \,e^{5}+540 x a \,c^{2} d^{3} e^{3}-390 c^{3} d^{5} e x +429 e^{6} a^{3}-1485 d^{2} e^{4} a^{2} c +1755 d^{4} e^{2} a \,c^{2}-715 d^{6} c^{3}\right ) \left (c d x +a e \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {7}{2}}}{6435 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{11}}\)	\(218\)
default	\(\frac {-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{15 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{12}}-\frac {2 d e c \left (-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{13 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{11}}-\frac {4 d e c \left (-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{11 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{10}}+\frac {4 d e c \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{99 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{9}}\right )}{13 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{5 \left (a \,e^{2}-c \,d^{2}\right )}}{e^{12}}\)	\(293\)
trager	\(-\frac {2 \left (-16 c^{7} d^{7} e^{3} x^{7}+8 a \,c^{6} d^{6} e^{4} x^{6}-120 c^{7} d^{8} e^{2} x^{6}-6 a^{2} c^{5} d^{5} e^{5} x^{5}+60 a \,c^{6} d^{7} e^{3} x^{5}-390 c^{7} d^{9} e \,x^{5}+5 a^{3} c^{4} d^{4} e^{6} x^{4}-45 a^{2} c^{5} d^{6} e^{4} x^{4}+195 a \,c^{6} d^{8} e^{2} x^{4}-715 c^{7} d^{10} x^{4}+800 a^{4} c^{3} d^{3} e^{7} x^{3}-3180 a^{3} c^{4} d^{5} e^{5} x^{3}+4680 a^{2} c^{5} d^{7} e^{3} x^{3}-2860 a \,c^{6} d^{9} e \,x^{3}+1854 a^{5} c^{2} d^{2} e^{8} x^{2}-6870 a^{4} c^{3} d^{4} e^{6} x^{2}+8970 a^{3} c^{4} d^{6} e^{4} x^{2}-4290 a^{2} c^{5} d^{8} e^{2} x^{2}+1518 a^{6} c d \,e^{9} x -5400 a^{5} c^{2} d^{3} e^{7} x +6630 a^{4} c^{3} d^{5} e^{5} x -2860 a^{3} c^{4} d^{7} e^{3} x +429 a^{7} e^{10}-1485 a^{6} d^{2} c \,e^{8}+1755 a^{5} c^{2} d^{4} e^{6}-715 a^{4} c^{3} d^{6} e^{4}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{6435 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{8}}\)	\(475\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(7/2)/(e*x+d)^12,x,method=_RETURNVER 
BOSE)
 

Output:

-2/6435*(c*d*x+a*e)*(-16*c^3*d^3*e^3*x^3+72*a*c^2*d^2*e^4*x^2-120*c^3*d^4* 
e^2*x^2-198*a^2*c*d*e^5*x+540*a*c^2*d^3*e^3*x-390*c^3*d^5*e*x+429*a^3*e^6- 
1485*a^2*c*d^2*e^4+1755*a*c^2*d^4*e^2-715*c^3*d^6)*(c*d*e*x^2+a*e^2*x+c*d^ 
2*x+a*d*e)^(7/2)/(e*x+d)^11/(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. 951 vs. \(2 (215) = 430\).

Time = 97.84 (sec) , antiderivative size = 951, normalized size of antiderivative = 4.12 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{12}} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^12,x, algorithm= 
"fricas")
 

Output:

2/6435*(16*c^7*d^7*e^3*x^7 + 715*a^4*c^3*d^6*e^4 - 1755*a^5*c^2*d^4*e^6 + 
1485*a^6*c*d^2*e^8 - 429*a^7*e^10 + 8*(15*c^7*d^8*e^2 - a*c^6*d^6*e^4)*x^6 
 + 6*(65*c^7*d^9*e - 10*a*c^6*d^7*e^3 + a^2*c^5*d^5*e^5)*x^5 + 5*(143*c^7* 
d^10 - 39*a*c^6*d^8*e^2 + 9*a^2*c^5*d^6*e^4 - a^3*c^4*d^4*e^6)*x^4 + 20*(1 
43*a*c^6*d^9*e - 234*a^2*c^5*d^7*e^3 + 159*a^3*c^4*d^5*e^5 - 40*a^4*c^3*d^ 
3*e^7)*x^3 + 6*(715*a^2*c^5*d^8*e^2 - 1495*a^3*c^4*d^6*e^4 + 1145*a^4*c^3* 
d^4*e^6 - 309*a^5*c^2*d^2*e^8)*x^2 + 2*(1430*a^3*c^4*d^7*e^3 - 3315*a^4*c^ 
3*d^5*e^5 + 2700*a^5*c^2*d^3*e^7 - 759*a^6*c*d*e^9)*x)*sqrt(c*d*e*x^2 + a* 
d*e + (c*d^2 + a*e^2)*x)/(c^4*d^16 - 4*a*c^3*d^14*e^2 + 6*a^2*c^2*d^12*e^4 
 - 4*a^3*c*d^10*e^6 + a^4*d^8*e^8 + (c^4*d^8*e^8 - 4*a*c^3*d^6*e^10 + 6*a^ 
2*c^2*d^4*e^12 - 4*a^3*c*d^2*e^14 + a^4*e^16)*x^8 + 8*(c^4*d^9*e^7 - 4*a*c 
^3*d^7*e^9 + 6*a^2*c^2*d^5*e^11 - 4*a^3*c*d^3*e^13 + a^4*d*e^15)*x^7 + 28* 
(c^4*d^10*e^6 - 4*a*c^3*d^8*e^8 + 6*a^2*c^2*d^6*e^10 - 4*a^3*c*d^4*e^12 + 
a^4*d^2*e^14)*x^6 + 56*(c^4*d^11*e^5 - 4*a*c^3*d^9*e^7 + 6*a^2*c^2*d^7*e^9 
 - 4*a^3*c*d^5*e^11 + a^4*d^3*e^13)*x^5 + 70*(c^4*d^12*e^4 - 4*a*c^3*d^10* 
e^6 + 6*a^2*c^2*d^8*e^8 - 4*a^3*c*d^6*e^10 + a^4*d^4*e^12)*x^4 + 56*(c^4*d 
^13*e^3 - 4*a*c^3*d^11*e^5 + 6*a^2*c^2*d^9*e^7 - 4*a^3*c*d^7*e^9 + a^4*d^5 
*e^11)*x^3 + 28*(c^4*d^14*e^2 - 4*a*c^3*d^12*e^4 + 6*a^2*c^2*d^10*e^6 - 4* 
a^3*c*d^8*e^8 + a^4*d^6*e^10)*x^2 + 8*(c^4*d^15*e - 4*a*c^3*d^13*e^3 + 6*a 
^2*c^2*d^11*e^5 - 4*a^3*c*d^9*e^7 + a^4*d^7*e^9)*x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{12}} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(7/2)/(e*x+d)**12,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{12}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^12,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. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{12}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^12,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,[0,0,8]%%%},[16]%%%}+%%%{%%{[%%%{-16,[0,1,7]%%%},0]: 
[1,0,%%%{
 

Mupad [B] (verification not implemented)

Time = 15.52 (sec) , antiderivative size = 8811, normalized size of antiderivative = 38.14 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{12}} \, dx=\text {Too large to display} \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(7/2)/(d + e*x)^12,x)
 

Output:

(((d*((8*c^6*d^7)/(195*e^2*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e)) - (4*c 
^5*d^5*(25*a*e^2 - 21*c*d^2))/(195*e^2*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^ 
2*e))))/e + (4*c^4*d^4*(376*a^2*e^4 - 1901*c^2*d^4 + 1723*a*c*d^2*e^2))/(1 
9305*e^3*(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 + (((d*((16*c^7*d^8)/(2145*e^2*(a*e^2 
- c*d^2)^3*(5*a*e^3 - 5*c*d^2*e)) - (8*c^6*d^6*(39*a*e^2 - 35*c*d^2))/(214 
5*e^2*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e))))/e - (8*c^5*d^5*(1011*a^2* 
e^4 + 1270*c^2*d^4 - 2295*a*c*d^2*e^2))/(15015*e^3*(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^8*d^9)/(19305*e^2*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e 
)) - (16*c^7*d^7*(51*a*e^2 - 47*c*d^2))/(19305*e^2*(a*e^2 - c*d^2)^4*(3*a* 
e^3 - 3*c*d^2*e))))/e - (16*c^6*d^6*(3259*a^2*e^4 + 3504*c^2*d^4 - 6773*a* 
c*d^2*e^2))/(96525*e^3*(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 - (((950*c^5*d^6 - 2666* 
a*c^4*d^4*e^2)/(6435*e^3*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)) + (4*c^5*d 
^6)/(15*e^3*(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 - (((18244*c^6*d^7 - 20092*a*c^5*d^5* 
e^2)/(45045*e^3*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)) + (8*c^6*d^7)/(19 
5*e^3*(a*e^2 - c*d^2)^2*(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*((d*((d*((256*c^11*d^12)/(20270...
 

Reduce [B] (verification not implemented)

Time = 2.31 (sec) , antiderivative size = 1595, normalized size of antiderivative = 6.90 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{12}} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^12,x)
 

Output:

(2*( - 429*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**7*e**15 + 1485*sqrt(d + e*x) 
*sqrt(a*e + c*d*x)*a**6*c*d**2*e**13 - 1518*sqrt(d + e*x)*sqrt(a*e + c*d*x 
)*a**6*c*d*e**14*x - 1755*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c**2*d**4*e 
**11 + 5400*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c**2*d**3*e**12*x - 1854* 
sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c**2*d**2*e**13*x**2 + 715*sqrt(d + e 
*x)*sqrt(a*e + c*d*x)*a**4*c**3*d**6*e**9 - 6630*sqrt(d + e*x)*sqrt(a*e + 
c*d*x)*a**4*c**3*d**5*e**10*x + 6870*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4* 
c**3*d**4*e**11*x**2 - 800*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**3*d**3* 
e**12*x**3 + 2860*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**4*d**7*e**8*x - 
8970*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**4*d**6*e**9*x**2 + 3180*sqrt( 
d + e*x)*sqrt(a*e + c*d*x)*a**3*c**4*d**5*e**10*x**3 - 5*sqrt(d + e*x)*sqr 
t(a*e + c*d*x)*a**3*c**4*d**4*e**11*x**4 + 4290*sqrt(d + e*x)*sqrt(a*e + c 
*d*x)*a**2*c**5*d**8*e**7*x**2 - 4680*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2 
*c**5*d**7*e**8*x**3 + 45*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**5*d**6*e 
**9*x**4 + 6*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**5*d**5*e**10*x**5 + 2 
860*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**6*d**9*e**6*x**3 - 195*sqrt(d + e 
*x)*sqrt(a*e + c*d*x)*a*c**6*d**8*e**7*x**4 - 60*sqrt(d + e*x)*sqrt(a*e + 
c*d*x)*a*c**6*d**7*e**8*x**5 - 8*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**6*d* 
*6*e**9*x**6 + 715*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**7*d**10*e**5*x**4 + 
390*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**7*d**9*e**6*x**5 + 120*sqrt(d + ...