\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{(d+e x)^6} \, dx\) [209]

Optimal result

Integrand size = 37, antiderivative size = 231 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^6} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 \left (c d^2-a e^2\right )^2 (d+e x)^5}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 \left (c d^2-a e^2\right )^3 (d+e x)^4}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{315 \left (c d^2-a e^2\right )^4 (d+e x)^3} \] Output:

2/9*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e^2+c*d^2)/(e*x+d)^6+4/21* 
c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e^2+c*d^2)^2/(e*x+d)^5+16/ 
105*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e^2+c*d^2)^3/(e*x+ 
d)^4+32/315*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e^2+c*d^2) 
^4/(e*x+d)^3
 

Mathematica [A] (verified)

Time = 10.06 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^6} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-35 a^4 e^7+5 a^3 c d e^5 (27 d-e x)+3 a^2 c^2 d^2 e^3 \left (-63 d^2+9 d e x+2 e^2 x^2\right )+a c^3 d^3 e \left (105 d^3-63 d^2 e x-36 d e^2 x^2-8 e^3 x^3\right )+c^4 d^4 x \left (105 d^3+126 d^2 e x+72 d e^2 x^2+16 e^3 x^3\right )\right )}{315 \left (c d^2-a e^2\right )^4 (d+e x)^5} \] Input:

Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^6,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.68 (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.

\(\displaystyle \int \frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^6} \, dx\)

\(\Big \downarrow \) 1129

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

\(\Big \downarrow \) 1129

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

\(\Big \downarrow \) 1129

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

\(\Big \downarrow \) 1123

\(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 (d+e x)^6 \left (c d^2-a e^2\right )}+\frac {2 c d \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 (d+e x)^5 \left (c d^2-a e^2\right )}+\frac {4 c d \left (\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 (d+e x)^3 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 (d+e x)^4 \left (c d^2-a e^2\right )}\right )}{7 \left (c d^2-a e^2\right )}\right )}{3 \left (c d^2-a e^2\right )}\)

Input:

Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^6,x]
 

Output:

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(9*(c*d^2 - a*e^2)*(d + 
e*x)^6) + (2*c*d*((2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(7*(c* 
d^2 - a*e^2)*(d + e*x)^5) + (4*c*d*((2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e* 
x^2)^(3/2))/(5*(c*d^2 - a*e^2)*(d + e*x)^4) + (4*c*d*(a*d*e + (c*d^2 + a*e 
^2)*x + c*d*e*x^2)^(3/2))/(15*(c*d^2 - a*e^2)^2*(d + e*x)^3)))/(7*(c*d^2 - 
 a*e^2))))/(3*(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 = 3.63 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.94

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-16 c^{3} d^{3} e^{3} x^{3}+24 x^{2} a \,c^{2} d^{2} e^{4}-72 c^{3} d^{4} e^{2} x^{2}-30 x \,a^{2} c d \,e^{5}+108 x a \,c^{2} d^{3} e^{3}-126 c^{3} d^{5} e x +35 e^{6} a^{3}-135 d^{2} e^{4} a^{2} c +189 d^{4} e^{2} a \,c^{2}-105 d^{6} c^{3}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{315 \left (e x +d \right )^{5} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}\)	\(217\)
orering	\(-\frac {2 \left (-16 c^{3} d^{3} e^{3} x^{3}+24 x^{2} a \,c^{2} d^{2} e^{4}-72 c^{3} d^{4} e^{2} x^{2}-30 x \,a^{2} c d \,e^{5}+108 x a \,c^{2} d^{3} e^{3}-126 c^{3} d^{5} e x +35 e^{6} a^{3}-135 d^{2} e^{4} a^{2} c +189 d^{4} e^{2} a \,c^{2}-105 d^{6} c^{3}\right ) \left (c d x +a e \right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{315 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{5}}\)	\(218\)
trager	\(-\frac {2 \left (-16 c^{4} d^{4} e^{3} x^{4}+8 a \,c^{3} d^{3} e^{4} x^{3}-72 c^{4} d^{5} e^{2} x^{3}-6 a^{2} c^{2} d^{2} e^{5} x^{2}+36 a \,c^{3} d^{4} e^{3} x^{2}-126 c^{4} d^{6} e \,x^{2}+5 d \,e^{6} c \,a^{3} x -27 d^{3} e^{4} a^{2} c^{2} x +63 d^{5} e^{2} a \,c^{3} x -105 d^{7} c^{4} x +35 e^{7} a^{4}-135 a^{3} c \,d^{2} e^{5}+189 a^{2} c^{2} d^{4} e^{3}-105 a \,c^{3} d^{6} e \right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{315 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{5}}\)	\(271\)
default	\(\frac {-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{9 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{6}}-\frac {2 d e c \left (-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{7 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{5}}-\frac {4 d e c \left (-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{4}}+\frac {4 d e c \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{7 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )}}{e^{6}}\)	\(293\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/(e*x+d)^6,x,method=_RETURNVERB 
OSE)
 

Output:

-2/315*(c*d*x+a*e)*(-16*c^3*d^3*e^3*x^3+24*a*c^2*d^2*e^4*x^2-72*c^3*d^4*e^ 
2*x^2-30*a^2*c*d*e^5*x+108*a*c^2*d^3*e^3*x-126*c^3*d^5*e*x+35*a^3*e^6-135* 
a^2*c*d^2*e^4+189*a*c^2*d^4*e^2-105*c^3*d^6)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a* 
d*e)^(1/2)/(e*x+d)^5/(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. 572 vs. \(2 (215) = 430\).

Time = 4.76 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.48 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^6} \, dx=\frac {2 \, {\left (16 \, c^{4} d^{4} e^{3} x^{4} + 105 \, a c^{3} d^{6} e - 189 \, a^{2} c^{2} d^{4} e^{3} + 135 \, a^{3} c d^{2} e^{5} - 35 \, a^{4} e^{7} + 8 \, {\left (9 \, c^{4} d^{5} e^{2} - a c^{3} d^{3} e^{4}\right )} x^{3} + 6 \, {\left (21 \, c^{4} d^{6} e - 6 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (105 \, c^{4} d^{7} - 63 \, a c^{3} d^{5} e^{2} + 27 \, a^{2} c^{2} d^{3} e^{4} - 5 \, a^{3} c d e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{315 \, {\left (c^{4} d^{13} - 4 \, a c^{3} d^{11} e^{2} + 6 \, a^{2} c^{2} d^{9} e^{4} - 4 \, a^{3} c d^{7} e^{6} + a^{4} d^{5} e^{8} + {\left (c^{4} d^{8} e^{5} - 4 \, a c^{3} d^{6} e^{7} + 6 \, a^{2} c^{2} d^{4} e^{9} - 4 \, a^{3} c d^{2} e^{11} + a^{4} e^{13}\right )} x^{5} + 5 \, {\left (c^{4} d^{9} e^{4} - 4 \, a c^{3} d^{7} e^{6} + 6 \, a^{2} c^{2} d^{5} e^{8} - 4 \, a^{3} c d^{3} e^{10} + a^{4} d e^{12}\right )} x^{4} + 10 \, {\left (c^{4} d^{10} e^{3} - 4 \, a c^{3} d^{8} e^{5} + 6 \, a^{2} c^{2} d^{6} e^{7} - 4 \, a^{3} c d^{4} e^{9} + a^{4} d^{2} e^{11}\right )} x^{3} + 10 \, {\left (c^{4} d^{11} e^{2} - 4 \, a c^{3} d^{9} e^{4} + 6 \, a^{2} c^{2} d^{7} e^{6} - 4 \, a^{3} c d^{5} e^{8} + a^{4} d^{3} e^{10}\right )} x^{2} + 5 \, {\left (c^{4} d^{12} e - 4 \, a c^{3} d^{10} e^{3} + 6 \, a^{2} c^{2} d^{8} e^{5} - 4 \, a^{3} c d^{6} e^{7} + a^{4} d^{4} e^{9}\right )} x\right )}} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^6,x, algorithm=" 
fricas")
 

Output:

2/315*(16*c^4*d^4*e^3*x^4 + 105*a*c^3*d^6*e - 189*a^2*c^2*d^4*e^3 + 135*a^ 
3*c*d^2*e^5 - 35*a^4*e^7 + 8*(9*c^4*d^5*e^2 - a*c^3*d^3*e^4)*x^3 + 6*(21*c 
^4*d^6*e - 6*a*c^3*d^4*e^3 + a^2*c^2*d^2*e^5)*x^2 + (105*c^4*d^7 - 63*a*c^ 
3*d^5*e^2 + 27*a^2*c^2*d^3*e^4 - 5*a^3*c*d*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e 
+ (c*d^2 + a*e^2)*x)/(c^4*d^13 - 4*a*c^3*d^11*e^2 + 6*a^2*c^2*d^9*e^4 - 4* 
a^3*c*d^7*e^6 + a^4*d^5*e^8 + (c^4*d^8*e^5 - 4*a*c^3*d^6*e^7 + 6*a^2*c^2*d 
^4*e^9 - 4*a^3*c*d^2*e^11 + a^4*e^13)*x^5 + 5*(c^4*d^9*e^4 - 4*a*c^3*d^7*e 
^6 + 6*a^2*c^2*d^5*e^8 - 4*a^3*c*d^3*e^10 + a^4*d*e^12)*x^4 + 10*(c^4*d^10 
*e^3 - 4*a*c^3*d^8*e^5 + 6*a^2*c^2*d^6*e^7 - 4*a^3*c*d^4*e^9 + a^4*d^2*e^1 
1)*x^3 + 10*(c^4*d^11*e^2 - 4*a*c^3*d^9*e^4 + 6*a^2*c^2*d^7*e^6 - 4*a^3*c* 
d^5*e^8 + a^4*d^3*e^10)*x^2 + 5*(c^4*d^12*e - 4*a*c^3*d^10*e^3 + 6*a^2*c^2 
*d^8*e^5 - 4*a^3*c*d^6*e^7 + a^4*d^4*e^9)*x)
 

Sympy [F]

\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^6} \, dx=\int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{\left (d + e x\right )^{6}}\, dx \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**6,x)
                                                                                    
                                                                                    
 

Output:

Integral(sqrt((d + e*x)*(a*e + c*d*x))/(d + e*x)**6, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^6,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. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^6} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^6,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,[0,0,5]%%%},[10]%%%}+%%%{%%{[%%%{-10,[0,1,4]%%%},0]: 
[1,0,%%%{
 

Mupad [B] (verification not implemented)

Time = 6.70 (sec) , antiderivative size = 1192, normalized size of antiderivative = 5.16 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(d + e*x)^6,x)
 

Output:

(((4*c^2*d^3)/(9*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)) - (4*a*c*d*e^2)/(9 
*(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 - (((2*a*e^2)/(9*a*e^3 - 9*c*d^2*e) - (2*c*d^2)/ 
(9*a*e^3 - 9*c*d^2*e))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + 
 e*x)^5 + (((4*c^3*d^4 + 4*a*c^2*d^2*e^2)/(63*(a*e^2 - c*d^2)^2*(5*a*e^3 - 
 5*c*d^2*e)) - (8*c^3*d^4)/(63*(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 + (((8*c^4*d^5 + 
 8*a*c^3*d^3*e^2)/(315*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)) - (16*c^4* 
d^5)/(315*(a*e^2 - c*d^2)^3*(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 + (((16*c^5*d^6 + 16*a*c^4*d^4*e^2)/( 
945*e*(a*e^2 - c*d^2)^5) - (32*c^5*d^6)/(945*e*(a*e^2 - c*d^2)^5))*(x*(a*e 
^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) + (((2*c^2*d^3 + 2*a*c*d 
*e^2)/(9*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)) - (4*c^2*d^3)/(9*(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 + (((8*c^3*d^4)/(63*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e 
)) - (8*a*c^2*d^2*e^2)/(63*(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 + (((16*c^4*d^5)/(31 
5*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)) - (16*a*c^3*d^3*e^2)/(315*(a*e^ 
2 - c*d^2)^3*(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 + (((32*c^5*d^6)/(945*e*(a*e^2 - c*d^2)^5) - (32*...
 

Reduce [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 926, normalized size of antiderivative = 4.01 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^6,x)
 

Output:

(2*( - 35*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*e**9 + 135*sqrt(d + e*x)*sq 
rt(a*e + c*d*x)*a**3*c*d**2*e**7 - 5*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3* 
c*d*e**8*x - 189*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**4*e**5 + 27* 
sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**3*e**6*x + 6*sqrt(d + e*x)*sq 
rt(a*e + c*d*x)*a**2*c**2*d**2*e**7*x**2 + 105*sqrt(d + e*x)*sqrt(a*e + c* 
d*x)*a*c**3*d**6*e**3 - 63*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**5*e** 
4*x - 36*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**4*e**5*x**2 - 8*sqrt(d 
+ e*x)*sqrt(a*e + c*d*x)*a*c**3*d**3*e**6*x**3 + 105*sqrt(d + e*x)*sqrt(a* 
e + c*d*x)*c**4*d**7*e**2*x + 126*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d** 
6*e**3*x**2 + 72*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d**5*e**4*x**3 + 16* 
sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d**4*e**5*x**4 - 16*sqrt(e)*sqrt(d)*s 
qrt(c)*c**4*d**9 - 80*sqrt(e)*sqrt(d)*sqrt(c)*c**4*d**8*e*x - 160*sqrt(e)* 
sqrt(d)*sqrt(c)*c**4*d**7*e**2*x**2 - 160*sqrt(e)*sqrt(d)*sqrt(c)*c**4*d** 
6*e**3*x**3 - 80*sqrt(e)*sqrt(d)*sqrt(c)*c**4*d**5*e**4*x**4 - 16*sqrt(e)* 
sqrt(d)*sqrt(c)*c**4*d**4*e**5*x**5))/(315*e**2*(a**4*d**5*e**8 + 5*a**4*d 
**4*e**9*x + 10*a**4*d**3*e**10*x**2 + 10*a**4*d**2*e**11*x**3 + 5*a**4*d* 
e**12*x**4 + a**4*e**13*x**5 - 4*a**3*c*d**7*e**6 - 20*a**3*c*d**6*e**7*x 
- 40*a**3*c*d**5*e**8*x**2 - 40*a**3*c*d**4*e**9*x**3 - 20*a**3*c*d**3*e** 
10*x**4 - 4*a**3*c*d**2*e**11*x**5 + 6*a**2*c**2*d**9*e**4 + 30*a**2*c**2* 
d**8*e**5*x + 60*a**2*c**2*d**7*e**6*x**2 + 60*a**2*c**2*d**6*e**7*x**3...