\(\int \frac {x^2}{(d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [125]

Optimal result

Integrand size = 40, antiderivative size = 225 \[ \int \frac {x^2}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 d^2}{5 e^2 \left (c d^2-a e^2\right ) (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {4 d \left (2 c d^2-5 a e^2\right )}{15 e^2 \left (c d^2-a e^2\right )^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 \left (c^2 d^4-10 a c d^2 e^2-15 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right )}{15 e^2 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \] Output:

2/5*d^2/e^2/(-a*e^2+c*d^2)/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/ 
2)-4/15*d*(-5*a*e^2+2*c*d^2)/e^2/(-a*e^2+c*d^2)^2/(e*x+d)/(a*d*e+(a*e^2+c* 
d^2)*x+c*d*e*x^2)^(1/2)+2/15*(-15*a^2*e^4-10*a*c*d^2*e^2+c^2*d^4)*(2*c*d*e 
*x+a*e^2+c*d^2)/e^2/(-a*e^2+c*d^2)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/ 
2)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.70 \[ \int \frac {x^2}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \left (-c^3 d^5 x^2 (5 d+2 e x)+a^3 e^4 \left (8 d^2+20 d e x+15 e^2 x^2\right )+a c^2 d^3 e x \left (20 d^2+49 d e x+20 e^2 x^2\right )+a^2 c d e^2 \left (40 d^3+104 d^2 e x+85 d e^2 x^2+30 e^3 x^3\right )\right )}{15 \left (c d^2-a e^2\right )^4 (d+e x)^2 \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

(-2*(-(c^3*d^5*x^2*(5*d + 2*e*x)) + a^3*e^4*(8*d^2 + 20*d*e*x + 15*e^2*x^2 
) + a*c^2*d^3*e*x*(20*d^2 + 49*d*e*x + 20*e^2*x^2) + a^2*c*d*e^2*(40*d^3 + 
 104*d^2*e*x + 85*d*e^2*x^2 + 30*e^3*x^3)))/(15*(c*d^2 - a*e^2)^4*(d + e*x 
)^2*Sqrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.28, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1267, 27, 1220, 1129, 1088}

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.

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

\(\Big \downarrow \) 1267

\(\displaystyle -\frac {\int \frac {e \left (d \left (c d^2+3 a e^2\right )+e \left (5 c d^2+3 a e^2\right ) x\right )}{2 (d+e x)^2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{2 c d e^3}-\frac {1}{2 c d e^2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {d \left (c d^2+3 a e^2\right )+e \left (5 c d^2+3 a e^2\right ) x}{(d+e x)^2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{4 c d e^2}-\frac {1}{2 c d e^2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1220

\(\displaystyle -\frac {\frac {\left (-15 a^2 e^4-10 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{(d+e x) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{5 \left (c d^2-a e^2\right )}-\frac {8 c d^3}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{4 c d e^2}-\frac {1}{2 c d e^2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1129

\(\displaystyle -\frac {\frac {\left (-15 a^2 e^4-10 a c d^2 e^2+c^2 d^4\right ) \left (\frac {4 c d \int \frac {1}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 \left (c d^2-a e^2\right )}+\frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{5 \left (c d^2-a e^2\right )}-\frac {8 c d^3}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{4 c d e^2}-\frac {1}{2 c d e^2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1088

\(\displaystyle -\frac {\frac {\left (-15 a^2 e^4-10 a c d^2 e^2+c^2 d^4\right ) \left (\frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {8 c d \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{5 \left (c d^2-a e^2\right )}-\frac {8 c d^3}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{4 c d e^2}-\frac {1}{2 c d e^2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

Input:

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

Output:

-1/2*1/(c*d*e^2*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - ( 
(-8*c*d^3)/(5*(c*d^2 - a*e^2)*(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + 
 c*d*e*x^2]) + ((c^2*d^4 - 10*a*c*d^2*e^2 - 15*a^2*e^4)*(2/(3*(c*d^2 - a*e 
^2)*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (8*c*d*(c*d^2 
 + a*e^2 + 2*c*d*e*x))/(3*(c*d^2 - a*e^2)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x 
 + c*d*e*x^2])))/(5*(c*d^2 - a*e^2)))/(4*c*d*e^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1088

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 
2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && 
 NeQ[b^2 - 4*a*c, 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]
 

rule 1220

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x 
^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e 
*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1))   Int[(d + e*x 
)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, 
 x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p + 1, 0 
]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 
]
 

rule 1267

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b 
*x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m 
 + n + 2*p + 1))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m 
 + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d 
+ e*x)^(n - 2)*(b*d*e*(p + 1) + a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) 
 - e*(2*c*d - b*e)*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
, m, p}, x] && IGtQ[n, 1] && IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
 

Maple [A] (verified)

Time = 2.38 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.11

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (30 a^{2} c d \,e^{5} x^{3}+20 a \,c^{2} d^{3} e^{3} x^{3}-2 c^{3} d^{5} e \,x^{3}+15 a^{3} e^{6} x^{2}+85 a^{2} c \,d^{2} e^{4} x^{2}+49 a \,c^{2} d^{4} e^{2} x^{2}-5 c^{3} d^{6} x^{2}+20 a^{3} d \,e^{5} x +104 a^{2} c \,d^{3} e^{3} x +20 a \,c^{2} d^{5} e x +8 a^{3} d^{2} e^{4}+40 a^{2} c \,d^{4} e^{2}\right )}{15 \left (e x +d \right ) \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\)	\(249\)
orering	\(-\frac {2 \left (30 a^{2} c d \,e^{5} x^{3}+20 a \,c^{2} d^{3} e^{3} x^{3}-2 c^{3} d^{5} e \,x^{3}+15 a^{3} e^{6} x^{2}+85 a^{2} c \,d^{2} e^{4} x^{2}+49 a \,c^{2} d^{4} e^{2} x^{2}-5 c^{3} d^{6} x^{2}+20 a^{3} d \,e^{5} x +104 a^{2} c \,d^{3} e^{3} x +20 a \,c^{2} d^{5} e x +8 a^{3} d^{2} e^{4}+40 a^{2} c \,d^{4} e^{2}\right ) \left (c d x +a e \right )}{15 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}\)	\(250\)
trager	\(-\frac {2 \left (30 a^{2} c d \,e^{5} x^{3}+20 a \,c^{2} d^{3} e^{3} x^{3}-2 c^{3} d^{5} e \,x^{3}+15 a^{3} e^{6} x^{2}+85 a^{2} c \,d^{2} e^{4} x^{2}+49 a \,c^{2} d^{4} e^{2} x^{2}-5 c^{3} d^{6} x^{2}+20 a^{3} d \,e^{5} x +104 a^{2} c \,d^{3} e^{3} x +20 a \,c^{2} d^{5} e x +8 a^{3} d^{2} e^{4}+40 a^{2} c \,d^{4} e^{2}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{15 \left (c d x +a e \right ) \left (a \,e^{2}-c \,d^{2}\right ) \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right ) \left (e x +d \right )^{3}}\)	\(252\)
default	\(\frac {4 c d x e +2 a \,e^{2}+2 c \,d^{2}}{e^{2} \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}+\frac {d^{2} \left (-\frac {2}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {6 d e c \left (-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 d e c \left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{e^{4}}-\frac {2 d \left (-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 d e c \left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{e^{3}}\)	\(455\)

Input:

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

Output:

-2/15*(c*d*x+a*e)*(30*a^2*c*d*e^5*x^3+20*a*c^2*d^3*e^3*x^3-2*c^3*d^5*e*x^3 
+15*a^3*e^6*x^2+85*a^2*c*d^2*e^4*x^2+49*a*c^2*d^4*e^2*x^2-5*c^3*d^6*x^2+20 
*a^3*d*e^5*x+104*a^2*c*d^3*e^3*x+20*a*c^2*d^5*e*x+8*a^3*d^2*e^4+40*a^2*c*d 
^4*e^2)/(e*x+d)/(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)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (213) = 426\).

Time = 3.88 (sec) , antiderivative size = 518, normalized size of antiderivative = 2.30 \[ \int \frac {x^2}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (40 \, a^{2} c d^{4} e^{2} + 8 \, a^{3} d^{2} e^{4} - 2 \, {\left (c^{3} d^{5} e - 10 \, a c^{2} d^{3} e^{3} - 15 \, a^{2} c d e^{5}\right )} x^{3} - {\left (5 \, c^{3} d^{6} - 49 \, a c^{2} d^{4} e^{2} - 85 \, a^{2} c d^{2} e^{4} - 15 \, a^{3} e^{6}\right )} x^{2} + 4 \, {\left (5 \, a c^{2} d^{5} e + 26 \, a^{2} c d^{3} e^{3} + 5 \, a^{3} d e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15 \, {\left (a c^{4} d^{11} e - 4 \, a^{2} c^{3} d^{9} e^{3} + 6 \, a^{3} c^{2} d^{7} e^{5} - 4 \, a^{4} c d^{5} e^{7} + a^{5} d^{3} e^{9} + {\left (c^{5} d^{9} e^{3} - 4 \, a c^{4} d^{7} e^{5} + 6 \, a^{2} c^{3} d^{5} e^{7} - 4 \, a^{3} c^{2} d^{3} e^{9} + a^{4} c d e^{11}\right )} x^{4} + {\left (3 \, c^{5} d^{10} e^{2} - 11 \, a c^{4} d^{8} e^{4} + 14 \, a^{2} c^{3} d^{6} e^{6} - 6 \, a^{3} c^{2} d^{4} e^{8} - a^{4} c d^{2} e^{10} + a^{5} e^{12}\right )} x^{3} + 3 \, {\left (c^{5} d^{11} e - 3 \, a c^{4} d^{9} e^{3} + 2 \, a^{2} c^{3} d^{7} e^{5} + 2 \, a^{3} c^{2} d^{5} e^{7} - 3 \, a^{4} c d^{3} e^{9} + a^{5} d e^{11}\right )} x^{2} + {\left (c^{5} d^{12} - a c^{4} d^{10} e^{2} - 6 \, a^{2} c^{3} d^{8} e^{4} + 14 \, a^{3} c^{2} d^{6} e^{6} - 11 \, a^{4} c d^{4} e^{8} + 3 \, a^{5} d^{2} e^{10}\right )} x\right )}} \] Input:

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

Output:

-2/15*(40*a^2*c*d^4*e^2 + 8*a^3*d^2*e^4 - 2*(c^3*d^5*e - 10*a*c^2*d^3*e^3 
- 15*a^2*c*d*e^5)*x^3 - (5*c^3*d^6 - 49*a*c^2*d^4*e^2 - 85*a^2*c*d^2*e^4 - 
 15*a^3*e^6)*x^2 + 4*(5*a*c^2*d^5*e + 26*a^2*c*d^3*e^3 + 5*a^3*d*e^5)*x)*s 
qrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a*c^4*d^11*e - 4*a^2*c^3*d^9*e 
^3 + 6*a^3*c^2*d^7*e^5 - 4*a^4*c*d^5*e^7 + a^5*d^3*e^9 + (c^5*d^9*e^3 - 4* 
a*c^4*d^7*e^5 + 6*a^2*c^3*d^5*e^7 - 4*a^3*c^2*d^3*e^9 + a^4*c*d*e^11)*x^4 
+ (3*c^5*d^10*e^2 - 11*a*c^4*d^8*e^4 + 14*a^2*c^3*d^6*e^6 - 6*a^3*c^2*d^4* 
e^8 - a^4*c*d^2*e^10 + a^5*e^12)*x^3 + 3*(c^5*d^11*e - 3*a*c^4*d^9*e^3 + 2 
*a^2*c^3*d^7*e^5 + 2*a^3*c^2*d^5*e^7 - 3*a^4*c*d^3*e^9 + a^5*d*e^11)*x^2 + 
 (c^5*d^12 - a*c^4*d^10*e^2 - 6*a^2*c^3*d^8*e^4 + 14*a^3*c^2*d^6*e^6 - 11* 
a^4*c*d^4*e^8 + 3*a^5*d^2*e^10)*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^2/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit 
hm="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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4176 vs. \(2 (213) = 426\).

Time = 0.24 (sec) , antiderivative size = 4176, normalized size of antiderivative = 18.56 \[ \int \frac {x^2}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-2/15*(15*a^2*c*d*e^3*abs(e)/((c^4*d^8*sgn(1/(e*x + d))*sgn(e) - 4*a*c^3*d 
^6*e^2*sgn(1/(e*x + d))*sgn(e) + 6*a^2*c^2*d^4*e^4*sgn(1/(e*x + d))*sgn(e) 
 - 4*a^3*c*d^2*e^6*sgn(1/(e*x + d))*sgn(e) + a^4*e^8*sgn(1/(e*x + d))*sgn( 
e))*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))) + 2*(c^3*d^5*abs(e) 
 - 10*a*c^2*d^3*e^2*abs(e) - 15*a^2*c*d*e^4*abs(e))*sgn(1/(e*x + d))*sgn(e 
)/(sqrt(c*d*e)*c^4*d^8*e - 4*sqrt(c*d*e)*a*c^3*d^6*e^3 + 6*sqrt(c*d*e)*a^2 
*c^2*d^4*e^5 - 4*sqrt(c*d*e)*a^3*c*d^2*e^7 + sqrt(c*d*e)*a^4*e^9) + (30*sq 
rt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a*c^17*d^34*e^20*abs(e)*sg 
n(1/(e*x + d))^4*sgn(e)^4 - 465*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e* 
x + d))*a^2*c^16*d^32*e^22*abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 + 3360*sqrt( 
c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^3*c^15*d^30*e^24*abs(e)*sgn 
(1/(e*x + d))^4*sgn(e)^4 - 15000*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e 
*x + d))*a^4*c^14*d^28*e^26*abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 + 46200*sqr 
t(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^5*c^13*d^26*e^28*abs(e)*s 
gn(1/(e*x + d))^4*sgn(e)^4 - 103740*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3 
/(e*x + d))*a^6*c^12*d^24*e^30*abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 + 174720 
*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^7*c^11*d^22*e^32*abs( 
e)*sgn(1/(e*x + d))^4*sgn(e)^4 - 223080*sqrt(c*d*e - c*d^2*e/(e*x + d) + a 
*e^3/(e*x + d))*a^8*c^10*d^20*e^34*abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 + 21 
4500*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^9*c^9*d^18*e^3...
 

Mupad [B] (verification not implemented)

Time = 7.42 (sec) , antiderivative size = 3062, normalized size of antiderivative = 13.61 \[ \int \frac {x^2}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

(((6*a*e^2 - 10*c*d^2)/(15*e*(a*e^2 - c*d^2)^3) - (4*c*d^2)/(5*e*(a*e^2 - 
c*d^2)^3))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((d 
*((d*((8*c^3*d^4*e)/(15*(a*e^2 - c*d^2)^5) - (8*c^2*d^2*e*(2*a*e^2 - c*d^2 
))/(15*(a*e^2 - c*d^2)^5)))/e + (2*c*d*(5*a^2*e^4 + c^2*d^4 - 10*a*c*d^2*e 
^2))/(5*(a*e^2 - c*d^2)^5)))/e - (12*a^3*e^6 + 6*c^3*d^6 - 24*a*c^2*d^4*e^ 
2 - 18*a^2*c*d^2*e^4)/(15*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*d^3 + 2*a*d*e^2)/(5*(a*e^2 - c*d^2 
)^2*(3*a*e^3 - 3*c*d^2*e)) + (d*((2*a*e^3 - 2*c*d^2*e)/(5*(a*e^2 - c*d^2)^ 
2*(3*a*e^3 - 3*c*d^2*e)) - (4*c*d^2*e)/(5*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*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)^2 + 
 (((d*((d*((e^2*(2*c^2*d^3 - 6*a*c*d*e^2))/(5*(a*e^2 - c*d^2)^2*(3*a^2*e^5 
 + 3*c^2*d^4*e - 6*a*c*d^2*e^3)) + (4*c^2*d^3*e^2)/(5*(a*e^2 - c*d^2)^2*(3 
*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3))))/e + (e^2*(12*a^2*e^3 - 24*a*c*d 
^2*e))/(5*(a*e^2 - c*d^2)^2*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3))))/e 
 + (12*a^2*d*e^4)/(5*(a*e^2 - c*d^2)^2*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^ 
2*e^3)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 + ((x* 
(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(x*((a*(((a*e^2 + c*d^2)*((8*c^ 
5*d^5*e^3*(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3* 
e^3 + a^2*c*d*e^5)) - (16*c^5*d^5*e^3*(3*a*e^2 - c*d^2))/(15*(a*e^2 - c*d^ 
2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (16*a*c^5...
 

Reduce [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 828, normalized size of antiderivative = 3.68 \[ \int \frac {x^2}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*(30*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*d**3*e**4 + 90*sqrt( 
e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*d**2*e**5*x + 90*sqrt(e)*sqrt(d) 
*sqrt(c)*sqrt(a*e + c*d*x)*a**2*d*e**6*x**2 + 30*sqrt(e)*sqrt(d)*sqrt(c)*s 
qrt(a*e + c*d*x)*a**2*e**7*x**3 + 20*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c* 
d*x)*a*c*d**5*e**2 + 60*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c*d**4 
*e**3*x + 60*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c*d**3*e**4*x**2 
+ 20*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c*d**2*e**5*x**3 - 2*sqrt 
(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**2*d**7 - 6*sqrt(e)*sqrt(d)*sqrt(c 
)*sqrt(a*e + c*d*x)*c**2*d**6*e*x - 6*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c 
*d*x)*c**2*d**5*e**2*x**2 - 2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c* 
*2*d**4*e**3*x**3 - 8*sqrt(d + e*x)*a**3*d**2*e**6 - 20*sqrt(d + e*x)*a**3 
*d*e**7*x - 15*sqrt(d + e*x)*a**3*e**8*x**2 - 40*sqrt(d + e*x)*a**2*c*d**4 
*e**4 - 104*sqrt(d + e*x)*a**2*c*d**3*e**5*x - 85*sqrt(d + e*x)*a**2*c*d** 
2*e**6*x**2 - 30*sqrt(d + e*x)*a**2*c*d*e**7*x**3 - 20*sqrt(d + e*x)*a*c** 
2*d**5*e**3*x - 49*sqrt(d + e*x)*a*c**2*d**4*e**4*x**2 - 20*sqrt(d + e*x)* 
a*c**2*d**3*e**5*x**3 + 5*sqrt(d + e*x)*c**3*d**6*e**2*x**2 + 2*sqrt(d + e 
*x)*c**3*d**5*e**3*x**3))/(15*sqrt(a*e + c*d*x)*e**2*(a**4*d**3*e**8 + 3*a 
**4*d**2*e**9*x + 3*a**4*d*e**10*x**2 + a**4*e**11*x**3 - 4*a**3*c*d**5*e* 
*6 - 12*a**3*c*d**4*e**7*x - 12*a**3*c*d**3*e**8*x**2 - 4*a**3*c*d**2*e**9 
*x**3 + 6*a**2*c**2*d**7*e**4 + 18*a**2*c**2*d**6*e**5*x + 18*a**2*c**2...