\(\int \frac {x^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^3} \, dx\) [40]

Optimal result

Integrand size = 40, antiderivative size = 322 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {(a e+c d x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d e^2}+\frac {\left (35 c^2 d^4-10 a c d^2 e^2-a^2 e^4\right ) \left (3 c d^2-5 a e^2-2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 c d e^4 \left (c d^2-a e^2\right )}+\frac {2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e^2 \left (c d^2-a e^2\right ) (d+e x)^3}-\frac {\left (c d^2-a e^2\right ) \left (35 c^2 d^4-10 a c d^2 e^2-a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{9/2}} \] Output:

1/3*(c*d*x+a*e)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/e^2+1/24*(-a 
^2*e^4-10*a*c*d^2*e^2+35*c^2*d^4)*(-2*c*d*e*x-5*a*e^2+3*c*d^2)*(a*d*e+(a*e 
^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/e^4/(-a*e^2+c*d^2)+2*d^2*(a*d*e+(a*e^2+c* 
d^2)*x+c*d*e*x^2)^(5/2)/e^2/(-a*e^2+c*d^2)/(e*x+d)^3-1/8*(-a*e^2+c*d^2)*(- 
a^2*e^4-10*a*c*d^2*e^2+35*c^2*d^4)*arctanh(c^(1/2)*d^(1/2)*(e*x+d)/e^(1/2) 
/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/c^(3/2)/d^(3/2)/e^(9/2)
 

Mathematica [A] (verified)

Time = 10.79 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.87 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (3 a^2 e^4 (d+e x)-2 a c d e^2 \left (50 d^2+19 d e x-7 e^2 x^2\right )+c^2 d^2 \left (105 d^3+35 d^2 e x-14 d e^2 x^2+8 e^3 x^3\right )\right )-\frac {3 \left (c d^2-a e^2\right )^{3/2} \left (35 c^2 d^4-10 a c d^2 e^2-a^2 e^4\right ) \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )}{\sqrt {c d} \sqrt {a e+c d x}}\right )}{24 c^{3/2} d^{3/2} e^{9/2} (d+e x)} \] Input:

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

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(3*a^2*e^4*(d + e* 
x) - 2*a*c*d*e^2*(50*d^2 + 19*d*e*x - 7*e^2*x^2) + c^2*d^2*(105*d^3 + 35*d 
^2*e*x - 14*d*e^2*x^2 + 8*e^3*x^3)) - (3*(c*d^2 - a*e^2)^(3/2)*(35*c^2*d^4 
 - 10*a*c*d^2*e^2 - a^2*e^4)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh 
[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2 
])])/(Sqrt[c*d]*Sqrt[a*e + c*d*x])))/(24*c^(3/2)*d^(3/2)*e^(9/2)*(d + e*x) 
)
 

Rubi [A] (verified)

Time = 1.54 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1213, 2192, 27, 2192, 27, 1160, 1092, 219}

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.

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

\(\Big \downarrow \) 1213

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

\(\Big \downarrow \) 2192

\(\displaystyle \frac {2 d^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^4 (d+e x)}-\frac {\frac {\int \frac {c^2 d^2 \left (11 c d^2-7 a e^2\right ) x^2 e^5-2 c d \left (3 c^2 d^4-8 a c e^2 d^2+3 a^2 e^4\right ) x e^4+6 c d^2 \left (c d^2-a e^2\right )^2 e^3}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 c d e}-\frac {1}{3} c d e^4 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 d^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^4 (d+e x)}-\frac {\frac {\int \frac {c^2 d^2 \left (11 c d^2-7 a e^2\right ) x^2 e^5-2 c d \left (3 c^2 d^4-8 a c e^2 d^2+3 a^2 e^4\right ) x e^4+6 c d^2 \left (c d^2-a e^2\right )^2 e^3}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 c d e}-\frac {1}{3} c d e^4 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {2 d^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^4 (d+e x)}-\frac {\frac {\frac {\int \frac {c^2 d^2 e^4 \left (2 d \left (12 c^2 d^4-35 a c e^2 d^2+19 a^2 e^4\right )-e \left (57 c^2 d^4-52 a c e^2 d^2+3 a^2 e^4\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d e}+\frac {1}{2} c d e^4 x \left (11 c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} c d e^4 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 d^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^4 (d+e x)}-\frac {\frac {\frac {1}{4} c d e^3 \int \frac {2 d \left (12 c^2 d^4-35 a c e^2 d^2+19 a^2 e^4\right )-e \left (57 c^2 d^4-52 a c e^2 d^2+3 a^2 e^4\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\frac {1}{2} c d e^4 x \left (11 c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} c d e^4 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {2 d^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^4 (d+e x)}-\frac {\frac {\frac {1}{4} c d e^3 \left (\frac {3 \left (c d^2-a e^2\right ) \left (-a^2 e^4-10 a c d^2 e^2+35 c^2 d^4\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d}-\frac {\left (3 a^2 e^4-52 a c d^2 e^2+57 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\right )+\frac {1}{2} c d e^4 x \left (11 c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} c d e^4 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {2 d^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^4 (d+e x)}-\frac {\frac {\frac {1}{4} c d e^3 \left (\frac {3 \left (c d^2-a e^2\right ) \left (-a^2 e^4-10 a c d^2 e^2+35 c^2 d^4\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{c d}-\frac {\left (3 a^2 e^4-52 a c d^2 e^2+57 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\right )+\frac {1}{2} c d e^4 x \left (11 c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} c d e^4 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 d^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^4 (d+e x)}-\frac {\frac {\frac {1}{4} c d e^3 \left (\frac {3 \left (c d^2-a e^2\right ) \left (-a^2 e^4-10 a c d^2 e^2+35 c^2 d^4\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{3/2} d^{3/2} \sqrt {e}}-\frac {\left (3 a^2 e^4-52 a c d^2 e^2+57 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\right )+\frac {1}{2} c d e^4 x \left (11 c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} c d e^4 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}\)

Input:

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

Output:

(2*d^2*(c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e^4*( 
d + e*x)) - (-1/3*(c*d*e^4*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2] 
) + ((c*d*e^4*(11*c*d^2 - 7*a*e^2)*x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d* 
e*x^2])/2 + (c*d*e^3*(-(((57*c^2*d^4 - 52*a*c*d^2*e^2 + 3*a^2*e^4)*Sqrt[a* 
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d)) + (3*(c*d^2 - a*e^2)*(35*c^2* 
d^4 - 10*a*c*d^2*e^2 - a^2*e^4)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqr 
t[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*c^( 
3/2)*d^(3/2)*Sqrt[e])))/4)/(6*c*d*e))/e^6
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1160

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

rule 1213

Int[(x_)^(n_.)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^ 
2)^(p_), x_Symbol] :> Simp[-2*(-d)^n*e^(2*m - n + 3)*(Sqrt[a + b*x + c*x^2] 
/((-2*c*d + b*e)^(m + 2)*(d + e*x))), x] - Simp[e^(2*m - n + 2)   Int[Expan 
dToSum[((-d)^n*(-2*c*d + b*e)^(-m - 1) - e^n*x^n*((-c)*d + b*e + c*e*x)^(-m 
 - 1))/(d + e*x), x]/Sqrt[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e} 
, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[m, 0] && IGtQ[n, 0] && EqQ[m 
+ p, -3/2]
 

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(936\) vs. \(2(296)=592\).

Time = 2.95 (sec) , antiderivative size = 937, normalized size of antiderivative = 2.91

method	result	size
default	\(\frac {\frac {\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}}}{3}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\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 )}}{4 d e c}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\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 )}{8 d e c \sqrt {d e c}}\right )}{2}}{e^{3}}+\frac {d^{2} \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 {5}{2}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}+\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 {5}{2}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {6 d e c \left (\frac {\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}}}{3}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\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 )}}{4 d e c}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\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 )}{8 d e c \sqrt {d e c}}\right )}{2}\right )}{a \,e^{2}-c \,d^{2}}\right )}{a \,e^{2}-c \,d^{2}}\right )}{e^{5}}-\frac {2 d \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 {5}{2}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {6 d e c \left (\frac {\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}}}{3}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\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 )}}{4 d e c}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\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 )}{8 d e c \sqrt {d e c}}\right )}{2}\right )}{a \,e^{2}-c \,d^{2}}\right )}{e^{4}}\)	\(937\)

Input:

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

Output:

1/e^3*(1/3*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)+1/2*(a*e^2-c*d^2) 
*(1/4*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/d/e/c*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*( 
x+d/e))^(1/2)-1/8*(a*e^2-c*d^2)^2/d/e/c*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d 
/e))/(d*e*c)^(1/2)+(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^ 
(1/2)))+d^2/e^5*(-2/(a*e^2-c*d^2)/(x+d/e)^3*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2) 
*(x+d/e))^(5/2)+4*d*e*c/(a*e^2-c*d^2)*(2/(a*e^2-c*d^2)/(x+d/e)^2*(d*e*c*(x 
+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)-6*d*e*c/(a*e^2-c*d^2)*(1/3*(d*e*c*(x+ 
d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)+1/2*(a*e^2-c*d^2)*(1/4*(2*d*e*c*(x+d/e 
)+a*e^2-c*d^2)/d/e/c*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/8*(a* 
e^2-c*d^2)^2/d/e/c*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d/e))/(d*e*c)^(1/2)+(d 
*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^(1/2)))))-2*d/e^4*(2/ 
(a*e^2-c*d^2)/(x+d/e)^2*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)-6*d* 
e*c/(a*e^2-c*d^2)*(1/3*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)+1/2*( 
a*e^2-c*d^2)*(1/4*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/d/e/c*(d*e*c*(x+d/e)^2+(a* 
e^2-c*d^2)*(x+d/e))^(1/2)-1/8*(a*e^2-c*d^2)^2/d/e/c*ln((1/2*a*e^2-1/2*c*d^ 
2+d*e*c*(x+d/e))/(d*e*c)^(1/2)+(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/ 
2))/(d*e*c)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 730, normalized size of antiderivative = 2.27 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=\left [\frac {3 \, {\left (35 \, c^{3} d^{7} - 45 \, a c^{2} d^{5} e^{2} + 9 \, a^{2} c d^{3} e^{4} + a^{3} d e^{6} + {\left (35 \, c^{3} d^{6} e - 45 \, a c^{2} d^{4} e^{3} + 9 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}\right )} x\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (8 \, c^{3} d^{3} e^{4} x^{3} + 105 \, c^{3} d^{6} e - 100 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - 14 \, {\left (c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{2} + {\left (35 \, c^{3} d^{5} e^{2} - 38 \, a c^{2} d^{3} e^{4} + 3 \, a^{2} 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}}{96 \, {\left (c^{2} d^{2} e^{6} x + c^{2} d^{3} e^{5}\right )}}, \frac {3 \, {\left (35 \, c^{3} d^{7} - 45 \, a c^{2} d^{5} e^{2} + 9 \, a^{2} c d^{3} e^{4} + a^{3} d e^{6} + {\left (35 \, c^{3} d^{6} e - 45 \, a c^{2} d^{4} e^{3} + 9 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}\right )} x\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (8 \, c^{3} d^{3} e^{4} x^{3} + 105 \, c^{3} d^{6} e - 100 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - 14 \, {\left (c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{2} + {\left (35 \, c^{3} d^{5} e^{2} - 38 \, a c^{2} d^{3} e^{4} + 3 \, a^{2} 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}}{48 \, {\left (c^{2} d^{2} e^{6} x + c^{2} d^{3} e^{5}\right )}}\right ] \] Input:

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

Output:

[1/96*(3*(35*c^3*d^7 - 45*a*c^2*d^5*e^2 + 9*a^2*c*d^3*e^4 + a^3*d*e^6 + (3 
5*c^3*d^6*e - 45*a*c^2*d^4*e^3 + 9*a^2*c*d^2*e^5 + a^3*e^7)*x)*sqrt(c*d*e) 
*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 - 4*sqrt(c*d*e* 
x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d*e) + 
 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*(8*c^3*d^3*e^4*x^3 + 105*c^3*d^6*e - 100 
*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - 14*(c^3*d^4*e^3 - a*c^2*d^2*e^5)*x^2 + 
(35*c^3*d^5*e^2 - 38*a*c^2*d^3*e^4 + 3*a^2*c*d*e^6)*x)*sqrt(c*d*e*x^2 + a* 
d*e + (c*d^2 + a*e^2)*x))/(c^2*d^2*e^6*x + c^2*d^3*e^5), 1/48*(3*(35*c^3*d 
^7 - 45*a*c^2*d^5*e^2 + 9*a^2*c*d^3*e^4 + a^3*d*e^6 + (35*c^3*d^6*e - 45*a 
*c^2*d^4*e^3 + 9*a^2*c*d^2*e^5 + a^3*e^7)*x)*sqrt(-c*d*e)*arctan(1/2*sqrt( 
c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c 
*d*e)/(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) + 2*(8* 
c^3*d^3*e^4*x^3 + 105*c^3*d^6*e - 100*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - 14 
*(c^3*d^4*e^3 - a*c^2*d^2*e^5)*x^2 + (35*c^3*d^5*e^2 - 38*a*c^2*d^3*e^4 + 
3*a^2*c*d*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^2*d^2*e^ 
6*x + c^2*d^3*e^5)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^3,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>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.99 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {1}{24} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (\frac {4 \, c d x}{e^{2}} - \frac {11 \, c^{3} d^{4} e^{10} - 7 \, a c^{2} d^{2} e^{12}}{c^{2} d^{2} e^{13}}\right )} x + \frac {57 \, c^{3} d^{5} e^{9} - 52 \, a c^{2} d^{3} e^{11} + 3 \, a^{2} c d e^{13}}{c^{2} d^{2} e^{13}}\right )} + \frac {2 \, {\left (c^{2} d^{6} - 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4}\right )}}{{\left ({\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} e + \sqrt {c d e} d\right )} e^{4}} + \frac {{\left (35 \, c^{3} d^{6} - 45 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \log \left ({\left | c d^{2} + a e^{2} + 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{16 \, \sqrt {c d e} c d e^{4}} \] Input:

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

Output:

1/24*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*c*d*x/e^2 - (11*c^3 
*d^4*e^10 - 7*a*c^2*d^2*e^12)/(c^2*d^2*e^13))*x + (57*c^3*d^5*e^9 - 52*a*c 
^2*d^3*e^11 + 3*a^2*c*d*e^13)/(c^2*d^2*e^13)) + 2*(c^2*d^6 - 2*a*c*d^4*e^2 
 + a^2*d^2*e^4)/(((sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a* 
d*e))*e + sqrt(c*d*e)*d)*e^4) + 1/16*(35*c^3*d^6 - 45*a*c^2*d^4*e^2 + 9*a^ 
2*c*d^2*e^4 + a^3*e^6)*log(abs(c*d^2 + a*e^2 + 2*sqrt(c*d*e)*(sqrt(c*d*e)* 
x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))))/(sqrt(c*d*e)*c*d*e^4)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 5.98 (sec) , antiderivative size = 862, normalized size of antiderivative = 2.68 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(24*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c*d**2*e**5 + 24*sqrt(d + e*x)*sq 
rt(a*e + c*d*x)*a**2*c*d*e**6*x - 800*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c* 
*2*d**4*e**3 - 304*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d**3*e**4*x + 11 
2*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d**2*e**5*x**2 + 840*sqrt(d + e*x 
)*sqrt(a*e + c*d*x)*c**3*d**6*e + 280*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**3 
*d**5*e**2*x - 112*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**4*e**3*x**2 + 6 
4*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**3*e**4*x**3 - 24*sqrt(e)*sqrt(d) 
*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/s 
qrt(a*e**2 - c*d**2))*a**3*d*e**6 - 24*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e 
)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2) 
)*a**3*e**7*x - 216*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) 
 + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*c*d**3*e**4 
- 216*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqr 
t(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*c*d**2*e**5*x + 1080*sqrt( 
e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d 
 + e*x))/sqrt(a*e**2 - c*d**2))*a*c**2*d**5*e**2 + 1080*sqrt(e)*sqrt(d)*sq 
rt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt 
(a*e**2 - c*d**2))*a*c**2*d**4*e**3*x - 840*sqrt(e)*sqrt(d)*sqrt(c)*log((s 
qrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c* 
d**2))*c**3*d**7 - 840*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + ...