\(\int \frac {x^2 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{(d+e x)^2} \, dx\) [21]

Optimal result

Integrand size = 40, antiderivative size = 256 \[ \int \frac {x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=-\frac {\left (15 c^2 d^4-8 a c d^2 e^2+a^2 e^4-2 c d e \left (c d^2-a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c d e^3 \left (c d^2-a e^2\right )}+\frac {2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{e^2 \left (c d^2-a e^2\right ) (d+e x)^2}+\frac {\left (15 c^2 d^4-6 a c d^2 e^2-a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 c^{3/2} d^{3/2} e^{7/2}} \] Output:

-1/4*(15*c^2*d^4-8*a*c*d^2*e^2+a^2*e^4-2*c*d*e*(-a*e^2+c*d^2)*x)*(a*d*e+(a 
*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/e^3/(-a*e^2+c*d^2)+2*d^2*(a*d*e+(a*e^2+ 
c*d^2)*x+c*d*e*x^2)^(3/2)/e^2/(-a*e^2+c*d^2)/(e*x+d)^2+1/4*(-a^2*e^4-6*a*c 
*d^2*e^2+15*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^(7/2)
 

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.71 \[ \int \frac {x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (a e^2 (d+e x)+c d \left (-15 d^2-5 d e x+2 e^2 x^2\right )\right )}{d+e x}+\frac {\left (15 c^2 d^4-6 a c d^2 e^2-a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{4 c^{3/2} d^{3/2} e^{7/2}} \] Input:

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

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(a*e^2*(d + e*x) 
+ c*d*(-15*d^2 - 5*d*e*x + 2*e^2*x^2)))/(d + e*x) + ((15*c^2*d^4 - 6*a*c*d 
^2*e^2 - a^2*e^4)*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqr 
t[d + e*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(4*c^(3/2)*d^(3/2)*e^(7/ 
2))
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1213, 25, 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.

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

\(\Big \downarrow \) 1213

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2192

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {1}{4} e \left (\frac {\left (-a^2 e^4-6 a c d^2 e^2+15 c^2 d^4\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d}-\left (7 d-\frac {a e^2}{c d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}\right )+\frac {1}{2} e^2 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^4}-\frac {2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 (d+e x)}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {1}{4} e \left (\frac {\left (-a^2 e^4-6 a c d^2 e^2+15 c^2 d^4\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{c d}-\left (7 d-\frac {a e^2}{c d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}\right )+\frac {1}{2} e^2 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^4}-\frac {2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 (d+e x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{4} e \left (\frac {\left (-a^2 e^4-6 a c d^2 e^2+15 c^2 d^4\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{3/2} d^{3/2} \sqrt {e}}-\left (7 d-\frac {a e^2}{c d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}\right )+\frac {1}{2} e^2 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^4}-\frac {2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 (d+e x)}\)

Input:

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

Output:

(-2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e^3*(d + e*x)) + ((e 
^2*x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/2 + (e*(-((7*d - (a*e^2) 
/(c*d))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + ((15*c^2*d^4 - 6*a* 
c*d^2*e^2 - a^2*e^4)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[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)/e^4
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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. \(504\) vs. \(2(234)=468\).

Time = 2.46 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.97

method	result	size
default	\(\frac {\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{8 d e c \sqrt {d e c}}}{e^{2}}+\frac {d^{2} \left (-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {2 d e c \left (\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{2 \sqrt {d e c}}\right )}{a \,e^{2}-c \,d^{2}}\right )}{e^{4}}-\frac {2 d \left (\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{2 \sqrt {d e c}}\right )}{e^{3}}\)	\(505\)

Input:

int(x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/(e*x+d)^2,x,method=_RETURN 
VERBOSE)
 

Output:

1/e^2*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2) 
/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*ln((1/2*a*e^2+1/2*c*d^2+c 
*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^(1/ 
2))+d^2/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))^(3/2)+2*d*e*c/(a*e^2-c*d^2)*((d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e)) 
^(1/2)+1/2*(a*e^2-c*d^2)*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^3 
*((d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+1/2*(a*e^2-c*d^2)*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 = 0.17 (sec) , antiderivative size = 564, normalized size of antiderivative = 2.20 \[ \int \frac {x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\left [-\frac {{\left (15 \, c^{2} d^{5} - 6 \, a c d^{3} e^{2} - a^{2} d e^{4} + {\left (15 \, c^{2} d^{4} e - 6 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} x\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (2 \, c^{2} d^{2} e^{3} x^{2} - 15 \, c^{2} d^{4} e + a c d^{2} e^{3} - {\left (5 \, c^{2} d^{3} e^{2} - a c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, {\left (c^{2} d^{2} e^{5} x + c^{2} d^{3} e^{4}\right )}}, -\frac {{\left (15 \, c^{2} d^{5} - 6 \, a c d^{3} e^{2} - a^{2} d e^{4} + {\left (15 \, c^{2} d^{4} e - 6 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} x\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (2 \, c^{2} d^{2} e^{3} x^{2} - 15 \, c^{2} d^{4} e + a c d^{2} e^{3} - {\left (5 \, c^{2} d^{3} e^{2} - a c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, {\left (c^{2} d^{2} e^{5} x + c^{2} d^{3} e^{4}\right )}}\right ] \] Input:

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

Output:

[-1/16*((15*c^2*d^5 - 6*a*c*d^3*e^2 - a^2*d*e^4 + (15*c^2*d^4*e - 6*a*c*d^ 
2*e^3 - a^2*e^5)*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*(2*c^2*d^ 
2*e^3*x^2 - 15*c^2*d^4*e + a*c*d^2*e^3 - (5*c^2*d^3*e^2 - a*c*d*e^4)*x)*sq 
rt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^2*d^2*e^5*x + c^2*d^3*e^4), 
-1/8*((15*c^2*d^5 - 6*a*c*d^3*e^2 - a^2*d*e^4 + (15*c^2*d^4*e - 6*a*c*d^2* 
e^3 - a^2*e^5)*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*(2*c^2*d^2*e^3*x^2 - 15*c^2*d^ 
4*e + a*c*d^2*e^3 - (5*c^2*d^3*e^2 - a*c*d*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e 
+ (c*d^2 + a*e^2)*x))/(c^2*d^2*e^5*x + c^2*d^3*e^4)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^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>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (234) = 468\).

Time = 0.28 (sec) , antiderivative size = 539, normalized size of antiderivative = 2.11 \[ \int \frac {x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=-\frac {1}{4} \, {\left (\frac {8 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} d^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{e^{4} {\left | e \right |}} + \frac {{\left (15 \, c^{2} d^{4} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 6 \, a c d^{2} e^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - a^{2} e^{4} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )\right )} \arctan \left (\frac {\sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}}}{\sqrt {-c d e}}\right )}{\sqrt {-c d e} c d e^{3} {\left | e \right |}} + \frac {7 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c^{3} d^{5} e \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 6 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} a c^{2} d^{3} e^{3} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} a^{2} c d e^{5} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 9 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}} c^{2} d^{4} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + 10 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}} a c d^{2} e^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}} a^{2} e^{4} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{{\left (\frac {c d^{2} e}{e x + d} - \frac {a e^{3}}{e x + d}\right )}^{2} c d e^{3} {\left | e \right |}}\right )} {\left | e \right |} \] Input:

integrate(x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^2,x, algorit 
hm="giac")
 

Output:

-1/4*(8*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*d^2*sgn(1/(e*x + 
 d))*sgn(e)/(e^4*abs(e)) + (15*c^2*d^4*sgn(1/(e*x + d))*sgn(e) - 6*a*c*d^2 
*e^2*sgn(1/(e*x + d))*sgn(e) - a^2*e^4*sgn(1/(e*x + d))*sgn(e))*arctan(sqr 
t(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))/sqrt(-c*d*e))/(sqrt(-c*d*e) 
*c*d*e^3*abs(e)) + (7*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*c^ 
3*d^5*e*sgn(1/(e*x + d))*sgn(e) - 6*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3 
/(e*x + d))*a*c^2*d^3*e^3*sgn(1/(e*x + d))*sgn(e) - sqrt(c*d*e - c*d^2*e/( 
e*x + d) + a*e^3/(e*x + d))*a^2*c*d*e^5*sgn(1/(e*x + d))*sgn(e) - 9*(c*d*e 
 - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*c^2*d^4*sgn(1/(e*x + d))*sgn 
(e) + 10*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a*c*d^2*e^2*s 
gn(1/(e*x + d))*sgn(e) - (c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/ 
2)*a^2*e^4*sgn(1/(e*x + d))*sgn(e))/((c*d^2*e/(e*x + d) - a*e^3/(e*x + d)) 
^2*c*d*e^3*abs(e)))*abs(e)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 548, normalized size of antiderivative = 2.14 \[ \int \frac {x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\frac {\sqrt {e x +d}\, \sqrt {c d x +a e}\, a c \,d^{2} e^{3}+\sqrt {e x +d}\, \sqrt {c d x +a e}\, a c d \,e^{4} x -15 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{4} e -5 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{3} e^{2} x +2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{2} e^{3} x^{2}-\sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} d \,e^{4}-\sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} e^{5} x -6 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a c \,d^{3} e^{2}-6 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a c \,d^{2} e^{3} x +15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{2} d^{5}+15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{2} d^{4} e x +2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, a c \,d^{3} e^{2}+2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, a c \,d^{2} e^{3} x -10 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{5}-10 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{4} e x}{4 c^{2} d^{2} e^{4} \left (e x +d \right )} \] Input:

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

Output:

(sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c*d**2*e**3 + sqrt(d + e*x)*sqrt(a*e + 
c*d*x)*a*c*d*e**4*x - 15*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**2*d**4*e - 5*s 
qrt(d + e*x)*sqrt(a*e + c*d*x)*c**2*d**3*e**2*x + 2*sqrt(d + e*x)*sqrt(a*e 
 + c*d*x)*c**2*d**2*e**3*x**2 - 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* 
d*e**4 - 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*e**5*x - 6*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*d**3*e**2 - 6*sqrt(e)*sqrt(d)*sqrt(c)*log((sq 
rt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d 
**2))*a*c*d**2*e**3*x + 15*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))*c**2*d**5 
+ 15*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))*c**2*d**4*e*x + 2*sqrt(e)*sqrt(d 
)*sqrt(c)*a*c*d**3*e**2 + 2*sqrt(e)*sqrt(d)*sqrt(c)*a*c*d**2*e**3*x - 10*s 
qrt(e)*sqrt(d)*sqrt(c)*c**2*d**5 - 10*sqrt(e)*sqrt(d)*sqrt(c)*c**2*d**4*e* 
x)/(4*c**2*d**2*e**4*(d + e*x))