\(\int \frac {x^3 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{(d+e x)^2} \, dx\) [20]

Optimal result

Integrand size = 40, antiderivative size = 338 \[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\frac {\left (35 c^3 d^6-25 a c^2 d^4 e^2+5 a^2 c d^2 e^4+a^3 e^6-2 c d e \left (c d^2-a e^2\right ) \left (5 c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^2 d^2 e^4 \left (c d^2-a e^2\right )}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 c d e^3}-\frac {2 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{e^3 \left (c d^2-a e^2\right ) (d+e x)^2}-\frac {\left (35 c^3 d^6-15 a c^2 d^4 e^2-3 a^2 c d^2 e^4-a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{5/2} d^{5/2} e^{9/2}} \] Output:

1/8*(35*c^3*d^6-25*a*c^2*d^4*e^2+5*a^2*c*d^2*e^4+a^3*e^6-2*c*d*e*(-a*e^2+c 
*d^2)*(a*e^2+5*c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2/e 
^4/(-a*e^2+c*d^2)+1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e^3-2*d^ 
3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e^3/(-a*e^2+c*d^2)/(e*x+d)^2-1/8 
*(-a^3*e^6-3*a^2*c*d^2*e^4-15*a*c^2*d^4*e^2+35*c^3*d^6)*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^(5/2)/d^( 
5/2)/e^(9/2)
 

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.71 \[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (-3 a^2 e^4 (d+e x)-2 a c d e^2 \left (5 d^2+4 d e x-e^2 x^2\right )+c^2 d^2 \left (105 d^3+35 d^2 e x-14 d e^2 x^2+8 e^3 x^3\right )\right )}{d+e x}-\frac {3 \left (35 c^3 d^6-15 a c^2 d^4 e^2-3 a^2 c d^2 e^4-a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{24 c^{5/2} d^{5/2} e^{9/2}} \] Input:

Integrate[(x^3*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]*(-3*a^2*e^4*(d + 
e*x) - 2*a*c*d*e^2*(5*d^2 + 4*d*e*x - 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)))/(d + e*x) - (3*(35*c^3*d^6 - 15*a*c^2*d 
^4*e^2 - 3*a^2*c*d^2*e^4 - a^3*e^6)*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/(S 
qrt[c]*Sqrt[d]*Sqrt[d + e*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(24*c^ 
(5/2)*d^(5/2)*e^(9/2))
 

Rubi [A] (verified)

Time = 1.43 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.04, 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.

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

\(\Big \downarrow \) 1213

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

\(\Big \downarrow \) 2192

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2192

\(\displaystyle \frac {2 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^4 (d+e x)}-\frac {\frac {\frac {\int \frac {c d e^3 \left (2 d \left (12 c^2 d^4-23 a c e^2 d^2+a^2 e^4\right )-e \left (3 c d^2-a e^2\right ) \left (19 c d^2+3 a e^2\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d e}+\frac {1}{2} e^3 x \left (11 c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} e^3 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^4 (d+e x)}-\frac {\frac {\frac {1}{4} e^2 \int \frac {2 d \left (12 c^2 d^4-23 a c e^2 d^2+a^2 e^4\right )-e \left (3 c d^2-a e^2\right ) \left (19 c d^2+3 a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\frac {1}{2} e^3 x \left (11 c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} e^3 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {2 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^4 (d+e x)}-\frac {\frac {\frac {1}{4} e^2 \left (\frac {3 \left (-a^3 e^6-3 a^2 c d^2 e^4-15 a c^2 d^4 e^2+35 c^3 d^6\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d}-\frac {\left (3 c d^2-a e^2\right ) \left (3 a e^2+19 c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\right )+\frac {1}{2} e^3 x \left (11 c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} e^3 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {2 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^4 (d+e x)}-\frac {\frac {\frac {1}{4} e^2 \left (\frac {3 \left (-a^3 e^6-3 a^2 c d^2 e^4-15 a c^2 d^4 e^2+35 c^3 d^6\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{c d}-\frac {\left (3 c d^2-a e^2\right ) \left (3 a e^2+19 c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\right )+\frac {1}{2} e^3 x \left (11 c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} e^3 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^4 (d+e x)}-\frac {\frac {\frac {1}{4} e^2 \left (\frac {3 \left (-a^3 e^6-3 a^2 c d^2 e^4-15 a c^2 d^4 e^2+35 c^3 d^6\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{3/2} d^{3/2} \sqrt {e}}-\frac {\left (3 c d^2-a e^2\right ) \left (3 a e^2+19 c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\right )+\frac {1}{2} e^3 x \left (11 c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} e^3 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5}\)

Input:

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

Output:

(2*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e^4*(d + e*x)) - (-1/ 
3*(e^3*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + ((e^3*(11*c*d^2 
- a*e^2)*x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/2 + (e^2*(-(((3*c* 
d^2 - a*e^2)*(19*c*d^2 + 3*a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x 
^2])/(c*d)) + (3*(35*c^3*d^6 - 15*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 - a^3*e^ 
6)*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)/(6 
*c*d*e))/e^5
 

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. \(727\) vs. \(2(312)=624\).

Time = 2.77 (sec) , antiderivative size = 728, normalized size of antiderivative = 2.15

method	result	size
default	\(\frac {\frac {{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}{3 d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\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}}\right )}{2 d e c}}{e^{2}}-\frac {2 d \left (\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}}\right )}{e^{3}}+\frac {3 d^{2} \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^{4}}-\frac {d^{3} \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^{5}}\)	\(728\)

Input:

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

Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 736, normalized size of antiderivative = 2.18 \[ \int \frac {x^3 \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 {3 \, {\left (35 \, c^{3} d^{7} - 15 \, a c^{2} d^{5} e^{2} - 3 \, a^{2} c d^{3} e^{4} - a^{3} d e^{6} + {\left (35 \, c^{3} d^{6} e - 15 \, a c^{2} d^{4} e^{3} - 3 \, 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 - 10 \, a c^{2} d^{4} e^{3} - 3 \, a^{2} c d^{2} e^{5} - 2 \, {\left (7 \, c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{2} + {\left (35 \, c^{3} d^{5} e^{2} - 8 \, 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^{3} d^{3} e^{6} x + c^{3} d^{4} e^{5}\right )}}, \frac {3 \, {\left (35 \, c^{3} d^{7} - 15 \, a c^{2} d^{5} e^{2} - 3 \, a^{2} c d^{3} e^{4} - a^{3} d e^{6} + {\left (35 \, c^{3} d^{6} e - 15 \, a c^{2} d^{4} e^{3} - 3 \, 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 - 10 \, a c^{2} d^{4} e^{3} - 3 \, a^{2} c d^{2} e^{5} - 2 \, {\left (7 \, c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{2} + {\left (35 \, c^{3} d^{5} e^{2} - 8 \, 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^{3} d^{3} e^{6} x + c^{3} d^{4} e^{5}\right )}}\right ] \] Input:

integrate(x^3*(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:

[-1/96*(3*(35*c^3*d^7 - 15*a*c^2*d^5*e^2 - 3*a^2*c*d^3*e^4 - a^3*d*e^6 + ( 
35*c^3*d^6*e - 15*a*c^2*d^4*e^3 - 3*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 - 10 
*a*c^2*d^4*e^3 - 3*a^2*c*d^2*e^5 - 2*(7*c^3*d^4*e^3 - a*c^2*d^2*e^5)*x^2 + 
 (35*c^3*d^5*e^2 - 8*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^3*d^3*e^6*x + c^3*d^4*e^5), 1/48*(3*(35*c^3*d 
^7 - 15*a*c^2*d^5*e^2 - 3*a^2*c*d^3*e^4 - a^3*d*e^6 + (35*c^3*d^6*e - 15*a 
*c^2*d^4*e^3 - 3*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 - 10*a*c^2*d^4*e^3 - 3*a^2*c*d^2*e^5 - 2*( 
7*c^3*d^4*e^3 - a*c^2*d^2*e^5)*x^2 + (35*c^3*d^5*e^2 - 8*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^3*d^3*e^6 
*x + c^3*d^4*e^5)]
 

Sympy [F]

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

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

Maxima [F(-2)]

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

integrate(x^3*(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. 901 vs. \(2 (312) = 624\).

Time = 0.40 (sec) , antiderivative size = 901, normalized size of antiderivative = 2.67 \[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\text {Too large to display} \] Input:

integrate(x^3*(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:

1/24*(48*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*d^3*sgn(1/(e*x 
+ d))*sgn(e)/(e^5*abs(e)) + 3*(35*c^3*d^6*sgn(1/(e*x + d))*sgn(e) - 15*a*c 
^2*d^4*e^2*sgn(1/(e*x + d))*sgn(e) - 3*a^2*c*d^2*e^4*sgn(1/(e*x + d))*sgn( 
e) - a^3*e^6*sgn(1/(e*x + d))*sgn(e))*arctan(sqrt(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^2*d^2*e^4*abs(e)) + (57 
*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*c^5*d^8*e^2*sgn(1/(e*x 
+ d))*sgn(e) - 45*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a*c^4* 
d^6*e^4*sgn(1/(e*x + d))*sgn(e) - 9*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3 
/(e*x + d))*a^2*c^3*d^4*e^6*sgn(1/(e*x + d))*sgn(e) - 3*sqrt(c*d*e - c*d^2 
*e/(e*x + d) + a*e^3/(e*x + d))*a^3*c^2*d^2*e^8*sgn(1/(e*x + d))*sgn(e) - 
136*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*c^4*d^7*e*sgn(1/(e 
*x + d))*sgn(e) + 120*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)* 
a*c^3*d^5*e^3*sgn(1/(e*x + d))*sgn(e) + 24*(c*d*e - c*d^2*e/(e*x + d) + a* 
e^3/(e*x + d))^(3/2)*a^2*c^2*d^3*e^5*sgn(1/(e*x + d))*sgn(e) - 8*(c*d*e - 
c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a^3*c*d*e^7*sgn(1/(e*x + d))*sg 
n(e) + 87*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*c^3*d^6*sgn( 
1/(e*x + d))*sgn(e) - 99*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/ 
2)*a*c^2*d^4*e^2*sgn(1/(e*x + d))*sgn(e) + 9*(c*d*e - c*d^2*e/(e*x + d) + 
a*e^3/(e*x + d))^(5/2)*a^2*c*d^2*e^4*sgn(1/(e*x + d))*sgn(e) + 3*(c*d*e - 
c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*a^3*e^6*sgn(1/(e*x + d))*sgn...
 

Mupad [F(-1)]

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

int((x^3*(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^3*(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 = 20.06 (sec) , antiderivative size = 862, normalized size of antiderivative = 2.55 \[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:

int(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^2,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) 
*sqrt(a*e + c*d*x)*a**2*c*d*e**6*x - 80*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a* 
c**2*d**4*e**3 - 64*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d**3*e**4*x + 1 
6*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 + 72*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 + 
 72*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**2*e**5*x + 360*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 + 360*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**4*e**3*x - 840*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**3*d**7 - 840*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*...