\(\int \frac {x^3 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{d+e x} \, dx\) [11]

Optimal result

Integrand size = 40, antiderivative size = 330 \[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {1}{24} \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}-\frac {\left (105 c^3 d^6-25 a c^2 d^4 e^2-17 a^2 c d^2 e^4-15 a^3 e^6-2 c d e \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^3 d^3 e^4}+\frac {\left (c d^2-a e^2\right ) \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{64 c^{7/2} d^{7/2} e^{9/2}} \] Output:

1/24*(a/c/d-7*d/e^2)*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/4*x^3*( 
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e-1/192*(105*c^3*d^6-25*a*c^2*d^4*e 
^2-17*a^2*c*d^2*e^4-15*a^3*e^6-2*c*d*e*(-5*a^2*e^4-6*a*c*d^2*e^2+35*c^2*d^ 
4)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/e^4+1/64*(-a*e^2+c*d 
^2)*(5*a^3*e^6+9*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^(7/2) 
/d^(7/2)/e^(9/2)
 

Mathematica [A] (verified)

Time = 11.80 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\sqrt {c} \sqrt {d} \sqrt {e} \left (-15 a^3 e^6+a^2 c d e^4 (-17 d+10 e x)+a c^2 d^2 e^2 \left (-25 d^2+12 d e x-8 e^2 x^2\right )+c^3 d^3 \left (105 d^3-70 d^2 e x+56 d e^2 x^2-48 e^3 x^3\right )\right )+\frac {3 \sqrt {c d} \sqrt {c d^2-a e^2} \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\right ) \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )}{\sqrt {a e+c d x} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}\right )}{192 c^{7/2} d^{7/2} e^{9/2}} \] Input:

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

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(Sqrt[c]*Sqrt[d]*Sqrt[e]*(-15*a^3*e^6 + a 
^2*c*d*e^4*(-17*d + 10*e*x) + a*c^2*d^2*e^2*(-25*d^2 + 12*d*e*x - 8*e^2*x^ 
2) + c^3*d^3*(105*d^3 - 70*d^2*e*x + 56*d*e^2*x^2 - 48*e^3*x^3))) + (3*Sqr 
t[c*d]*Sqrt[c*d^2 - a*e^2]*(35*c^3*d^6 + 15*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^ 
4 + 5*a^3*e^6)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c 
*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 
- a*e^2)])))/(192*c^(7/2)*d^(7/2)*e^(9/2))
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1215, 1236, 27, 1236, 27, 1225, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1236

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}-\frac {\frac {\int -\frac {x \left (4 a d e \left (7 c d^2-a e^2\right )+\left (35 c^2 d^4-6 a c e^2 d^2-5 a^2 e^4\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 c d e}+\frac {1}{3} x^2 \left (\frac {7 d}{e}-\frac {a e}{c d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}-\frac {\frac {1}{3} x^2 \left (\frac {7 d}{e}-\frac {a e}{c d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}-\frac {\int \frac {x \left (4 a d e \left (7 c d^2-a e^2\right )+\left (35 c^2 d^4-6 a c e^2 d^2-5 a^2 e^4\right ) x\right )}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 c d e}}{8 e}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}-\frac {\frac {1}{3} x^2 \left (\frac {7 d}{e}-\frac {a e}{c d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}-\frac {\frac {3 \left (c d^2-a e^2\right ) \left (5 a^3 e^6+9 a^2 c d^2 e^4+15 a c^2 d^4 e^2+35 c^3 d^6\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c^2 d^2 e^2}-\frac {\left (-15 a^3 e^6-2 c d e x \left (-5 a^2 e^4-6 a c d^2 e^2+35 c^2 d^4\right )-17 a^2 c d^2 e^4-25 a c^2 d^4 e^2+105 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e^2}}{6 c d e}}{8 e}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}-\frac {\frac {1}{3} x^2 \left (\frac {7 d}{e}-\frac {a e}{c d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}-\frac {\frac {3 \left (c d^2-a e^2\right ) \left (5 a^3 e^6+9 a^2 c d^2 e^4+15 a c^2 d^4 e^2+35 c^3 d^6\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c^2 d^2 e^2}-\frac {\left (-15 a^3 e^6-2 c d e x \left (-5 a^2 e^4-6 a c d^2 e^2+35 c^2 d^4\right )-17 a^2 c d^2 e^4-25 a c^2 d^4 e^2+105 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e^2}}{6 c d e}}{8 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}-\frac {\frac {1}{3} x^2 \left (\frac {7 d}{e}-\frac {a e}{c d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}-\frac {\frac {3 \left (c d^2-a e^2\right ) \left (5 a^3 e^6+9 a^2 c d^2 e^4+15 a c^2 d^4 e^2+35 c^3 d^6\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{5/2} d^{5/2} e^{5/2}}-\frac {\left (-15 a^3 e^6-2 c d e x \left (-5 a^2 e^4-6 a c d^2 e^2+35 c^2 d^4\right )-17 a^2 c d^2 e^4-25 a c^2 d^4 e^2+105 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e^2}}{6 c d e}}{8 e}\)

Input:

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

Output:

(x^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*e) - ((((7*d)/e - (a* 
e)/(c*d))*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/3 - (-1/4*((105 
*c^3*d^6 - 25*a*c^2*d^4*e^2 - 17*a^2*c*d^2*e^4 - 15*a^3*e^6 - 2*c*d*e*(35* 
c^2*d^4 - 6*a*c*d^2*e^2 - 5*a^2*e^4)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c 
*d*e*x^2])/(c^2*d^2*e^2) + (3*(c*d^2 - a*e^2)*(35*c^3*d^6 + 15*a*c^2*d^4*e 
^2 + 9*a^2*c*d^2*e^4 + 5*a^3*e^6)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*S 
qrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(8*c 
^(5/2)*d^(5/2)*e^(5/2)))/(6*c*d*e))/(8*e)
 

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 1215

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

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 1236

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(954\) vs. \(2(302)=604\).

Time = 2.47 (sec) , antiderivative size = 955, normalized size of antiderivative = 2.89

method	result	size
default	\(\frac {d^{2} \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 {\frac {x {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}{4 d e c}-\frac {5 \left (a \,e^{2}+c \,d^{2}\right ) \left (\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}\right )}{8 d e c}-\frac {a \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 )}{4 c}}{e}-\frac {d \left (\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}\right )}{e^{2}}-\frac {d^{3} \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}}\)	\(955\)

Input:

int(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/(e*x+d),x,method=_RETURNVE 
RBOSE)
 

Output:

d^2/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))+1/e*(1/4*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/d/e/c-5/8*(a*e^2+ 
c*d^2)/d/e/c*(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)))-1/4*a/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)))-d/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)))-d^3/e^4*((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 = 0.18 (sec) , antiderivative size = 678, normalized size of antiderivative = 2.05 \[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\left [-\frac {3 \, {\left (35 \, c^{4} d^{8} - 20 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (48 \, c^{4} d^{4} e^{4} x^{3} - 105 \, c^{4} d^{7} e + 25 \, a c^{3} d^{5} e^{3} + 17 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7} - 8 \, {\left (7 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (35 \, c^{4} d^{6} e^{2} - 6 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{768 \, c^{4} d^{4} e^{5}}, -\frac {3 \, {\left (35 \, c^{4} d^{8} - 20 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (48 \, c^{4} d^{4} e^{4} x^{3} - 105 \, c^{4} d^{7} e + 25 \, a c^{3} d^{5} e^{3} + 17 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7} - 8 \, {\left (7 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (35 \, c^{4} d^{6} e^{2} - 6 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{384 \, c^{4} d^{4} e^{5}}\right ] \] Input:

integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x, algorithm 
="fricas")
 

Output:

[-1/768*(3*(35*c^4*d^8 - 20*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^ 
2*e^6 - 5*a^4*e^8)*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*(48*c^4*d^ 
4*e^4*x^3 - 105*c^4*d^7*e + 25*a*c^3*d^5*e^3 + 17*a^2*c^2*d^3*e^5 + 15*a^3 
*c*d*e^7 - 8*(7*c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^2 + 2*(35*c^4*d^6*e^2 - 6*a 
*c^3*d^4*e^4 - 5*a^2*c^2*d^2*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e 
^2)*x))/(c^4*d^4*e^5), -1/384*(3*(35*c^4*d^8 - 20*a*c^3*d^6*e^2 - 6*a^2*c^ 
2*d^4*e^4 - 4*a^3*c*d^2*e^6 - 5*a^4*e^8)*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*(48*c^4 
*d^4*e^4*x^3 - 105*c^4*d^7*e + 25*a*c^3*d^5*e^3 + 17*a^2*c^2*d^3*e^5 + 15* 
a^3*c*d*e^7 - 8*(7*c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^2 + 2*(35*c^4*d^6*e^2 - 
6*a*c^3*d^4*e^4 - 5*a^2*c^2*d^2*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + 
a*e^2)*x))/(c^4*d^4*e^5)]
 

Sympy [F]

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

integrate(x**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d),x)
 

Output:

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

Maxima [F(-2)]

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

integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),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>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {1}{192} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, x {\left (\frac {6 \, x}{e} - \frac {7 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}}{c^{3} d^{3} e^{4}}\right )} + \frac {35 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 5 \, a^{2} c d e^{5}}{c^{3} d^{3} e^{4}}\right )} x - \frac {105 \, c^{3} d^{6} - 25 \, a c^{2} d^{4} e^{2} - 17 \, a^{2} c d^{2} e^{4} - 15 \, a^{3} e^{6}}{c^{3} d^{3} e^{4}}\right )} - \frac {{\left (35 \, c^{4} d^{8} - 20 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{128 \, \sqrt {c d e} c^{3} d^{3} e^{4}} \] Input:

integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x, algorithm 
="giac")
 

Output:

1/192*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*x*(6*x/e - (7*c^3* 
d^4*e^2 - a*c^2*d^2*e^4)/(c^3*d^3*e^4)) + (35*c^3*d^5*e - 6*a*c^2*d^3*e^3 
- 5*a^2*c*d*e^5)/(c^3*d^3*e^4))*x - (105*c^3*d^6 - 25*a*c^2*d^4*e^2 - 17*a 
^2*c*d^2*e^4 - 15*a^3*e^6)/(c^3*d^3*e^4)) - 1/128*(35*c^4*d^8 - 20*a*c^3*d 
^6*e^2 - 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 - 5*a^4*e^8)*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^3*d^3*e^4)
 

Mupad [F(-1)]

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

int((x^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x),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), x)
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.75 \[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {15 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{3} c d \,e^{7}+17 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{3} e^{5}-10 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{6} x +25 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{5} e^{3}-12 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{4} e^{4} x +8 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{3} e^{5} x^{2}-105 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{7} e +70 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{6} e^{2} x -56 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{5} e^{3} x^{2}+48 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{4} e^{4} x^{3}-15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{4} e^{8}-12 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{3} c \,d^{2} e^{6}-18 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} c^{2} d^{4} e^{4}-60 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,c^{3} d^{6} e^{2}+105 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{4} d^{8}}{192 c^{4} d^{4} e^{5}} \] Input:

int(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x)
 

Output:

(15*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c*d*e**7 + 17*sqrt(d + e*x)*sqrt( 
a*e + c*d*x)*a**2*c**2*d**3*e**5 - 10*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2 
*c**2*d**2*e**6*x + 25*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**5*e**3 - 
12*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**4*e**4*x + 8*sqrt(d + e*x)*sq 
rt(a*e + c*d*x)*a*c**3*d**3*e**5*x**2 - 105*sqrt(d + e*x)*sqrt(a*e + c*d*x 
)*c**4*d**7*e + 70*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d**6*e**2*x - 56*s 
qrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d**5*e**3*x**2 + 48*sqrt(d + e*x)*sqrt 
(a*e + c*d*x)*c**4*d**4*e**4*x**3 - 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) 
)*a**4*e**8 - 12*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*c*d**2*e**6 - 1 
8*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**2*d**4*e**4 - 60*sqrt(e)*sq 
rt(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**3*d**6*e**2 + 105*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**4*d**8)/(192*c**4*d**4*e**5)