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

Optimal result

Integrand size = 36, antiderivative size = 243 \[ \int x (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=-\frac {\left (c d^2-a e^2\right ) \left (3 c d^2+5 a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c^3 d^3 e^2}+\frac {\left (3 c d^2-5 a e^2+6 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 c^2 d^2 e}+\frac {\left (c d^2-a e^2\right )^3 \left (3 c d^2+5 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{64 c^{7/2} d^{7/2} e^{5/2}} \] Output:

-1/64*(-a*e^2+c*d^2)*(5*a*e^2+3*c*d^2)*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e 
^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/e^2+1/24*(6*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)^(3/2)/c^2/d^2/e+1/64*(-a*e^2+c*d^2)^3*(5 
*a*e^2+3*c*d^2)*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^(5/2)
 

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.98 \[ \int x (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (15 a^3 e^6-a^2 c d e^4 (31 d+10 e x)+a c^2 d^2 e^2 \left (9 d^2+20 d e x+8 e^2 x^2\right )+c^3 \left (-9 d^6+6 d^5 e x+72 d^4 e^2 x^2+48 d^3 e^3 x^3\right )\right )+\frac {3 \left (c d^2-a e^2\right )^3 \left (3 c d^2+5 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{192 c^{7/2} d^{7/2} e^{5/2}} \] Input:

Integrate[x*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],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*(31*d + 10*e*x) + a*c^2*d^2*e^2*(9*d^2 + 20*d*e*x + 8*e^2*x^2) + c 
^3*(-9*d^6 + 6*d^5*e*x + 72*d^4*e^2*x^2 + 48*d^3*e^3*x^3)) + (3*(c*d^2 - a 
*e^2)^3*(3*c*d^2 + 5*a*e^2)*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c]*S 
qrt[d]*Sqrt[d + e*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(192*c^(7/2)*d 
^(7/2)*e^(5/2))
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1225, 1087, 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.

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

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\left (-5 a e^2+3 c d^2+6 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 c^2 d^2 e}-\frac {\left (-\frac {5 a^2 e^4}{c^2 d^2}+\frac {2 a e^2}{c}+3 d^2\right ) \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{16 e}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (-5 a e^2+3 c d^2+6 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 c^2 d^2 e}-\frac {\left (-\frac {5 a^2 e^4}{c^2 d^2}+\frac {2 a e^2}{c}+3 d^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{16 e}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\left (-5 a e^2+3 c d^2+6 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 c^2 d^2 e}-\frac {\left (-\frac {5 a^2 e^4}{c^2 d^2}+\frac {2 a e^2}{c}+3 d^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{16 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (-5 a e^2+3 c d^2+6 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 c^2 d^2 e}-\frac {\left (-\frac {5 a^2 e^4}{c^2 d^2}+\frac {2 a e^2}{c}+3 d^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right )}{16 e}\)

Input:

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

Output:

((3*c*d^2 - 5*a*e^2 + 6*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^( 
3/2))/(24*c^2*d^2*e) - ((3*d^2 + (2*a*e^2)/c - (5*a^2*e^4)/(c^2*d^2))*(((c 
*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4* 
c*d*e) - ((c*d^2 - a*e^2)^2*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])])/(8*c^(3/2) 
*d^(3/2)*e^(3/2))))/(16*e)
 

Defintions of rubi rules used

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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(654\) vs. \(2(219)=438\).

Time = 2.19 (sec) , antiderivative size = 655, normalized size of antiderivative = 2.70

method	result	size
default	\(d \left (\frac {{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}{3 d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{8 d e c \sqrt {d e c}}\right )}{2 d e c}\right )+e \left (\frac {x {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}{4 d e c}-\frac {5 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}{3 d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{8 d e c \sqrt {d e c}}\right )}{2 d e c}\right )}{8 d e c}-\frac {a \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{8 d e c \sqrt {d e c}}\right )}{4 c}\right )\)	\(655\)

Input:

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

Output:

d*(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)) 
)+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)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 676, normalized size of antiderivative = 2.78 \[ \int x (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\left [-\frac {3 \, {\left (3 \, c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} + 12 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (48 \, c^{4} d^{4} e^{4} x^{3} - 9 \, c^{4} d^{7} e + 9 \, a c^{3} d^{5} e^{3} - 31 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7} + 8 \, {\left (9 \, c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (3 \, c^{4} d^{6} e^{2} + 10 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{768 \, c^{4} d^{4} e^{3}}, -\frac {3 \, {\left (3 \, c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} + 12 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (48 \, c^{4} d^{4} e^{4} x^{3} - 9 \, c^{4} d^{7} e + 9 \, a c^{3} d^{5} e^{3} - 31 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7} + 8 \, {\left (9 \, c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (3 \, c^{4} d^{6} e^{2} + 10 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{384 \, c^{4} d^{4} e^{3}}\right ] \] Input:

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

Output:

[-1/768*(3*(3*c^4*d^8 - 4*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4*e^4 + 12*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 - 9*c^4*d^7*e + 9*a*c^3*d^5*e^3 - 31*a^2*c^2*d^3*e^5 + 15*a^3*c*d 
*e^7 + 8*(9*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^2 + 2*(3*c^4*d^6*e^2 + 10*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^3), -1/384*(3*(3*c^4*d^8 - 4*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4* 
e^4 + 12*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 - 9*c^4*d^7*e + 9*a*c^3*d^5*e^3 - 31*a^2*c^2*d^3*e^5 + 15*a^3*c*d* 
e^7 + 8*(9*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^2 + 2*(3*c^4*d^6*e^2 + 10*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^3)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 887 vs. \(2 (238) = 476\).

Time = 1.47 (sec) , antiderivative size = 887, normalized size of antiderivative = 3.65 \[ \int x (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise(((-a*(5*a*d*e**2/4 + c*d**3 - (5*a*e**2/2 + 5*c*d**2/2)*(a*e**3 
+ 2*c*d**2*e - e*(7*a*e**2/2 + 7*c*d**2/2)/4)/(3*c*d*e))/(2*c) - (a*e**2 + 
 c*d**2)*(a*d**2*e - 2*a*(a*e**3 + 2*c*d**2*e - e*(7*a*e**2/2 + 7*c*d**2/2 
)/4)/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(5*a*d*e**2/4 + c*d**3 - (5*a*e**2/ 
2 + 5*c*d**2/2)*(a*e**3 + 2*c*d**2*e - e*(7*a*e**2/2 + 7*c*d**2/2)/4)/(3*c 
*d*e))/(2*c*d*e))/(2*c*d*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x + 
2*sqrt(c*d*e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e), 
 Ne(a*d*e - (a*e**2 + c*d**2)**2/(4*c*d*e), 0)), ((x - (-a*e**2 - c*d**2)/ 
(2*c*d*e))*log(x - (-a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*(x - (-a*e**2 
- c*d**2)/(2*c*d*e))**2), True)) + sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c 
*d**2))*(e*x**3/4 + x**2*(a*e**3 + 2*c*d**2*e - e*(7*a*e**2/2 + 7*c*d**2/2 
)/4)/(3*c*d*e) + x*(5*a*d*e**2/4 + c*d**3 - (5*a*e**2/2 + 5*c*d**2/2)*(a*e 
**3 + 2*c*d**2*e - e*(7*a*e**2/2 + 7*c*d**2/2)/4)/(3*c*d*e))/(2*c*d*e) + ( 
a*d**2*e - 2*a*(a*e**3 + 2*c*d**2*e - e*(7*a*e**2/2 + 7*c*d**2/2)/4)/(3*c) 
 - (3*a*e**2/2 + 3*c*d**2/2)*(5*a*d*e**2/4 + c*d**3 - (5*a*e**2/2 + 5*c*d* 
*2/2)*(a*e**3 + 2*c*d**2*e - e*(7*a*e**2/2 + 7*c*d**2/2)/4)/(3*c*d*e))/(2* 
c*d*e))/(c*d*e)), Ne(c*d*e, 0)), (2*(-a*c*d**4*e*(a*d*e + x*(a*e**2 + c*d* 
*2))**(3/2)/(3*(a*e**2 + c*d**2)) + e*(a*d*e + x*(a*e**2 + c*d**2))**(7/2) 
/(7*(a*e**2 + c*d**2)) + (a*d*e + x*(a*e**2 + c*d**2))**(5/2)*(-a*d*e**2 + 
 c*d**3)/(5*(a*e**2 + c*d**2)))/(a*e**2 + c*d**2)**2, Ne(a*e**2 + c*d**...
 

Maxima [F(-2)]

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

integrate(x*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),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 = 0.14 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.25 \[ \int x (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {1}{192} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (6 \, e x + \frac {9 \, c^{3} d^{4} e^{3} + a c^{2} d^{2} e^{5}}{c^{3} d^{3} e^{3}}\right )} x + \frac {3 \, c^{3} d^{5} e^{2} + 10 \, a c^{2} d^{3} e^{4} - 5 \, a^{2} c d e^{6}}{c^{3} d^{3} e^{3}}\right )} x - \frac {9 \, c^{3} d^{6} e - 9 \, a c^{2} d^{4} e^{3} + 31 \, a^{2} c d^{2} e^{5} - 15 \, a^{3} e^{7}}{c^{3} d^{3} e^{3}}\right )} - \frac {{\left (3 \, c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} + 12 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{128 \, \sqrt {c d e} c^{3} d^{3} e^{2}} \] Input:

integrate(x*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),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*(6*e*x + (9*c^3*d^ 
4*e^3 + a*c^2*d^2*e^5)/(c^3*d^3*e^3))*x + (3*c^3*d^5*e^2 + 10*a*c^2*d^3*e^ 
4 - 5*a^2*c*d*e^6)/(c^3*d^3*e^3))*x - (9*c^3*d^6*e - 9*a*c^2*d^4*e^3 + 31* 
a^2*c*d^2*e^5 - 15*a^3*e^7)/(c^3*d^3*e^3)) - 1/128*(3*c^4*d^8 - 4*a*c^3*d^ 
6*e^2 - 6*a^2*c^2*d^4*e^4 + 12*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^2)
 

Mupad [B] (verification not implemented)

Time = 7.17 (sec) , antiderivative size = 544, normalized size of antiderivative = 2.24 \[ \int x (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {x\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{4\,c\,d}-\frac {a\,e\,\left (\left (\frac {x}{2}+\frac {c\,d^2+a\,e^2}{4\,c\,d\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}-\frac {\ln \left (2\,\sqrt {\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )}\,\sqrt {c\,d\,e}+a\,e^2+c\,d^2+2\,c\,d\,e\,x\right )\,\left (\frac {{\left (c\,d^2+a\,e^2\right )}^2}{4}-a\,c\,d^2\,e^2\right )}{2\,{\left (c\,d\,e\right )}^{3/2}}\right )}{4\,c}-\frac {\left (\frac {\ln \left (2\,\sqrt {\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )}\,\sqrt {c\,d\,e}+a\,e^2+c\,d^2+2\,c\,d\,e\,x\right )\,\left ({\left (c\,d^2+a\,e^2\right )}^3-4\,a\,c\,d^2\,e^2\,\left (c\,d^2+a\,e^2\right )\right )}{16\,{\left (c\,d\,e\right )}^{5/2}}+\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (8\,c\,d\,e\,\left (c\,d\,e\,x^2+a\,d\,e\right )-3\,{\left (c\,d^2+a\,e^2\right )}^2+2\,c\,d\,e\,x\,\left (c\,d^2+a\,e^2\right )\right )}{24\,c^2\,d^2\,e^2}\right )\,\left (5\,c\,d^2+5\,a\,e^2\right )}{8\,c\,d}+\frac {d\,\ln \left (2\,\sqrt {\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )}\,\sqrt {c\,d\,e}+a\,e^2+c\,d^2+2\,c\,d\,e\,x\right )\,\left ({\left (c\,d^2+a\,e^2\right )}^3-4\,a\,c\,d^2\,e^2\,\left (c\,d^2+a\,e^2\right )\right )}{16\,{\left (c\,d\,e\right )}^{5/2}}+\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (8\,c\,d\,e\,\left (c\,d\,e\,x^2+a\,d\,e\right )-3\,{\left (c\,d^2+a\,e^2\right )}^2+2\,c\,d\,e\,x\,\left (c\,d^2+a\,e^2\right )\right )}{24\,c^2\,d\,e^2} \] Input:

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

Output:

(x*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2))/(4*c*d) - (a*e*((x/2 + ( 
a*e^2 + c*d^2)/(4*c*d*e))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2) - 
(log(2*((a*e + c*d*x)*(d + e*x))^(1/2)*(c*d*e)^(1/2) + a*e^2 + c*d^2 + 2*c 
*d*e*x)*((a*e^2 + c*d^2)^2/4 - a*c*d^2*e^2))/(2*(c*d*e)^(3/2))))/(4*c) - ( 
((log(2*((a*e + c*d*x)*(d + e*x))^(1/2)*(c*d*e)^(1/2) + a*e^2 + c*d^2 + 2* 
c*d*e*x)*((a*e^2 + c*d^2)^3 - 4*a*c*d^2*e^2*(a*e^2 + c*d^2)))/(16*(c*d*e)^ 
(5/2)) + ((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(8*c*d*e*(a*d*e + 
c*d*e*x^2) - 3*(a*e^2 + c*d^2)^2 + 2*c*d*e*x*(a*e^2 + c*d^2)))/(24*c^2*d^2 
*e^2))*(5*a*e^2 + 5*c*d^2))/(8*c*d) + (d*log(2*((a*e + c*d*x)*(d + e*x))^( 
1/2)*(c*d*e)^(1/2) + a*e^2 + c*d^2 + 2*c*d*e*x)*((a*e^2 + c*d^2)^3 - 4*a*c 
*d^2*e^2*(a*e^2 + c*d^2)))/(16*(c*d*e)^(5/2)) + ((x*(a*e^2 + c*d^2) + a*d* 
e + c*d*e*x^2)^(1/2)*(8*c*d*e*(a*d*e + c*d*e*x^2) - 3*(a*e^2 + c*d^2)^2 + 
2*c*d*e*x*(a*e^2 + c*d^2)))/(24*c^2*d*e^2)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 579, normalized size of antiderivative = 2.38 \[ \int x (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {15 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{3} c d \,e^{7}-31 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{3} e^{5}-10 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{6} x +9 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{5} e^{3}+20 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{4} e^{4} x +8 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{3} e^{5} x^{2}-9 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{7} e +6 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{6} e^{2} x +72 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{5} e^{3} x^{2}+48 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{4} e^{4} x^{3}-15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{4} e^{8}+36 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{3} c \,d^{2} e^{6}-18 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} c^{2} d^{4} e^{4}-12 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,c^{3} d^{6} e^{2}+9 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{4} d^{8}}{192 c^{4} d^{4} e^{3}} \] Input:

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

Output:

(15*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c*d*e**7 - 31*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 + 9*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**5*e**3 + 2 
0*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**4*e**4*x + 8*sqrt(d + e*x)*sqr 
t(a*e + c*d*x)*a*c**3*d**3*e**5*x**2 - 9*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c 
**4*d**7*e + 6*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d**6*e**2*x + 72*sqrt( 
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)*sq 
rt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a* 
*4*e**8 + 36*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 - 18*sq 
rt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqr 
t(d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*c**2*d**4*e**4 - 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*c**3*d**6*e**2 + 9*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**4*d**8)/(192*c**4*d**4*e**3)