\(\int (d+e x)^2 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2} \, dx\) [201]

Optimal result

Integrand size = 37, antiderivative size = 298 \[ \int (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {5 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c^3 d^3 e}+\frac {5 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 c^2 d^2}+\frac {5 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{32 c^3 d^3 (d+e x)}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 c d}-\frac {5 \left (c d^2-a e^2\right )^4 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{64 c^{7/2} d^{7/2} e^{3/2}} \] Output:

5/64*(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/e+5/ 
24*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^2/d^2+5/32*(-a 
*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^3/d^3/(e*x+d)+1/4* 
(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d-5/64*(-a*e^2+c*d^2)^4* 
arctanh(e^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^(1/2)/d^(1/2)/(e 
*x+d))/c^(7/2)/d^(7/2)/e^(3/2)
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.75 \[ \int (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (15 a^3 e^6-5 a^2 c d e^4 (11 d+2 e x)+a c^2 d^2 e^2 \left (73 d^2+36 d e x+8 e^2 x^2\right )+c^3 d^3 \left (15 d^3+118 d^2 e x+136 d e^2 x^2+48 e^3 x^3\right )\right )-\frac {15 \left (c d^2-a e^2\right )^4 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{192 c^{7/2} d^{7/2} e^{3/2}} \] Input:

Integrate[(d + e*x)^2*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 - 5*a^ 
2*c*d*e^4*(11*d + 2*e*x) + a*c^2*d^2*e^2*(73*d^2 + 36*d*e*x + 8*e^2*x^2) + 
 c^3*d^3*(15*d^3 + 118*d^2*e*x + 136*d*e^2*x^2 + 48*e^3*x^3)) - (15*(c*d^2 
 - a*e^2)^4*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c* 
d*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(192*c^(7/2)*d^(7/2)*e^(3/2))
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {1134, 1160, 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.

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

\(\Big \downarrow \) 1134

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 1134

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1)))   Int[(d + e*x)^ 
(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[ 
c*d^2 - b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2 
*p]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(815\) vs. \(2(266)=532\).

Time = 1.69 (sec) , antiderivative size = 816, normalized size of antiderivative = 2.74

method	result	size
default	\(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^{2} \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 )+2 d e \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 )\)	\(816\)

Input:

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

Output:

d^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) 
)+e^2*(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)))+2*d*e*(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)))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 672, normalized size of antiderivative = 2.26 \[ \int (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\left [\frac {15 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (48 \, c^{4} d^{4} e^{4} x^{3} + 15 \, c^{4} d^{7} e + 73 \, a c^{3} d^{5} e^{3} - 55 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7} + 8 \, {\left (17 \, c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (59 \, c^{4} d^{6} e^{2} + 18 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{768 \, c^{4} d^{4} e^{2}}, \frac {15 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (48 \, c^{4} d^{4} e^{4} x^{3} + 15 \, c^{4} d^{7} e + 73 \, a c^{3} d^{5} e^{3} - 55 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7} + 8 \, {\left (17 \, c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (59 \, c^{4} d^{6} e^{2} + 18 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{384 \, c^{4} d^{4} e^{2}}\right ] \] Input:

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

Output:

[1/768*(15*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^ 
6 + 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 + 15*c^4*d^7*e + 73*a*c^3*d^5*e^3 - 55*a^2*c^2*d^3*e^5 + 15*a^3*c*d*e^ 
7 + 8*(17*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^2 + 2*(59*c^4*d^6*e^2 + 18*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^2), 1/384*(15*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 
 - 4*a^3*c*d^2*e^6 + 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 + 15*c^4*d^7*e + 73*a*c^3*d^5*e^3 - 55*a^2*c^2*d^3*e^5 + 15*a^3*c*d*e^7 
+ 8*(17*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^2 + 2*(59*c^4*d^6*e^2 + 18*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^2)]
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 989 vs. \(2 (282) = 564\).

Time = 0.83 (sec) , antiderivative size = 989, normalized size of antiderivative = 3.32 \[ \int (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(e**2*x**3/4 + x 
**2*(a*e**4 + 3*c*d**2*e**2 - e**2*(7*a*e**2/2 + 7*c*d**2/2)/4)/(3*c*d*e) 
+ x*(9*a*d*e**3/4 + 3*c*d**3*e - (5*a*e**2/2 + 5*c*d**2/2)*(a*e**4 + 3*c*d 
**2*e**2 - e**2*(7*a*e**2/2 + 7*c*d**2/2)/4)/(3*c*d*e))/(2*c*d*e) + (3*a*d 
**2*e**2 - 2*a*(a*e**4 + 3*c*d**2*e**2 - e**2*(7*a*e**2/2 + 7*c*d**2/2)/4) 
/(3*c) + c*d**4 - (3*a*e**2/2 + 3*c*d**2/2)*(9*a*d*e**3/4 + 3*c*d**3*e - ( 
5*a*e**2/2 + 5*c*d**2/2)*(a*e**4 + 3*c*d**2*e**2 - e**2*(7*a*e**2/2 + 7*c* 
d**2/2)/4)/(3*c*d*e))/(2*c*d*e))/(c*d*e)) + (a*d**3*e - a*(9*a*d*e**3/4 + 
3*c*d**3*e - (5*a*e**2/2 + 5*c*d**2/2)*(a*e**4 + 3*c*d**2*e**2 - e**2*(7*a 
*e**2/2 + 7*c*d**2/2)/4)/(3*c*d*e))/(2*c) - (a*e**2 + c*d**2)*(3*a*d**2*e* 
*2 - 2*a*(a*e**4 + 3*c*d**2*e**2 - e**2*(7*a*e**2/2 + 7*c*d**2/2)/4)/(3*c) 
 + c*d**4 - (3*a*e**2/2 + 3*c*d**2/2)*(9*a*d*e**3/4 + 3*c*d**3*e - (5*a*e* 
*2/2 + 5*c*d**2/2)*(a*e**4 + 3*c*d**2*e**2 - e**2*(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)))/sqr 
t(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)), Ne(c*d*e, 0)), (2*(c**2*d**6*(a 
*d*e + x*(a*e**2 + c*d**2))**(3/2)/(3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2* 
d**4)) + 2*c*d**3*e*(a*d*e + x*(a*e**2 + c*d**2))**(5/2)/(5*(a**2*e**4 ...
 

Maxima [F(-2)]

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

integrate((e*x+d)^2*(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 = 0.21 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.02 \[ \int (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {1}{192} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (6 \, e^{2} x + \frac {17 \, c^{3} d^{4} e^{4} + a c^{2} d^{2} e^{6}}{c^{3} d^{3} e^{3}}\right )} x + \frac {59 \, c^{3} d^{5} e^{3} + 18 \, a c^{2} d^{3} e^{5} - 5 \, a^{2} c d e^{7}}{c^{3} d^{3} e^{3}}\right )} x + \frac {15 \, c^{3} d^{6} e^{2} + 73 \, a c^{2} d^{4} e^{4} - 55 \, a^{2} c d^{2} e^{6} + 15 \, a^{3} e^{8}}{c^{3} d^{3} e^{3}}\right )} + \frac {5 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{128 \, \sqrt {c d e} c^{3} d^{3} e} \] Input:

integrate((e*x+d)^2*(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^2*x + (17*c^3 
*d^4*e^4 + a*c^2*d^2*e^6)/(c^3*d^3*e^3))*x + (59*c^3*d^5*e^3 + 18*a*c^2*d^ 
3*e^5 - 5*a^2*c*d*e^7)/(c^3*d^3*e^3))*x + (15*c^3*d^6*e^2 + 73*a*c^2*d^4*e 
^4 - 55*a^2*c*d^2*e^6 + 15*a^3*e^8)/(c^3*d^3*e^3)) + 5/128*(c^4*d^8 - 4*a* 
c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + 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)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.94 \[ \int (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {15 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{3} c d \,e^{7}-55 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{3} e^{5}-10 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{6} x +73 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{5} e^{3}+36 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{4} e^{4} x +8 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{3} e^{5} x^{2}+15 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{7} e +118 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{6} e^{2} x +136 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{5} e^{3} x^{2}+48 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{4} e^{4} x^{3}-15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{4} e^{8}+60 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{3} c \,d^{2} e^{6}-90 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} c^{2} d^{4} e^{4}+60 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,c^{3} d^{6} e^{2}-15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{4} d^{8}}{192 c^{4} d^{4} e^{2}} \] Input:

int((e*x+d)^2*(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 - 55*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 + 73*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**5*e**3 + 
36*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 + 15*sqrt(d + e*x)*sqrt(a*e + c*d*x) 
*c**4*d**7*e + 118*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d**6*e**2*x + 136* 
sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d**5*e**3*x**2 + 48*sqrt(d + e*x)*sqr 
t(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 + 60*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 - 
90*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)*s 
qrt(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 - 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**4*d**8)/(192*c**4*d**4*e**2)