\(\int (d+e x) \sqrt {a d e+(c d^2+a e^2) x+c d e x^2} \, dx\) [1910]

Optimal result

Integrand size = 35, antiderivative size = 210 \[ \int (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 c d}-\frac {\left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 c^{5/2} d^{5/2} e^{3/2}} \]

[Out]

1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d-1/16*(-a*e^2+c*d^2)^3*arctanh(1/2*(2*c*d*e*x+a*e^2+c*d^2)/c^(1
/2)/d^(1/2)/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^(3/2)+1/8*(-a*e^2+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^2/d^2/e

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {654, 626, 635, 212} \[ \int (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=-\frac {\left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 c^{5/2} d^{5/2} e^{3/2}}+\frac {\left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c^2 d^2 e}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 c d} \]

[In]

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

[Out]

((c*d^2 - a*e^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])/(8*c^2*d^2*e) + (a*d
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(3*c*d) - ((c*d^2 - a*e^2)^3*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])])/(16*c^(5/2)*d^(5/2)*e^(3/2))

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

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] - Dist[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]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 654

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] + Dist[(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[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \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}+\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{2 d} \\ & = \frac {\left (c d^2-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}}{8 c^2 d^2 e}+\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}-\frac {\left (c d^2-a e^2\right )^3 \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 c^2 d^2 e} \\ & = \frac {\left (c d^2-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}}{8 c^2 d^2 e}+\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}-\frac {\left (c d^2-a e^2\right )^3 \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^2 d^2 e} \\ & = \frac {\left (c d^2-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}}{8 c^2 d^2 e}+\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}-\frac {\left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 c^{5/2} d^{5/2} e^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.85 \[ \int (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (-3 a^2 e^4+2 a c d e^2 (4 d+e x)+c^2 d^2 \left (3 d^2+14 d e x+8 e^2 x^2\right )\right )-\frac {3 \left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{24 c^{5/2} d^{5/2} e^{3/2}} \]

[In]

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

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(-3*a^2*e^4 + 2*a*c*d*e^2*(4*d + e*x) + c^2*d^2*(3*d^2
+ 14*d*e*x + 8*e^2*x^2)) - (3*(c*d^2 - a*e^2)^3*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])))/(24*c^(5/2)*d^(5/2)*e^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(372\) vs. \(2(184)=368\).

Time = 2.71 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.78

method	result	size
default	\(d \left (\frac {\left (2 x c d e +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+x c d e}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )+e \left (\frac {{\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}{3 c d e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \left (\frac {\left (2 x c d e +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+x c d e}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{2 c d e}\right )\)	\(373\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.53 \[ \int (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\left [-\frac {3 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (8 \, c^{3} d^{3} e^{3} x^{2} + 3 \, c^{3} d^{5} e + 8 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5} + 2 \, {\left (7 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{96 \, c^{3} d^{3} e^{2}}, \frac {3 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (8 \, c^{3} d^{3} e^{3} x^{2} + 3 \, c^{3} d^{5} e + 8 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5} + 2 \, {\left (7 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{48 \, c^{3} d^{3} e^{2}}\right ] \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (197) = 394\).

Time = 0.90 (sec) , antiderivative size = 571, normalized size of antiderivative = 2.72 \[ \int (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\begin {cases} \left (\frac {e x^{2}}{3} + \frac {x \left (a e^{3} + 2 c d^{2} e - \frac {e \left (\frac {5 a e^{2}}{2} + \frac {5 c d^{2}}{2}\right )}{3}\right )}{2 c d e} + \frac {\frac {4 a d e^{2}}{3} + c d^{3} - \frac {\left (\frac {3 a e^{2}}{2} + \frac {3 c d^{2}}{2}\right ) \left (a e^{3} + 2 c d^{2} e - \frac {e \left (\frac {5 a e^{2}}{2} + \frac {5 c d^{2}}{2}\right )}{3}\right )}{2 c d e}}{c d e}\right ) \sqrt {a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} + \left (a d^{2} e - \frac {a \left (a e^{3} + 2 c d^{2} e - \frac {e \left (\frac {5 a e^{2}}{2} + \frac {5 c d^{2}}{2}\right )}{3}\right )}{2 c} - \frac {\left (a e^{2} + c d^{2}\right ) \left (\frac {4 a d e^{2}}{3} + c d^{3} - \frac {\left (\frac {3 a e^{2}}{2} + \frac {3 c d^{2}}{2}\right ) \left (a e^{3} + 2 c d^{2} e - \frac {e \left (\frac {5 a e^{2}}{2} + \frac {5 c d^{2}}{2}\right )}{3}\right )}{2 c d e}\right )}{2 c d e}\right ) \left (\begin {cases} \frac {\log {\left (a e^{2} + c d^{2} + 2 c d e x + 2 \sqrt {c d e} \sqrt {a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \right )}}{\sqrt {c d e}} & \text {for}\: a d e - \frac {\left (a e^{2} + c d^{2}\right )^{2}}{4 c d e} \neq 0 \\\frac {\left (x - \frac {- a e^{2} - c d^{2}}{2 c d e}\right ) \log {\left (x - \frac {- a e^{2} - c d^{2}}{2 c d e} \right )}}{\sqrt {c d e \left (x - \frac {- a e^{2} - c d^{2}}{2 c d e}\right )^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c d e \neq 0 \\\frac {2 \left (\frac {c d^{3} \left (a d e + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {3}{2}}}{3 \left (a e^{2} + c d^{2}\right )} + \frac {e \left (a d e + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {5}{2}}}{5 \left (a e^{2} + c d^{2}\right )}\right )}{a e^{2} + c d^{2}} & \text {for}\: a e^{2} + c d^{2} \neq 0 \\\sqrt {a d e} \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.06 \[ \int (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {1}{24} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, e x + \frac {7 \, c^{2} d^{3} e^{2} + a c d e^{4}}{c^{2} d^{2} e^{2}}\right )} x + \frac {3 \, c^{2} d^{4} e + 8 \, a c d^{2} e^{3} - 3 \, a^{2} e^{5}}{c^{2} d^{2} e^{2}}\right )} + \frac {{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{16 \, \sqrt {c d e} c^{2} d^{2} e} \]

[In]

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

[Out]

1/24*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*e*x + (7*c^2*d^3*e^2 + a*c*d*e^4)/(c^2*d^2*e^2))*x + (3
*c^2*d^4*e + 8*a*c*d^2*e^3 - 3*a^2*e^5)/(c^2*d^2*e^2)) + 1/16*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a
^3*e^6)*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^2*d^2*e)

Mupad [B] (verification not implemented)

Time = 10.21 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.46 \[ \int (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=d\,\left (\frac {x}{2}+\frac {c\,d^2+a\,e^2}{4\,c\,d\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}-\frac {d\,\ln \left (2\,\sqrt {\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )}\,\sqrt {c\,d\,e}+a\,e^2+c\,d^2+2\,c\,d\,e\,x\right )\,\left (\frac {{\left (c\,d^2+a\,e^2\right )}^2}{4}-a\,c\,d^2\,e^2\right )}{2\,{\left (c\,d\,e\right )}^{3/2}}+\frac {e\,\ln \left (2\,\sqrt {\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )}\,\sqrt {c\,d\,e}+a\,e^2+c\,d^2+2\,c\,d\,e\,x\right )\,\left ({\left (c\,d^2+a\,e^2\right )}^3-4\,a\,c\,d^2\,e^2\,\left (c\,d^2+a\,e^2\right )\right )}{16\,{\left (c\,d\,e\right )}^{5/2}}+\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (8\,c\,d\,e\,\left (c\,d\,e\,x^2+a\,d\,e\right )-3\,{\left (c\,d^2+a\,e^2\right )}^2+2\,c\,d\,e\,x\,\left (c\,d^2+a\,e^2\right )\right )}{24\,c^2\,d^2\,e} \]

[In]

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

[Out]

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