\(\int (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2} \, dx\) [1922]

Optimal result

Integrand size = 29, antiderivative size = 232 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=-\frac {3 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c^2 d^2 e^2}+\frac {\left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 c d e}+\frac {3 \left (c d^2-a e^2\right )^4 \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 c^{5/2} d^{5/2} e^{5/2}} \]

[Out]

1/8*(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)^(3/2)/c/d/e+3/128*(-a*e^2+c*d^2)^4*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^(5/2
)-3/64*(-a*e^2+c*d^2)^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^2

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {626, 635, 212} \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {3 \left (c d^2-a e^2\right )^4 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{5/2} d^{5/2} e^{5/2}}-\frac {3 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 c^2 d^2 e^2}+\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e} \]

[In]

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

[Out]

(-3*(c*d^2 - a*e^2)^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])/(64*c^2*d^2*e^2
) + ((c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(8*c*d*e) + (3*(c*d^2 - a*e^2)
^4*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])
])/(128*c^(5/2)*d^(5/2)*e^(5/2))

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (c d^2+a e^2+2 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}}{8 c d e}-\frac {\left (3 \left (c d^2-a e^2\right )^2\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{16 c d e} \\ & = -\frac {3 \left (c d^2-a e^2\right )^2 \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^2 d^2 e^2}+\frac {\left (c d^2+a e^2+2 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}}{8 c d e}+\frac {\left (3 \left (c d^2-a e^2\right )^4\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 c^2 d^2 e^2} \\ & = -\frac {3 \left (c d^2-a e^2\right )^2 \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^2 d^2 e^2}+\frac {\left (c d^2+a e^2+2 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}}{8 c d e}+\frac {\left (3 \left (c d^2-a e^2\right )^4\right ) \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 )}{64 c^2 d^2 e^2} \\ & = -\frac {3 \left (c d^2-a e^2\right )^2 \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^2 d^2 e^2}+\frac {\left (c d^2+a e^2+2 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}}{8 c d e}+\frac {3 \left (c d^2-a e^2\right )^4 \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 )}{128 c^{5/2} d^{5/2} e^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.03 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (-3 a^3 e^6+a^2 c d e^4 (11 d+2 e x)+a c^2 d^2 e^2 \left (11 d^2+44 d e x+24 e^2 x^2\right )+c^3 d^3 \left (-3 d^3+2 d^2 e x+24 d e^2 x^2+16 e^3 x^3\right )\right )}{(a e+c d x) (d+e x)}+\frac {3 \left (c d^2-a e^2\right )^4 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{64 c^{5/2} d^{5/2} e^{5/2}} \]

[In]

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

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(-3*a^3*e^6 + a^2*c*d*e^4*(11*d + 2*e*x) + a*c^2*d^
2*e^2*(11*d^2 + 44*d*e*x + 24*e^2*x^2) + c^3*d^3*(-3*d^3 + 2*d^2*e*x + 24*d*e^2*x^2 + 16*e^3*x^3)))/((a*e + c*
d*x)*(d + e*x)) + (3*(c*d^2 - a*e^2)^4*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(
(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(64*c^(5/2)*d^(5/2)*e^(5/2))

Maple [A] (verified)

Time = 2.35 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.06

method	result	size
default	\(\frac {\left (2 x c d e +e^{2} a +c \,d^{2}\right ) {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 x c d e +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+x c d e}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\)	\(247\)

[In]

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

[Out]

1/8*(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)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^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 = 0.37 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.87 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\left [\frac {3 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (16 \, c^{4} d^{4} e^{4} x^{3} - 3 \, c^{4} d^{7} e + 11 \, a c^{3} d^{5} e^{3} + 11 \, a^{2} c^{2} d^{3} e^{5} - 3 \, a^{3} c d e^{7} + 24 \, {\left (c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (c^{4} d^{6} e^{2} + 22 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{256 \, c^{3} d^{3} e^{3}}, -\frac {3 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (16 \, c^{4} d^{4} e^{4} x^{3} - 3 \, c^{4} d^{7} e + 11 \, a c^{3} d^{5} e^{3} + 11 \, a^{2} c^{2} d^{3} e^{5} - 3 \, a^{3} c d e^{7} + 24 \, {\left (c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (c^{4} d^{6} e^{2} + 22 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{128 \, c^{3} d^{3} e^{3}}\right ] \]

[In]

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

[Out]

[1/256*(3*(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*(16*c^4*d^4*e^4*x^3 - 3*c^4*d^7*e + 11*a*c^3*d^5*e^
3 + 11*a^2*c^2*d^3*e^5 - 3*a^3*c*d*e^7 + 24*(c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^2 + 2*(c^4*d^6*e^2 + 22*a*c^3*d^4*
e^4 + 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^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 - 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*(16*c^4*d^4*e^4*x^3 - 3*c^4*d^7*e + 11*a*c^3*d^5*e^3 + 11*a^2*c^2*d^3*e^5 - 3*a^3*c*d*e^7 + 24
*(c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^2 + 2*(c^4*d^6*e^2 + 22*a*c^3*d^4*e^4 + 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^3*d^3*e^3)]

Sympy [A] (verification not implemented)

Time = 1.71 (sec) , antiderivative size = 1853, normalized size of antiderivative = 7.99 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

a*d*e*Piecewise(((x/2 + (a*e**2/4 + c*d**2/4)/(c*d*e))*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)) + (a*d*e
/2 - (a*e**2/4 + c*d**2/4)*(a*e**2 + c*d**2)/(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*(a*d*e + x*(a*e**2 + c*d**2))**(3/2)/(3*(a*e**2 + c*d**2)), Ne(a*
e**2 + c*d**2, 0)), (x*sqrt(a*d*e), True)) + a*e**2*Piecewise(((-a*(a*e**2/6 + c*d**2/6)/(2*c) - (a*e**2 + c*d
**2)*(a*d*e/3 - (a*e**2/6 + c*d**2/6)*(3*a*e**2/2 + 3*c*d**2/2)/(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))/sq
rt(c*d*e*(x - (-a*e**2 - c*d**2)/(2*c*d*e))**2), True)) + (x**2/3 + x*(a*e**2/6 + c*d**2/6)/(2*c*d*e) + (a*d*e
/3 - (a*e**2/6 + c*d**2/6)*(3*a*e**2/2 + 3*c*d**2/2)/(2*c*d*e))/(c*d*e))*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 +
c*d**2)), Ne(c*d*e, 0)), (2*(-a*d*e*(a*d*e + x*(a*e**2 + c*d**2))**(3/2)/3 + (a*d*e + x*(a*e**2 + c*d**2))**(
5/2)/5)/(a*e**2 + c*d**2)**2, Ne(a*e**2 + c*d**2, 0)), (x**2*sqrt(a*d*e)/2, True)) + c*d**2*Piecewise(((-a*(a*
e**2/6 + c*d**2/6)/(2*c) - (a*e**2 + c*d**2)*(a*d*e/3 - (a*e**2/6 + c*d**2/6)*(3*a*e**2/2 + 3*c*d**2/2)/(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)) + (x**2/3 + x*
(a*e**2/6 + c*d**2/6)/(2*c*d*e) + (a*d*e/3 - (a*e**2/6 + c*d**2/6)*(3*a*e**2/2 + 3*c*d**2/2)/(2*c*d*e))/(c*d*e
))*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)), Ne(c*d*e, 0)), (2*(-a*d*e*(a*d*e + x*(a*e**2 + c*d**2))**(3
/2)/3 + (a*d*e + x*(a*e**2 + c*d**2))**(5/2)/5)/(a*e**2 + c*d**2)**2, Ne(a*e**2 + c*d**2, 0)), (x**2*sqrt(a*d*
e)/2, True)) + c*d*e*Piecewise(((-a*(a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d*e))/(2*c
) - (a*e**2 + c*d**2)*(-2*a*(a*e**2/8 + c*d**2/8)/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(a*d*e/4 - (a*e**2/8 + c*d
**2/8)*(5*a*e**2/2 + 5*c*d**2/2)/(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))*(x**3/4 + x**2*(a*e**2/8
+ c*d**2/8)/(3*c*d*e) + x*(a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d*e))/(2*c*d*e) + (-
2*a*(a*e**2/8 + c*d**2/8)/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e**2/2 + 5*c
*d**2/2)/(3*c*d*e))/(2*c*d*e))/(c*d*e)), Ne(c*d*e, 0)), (2*(a**2*d**2*e**2*(a*d*e + x*(a*e**2 + c*d**2))**(3/2
)/3 - 2*a*d*e*(a*d*e + x*(a*e**2 + c*d**2))**(5/2)/5 + (a*d*e + x*(a*e**2 + c*d**2))**(7/2)/7)/(a*e**2 + c*d**
2)**3, Ne(a*e**2 + c*d**2, 0)), (x**3*sqrt(a*d*e)/3, True))

Maxima [F(-2)]

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

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),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 = 0.33 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.34 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {1}{64} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, c d e x + \frac {3 \, {\left (c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )}}{c^{3} d^{3} e^{3}}\right )} x + \frac {c^{4} d^{6} e^{2} + 22 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}}{c^{3} d^{3} e^{3}}\right )} x - \frac {3 \, c^{4} d^{7} e - 11 \, a c^{3} d^{5} e^{3} - 11 \, a^{2} c^{2} d^{3} e^{5} + 3 \, a^{3} c d e^{7}}{c^{3} d^{3} e^{3}}\right )} - \frac {3 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{128 \, \sqrt {c d e} c^{2} d^{2} e^{2}} \]

[In]

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

[Out]

1/64*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*c*d*e*x + 3*(c^4*d^5*e^3 + a*c^3*d^3*e^5)/(c^3*d^3*e
^3))*x + (c^4*d^6*e^2 + 22*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)/(c^3*d^3*e^3))*x - (3*c^4*d^7*e - 11*a*c^3*d^5*e^3
- 11*a^2*c^2*d^3*e^5 + 3*a^3*c*d*e^7)/(c^3*d^3*e^3)) - 3/128*(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^2*d^2*e^2)

Mupad [B] (verification not implemented)

Time = 10.19 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.97 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {\left (\frac {c\,d^2}{2}+c\,x\,d\,e+\frac {a\,e^2}{2}\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{4\,c\,d\,e}-\frac {\left (\frac {3\,{\left (c\,d^2+a\,e^2\right )}^2}{4}-3\,a\,c\,d^2\,e^2\right )\,\left (\left (\frac {x}{2}+\frac {c\,d^2+a\,e^2}{4\,c\,d\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}-\frac {\ln \left (2\,\sqrt {\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )}\,\sqrt {c\,d\,e}+a\,e^2+c\,d^2+2\,c\,d\,e\,x\right )\,\left (\frac {{\left (c\,d^2+a\,e^2\right )}^2}{4}-a\,c\,d^2\,e^2\right )}{2\,{\left (c\,d\,e\right )}^{3/2}}\right )}{4\,c\,d\,e} \]

[In]

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

[Out]

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