\(\int \frac {x^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{d+e x} \, dx\) [447]

Optimal result

Integrand size = 40, antiderivative size = 352 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=-\frac {\left (c d^2-a e^2\right ) \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 e}+\frac {\left (35 c^2 d^4-12 a c d^2 e^2-15 a^2 e^4-6 c d e \left (7 c d^2-3 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}+\frac {\left (c d^2-a e^2\right )^3 \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{256 c^{7/2} d^{7/2} e^{9/2}} \]

[Out]

1/5*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e+1/240*(35*c^2*d^4-12*a*c*d^2*e^2-15*a^2*e^4-6*c*d*e*(-3*a*e^
2+7*c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^2/d^2/e^3+1/256*(-a*e^2+c*d^2)^3*(3*a^2*e^4+6*a*c*d^2*
e^2+7*c^2*d^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^(7/2)/d^(7/2)/e^(9/2)-1/128*(-a*e^2+c*d^2)*(3*a^2*e^4+6*a*c*d^2*e^2+7*c^2*d^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^3/d^3/e^4

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {865, 846, 793, 626, 635, 212} \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {\left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{7/2} d^{7/2} e^{9/2}}+\frac {\left (-15 a^2 e^4-6 c d e x \left (7 c d^2-3 a e^2\right )-12 a c d^2 e^2+35 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}-\frac {\left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^3 d^3 e^4}+\frac {x^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e} \]

[In]

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

[Out]

-1/128*((c*d^2 - a*e^2)*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*(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])/(c^3*d^3*e^4) + (x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*e) + ((35*c^
2*d^4 - 12*a*c*d^2*e^2 - 15*a^2*e^4 - 6*c*d*e*(7*c*d^2 - 3*a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(
3/2))/(240*c^2*d^2*e^3) + ((c*d^2 - a*e^2)^3*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^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])])/(256*c^(7/2)*d^(7/2)*e^(9
/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 793

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] + Dist[(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] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 846

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 865

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Int[((f + g*x)^n*(a + b*x + c*x^2)^(m + p))/(a/d + c*(x/e))^m, x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] &&
NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[m, 0] && In
tegerQ[n] && (LtQ[n, 0] || GtQ[p, 0])

Rubi steps \begin{align*} \text {integral}& = \int x^2 (a e+c d x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx \\ & = \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 e}+\frac {\int x \left (-2 a c d^2 e-\frac {1}{2} c d \left (7 c d^2-3 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{5 c d e} \\ & = \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 e}+\frac {\left (35 c^2 d^4-12 a c d^2 e^2-15 a^2 e^4-6 c d e \left (7 c d^2-3 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}-\frac {\left (\left (c d^2-a e^2\right ) \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right )\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{32 c^2 d^2 e^3} \\ & = -\frac {\left (c d^2-a e^2\right ) \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\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}}{128 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 e}+\frac {\left (35 c^2 d^4-12 a c d^2 e^2-15 a^2 e^4-6 c d e \left (7 c d^2-3 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}+\frac {\left (\left (c d^2-a e^2\right )^3 \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right )\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 c^3 d^3 e^4} \\ & = -\frac {\left (c d^2-a e^2\right ) \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\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}}{128 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 e}+\frac {\left (35 c^2 d^4-12 a c d^2 e^2-15 a^2 e^4-6 c d e \left (7 c d^2-3 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}+\frac {\left (\left (c d^2-a e^2\right )^3 \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right )\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 )}{128 c^3 d^3 e^4} \\ & = -\frac {\left (c d^2-a e^2\right ) \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\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}}{128 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 e}+\frac {\left (35 c^2 d^4-12 a c d^2 e^2-15 a^2 e^4-6 c d e \left (7 c d^2-3 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}+\frac {\left (c d^2-a e^2\right )^3 \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \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 )}{256 c^{7/2} d^{7/2} e^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (45 a^4 e^8-30 a^3 c d e^6 (d+e x)-6 a^2 c^2 d^2 e^4 \left (6 d^2-3 d e x-4 e^2 x^2\right )+2 a c^3 d^3 e^2 \left (95 d^3-61 d^2 e x+48 d e^2 x^2+264 e^3 x^3\right )+c^4 d^4 \left (-105 d^4+70 d^3 e x-56 d^2 e^2 x^2+48 d e^3 x^3+384 e^4 x^4\right )\right )+\frac {15 \left (c d^2-a e^2\right )^3 \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{1920 c^{7/2} d^{7/2} e^{9/2}} \]

[In]

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

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(45*a^4*e^8 - 30*a^3*c*d*e^6*(d + e*x) - 6*a^2*c^2*d^2
*e^4*(6*d^2 - 3*d*e*x - 4*e^2*x^2) + 2*a*c^3*d^3*e^2*(95*d^3 - 61*d^2*e*x + 48*d*e^2*x^2 + 264*e^3*x^3) + c^4*
d^4*(-105*d^4 + 70*d^3*e*x - 56*d^2*e^2*x^2 + 48*d*e^3*x^3 + 384*e^4*x^4)) + (15*(c*d^2 - a*e^2)^3*(7*c^2*d^4
+ 6*a*c*d^2*e^2 + 3*a^2*e^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])))/(1920*c^(7/2)*d^(7/2)*e^(9/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(796\) vs. \(2(322)=644\).

Time = 0.64 (sec) , antiderivative size = 797, normalized size of antiderivative = 2.26

method	result	size
default	\(\frac {\frac {{\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {5}{2}}}{5 c d e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \left (\frac {\left (2 c d e x +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 c d e x +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}+c d e x}{\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}\right )}{2 c d e}}{e}-\frac {d \left (\frac {\left (2 c d e x +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 c d e x +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}+c d e x}{\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}\right )}{e^{2}}+\frac {d^{2} \left (\frac {\left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+\frac {\left (e^{2} a -c \,d^{2}\right ) \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 c d e}-\frac {\left (e^{2} a -c \,d^{2}\right )^{2} \ln \left (\frac {\frac {e^{2} a}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 c d e \sqrt {c d e}}\right )}{2}\right )}{e^{3}}\)	\(797\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 846, normalized size of antiderivative = 2.40 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\left [-\frac {15 \, {\left (7 \, c^{5} d^{10} - 15 \, a c^{4} d^{8} e^{2} + 6 \, a^{2} c^{3} d^{6} e^{4} + 2 \, a^{3} c^{2} d^{4} e^{6} + 3 \, a^{4} c d^{2} e^{8} - 3 \, a^{5} e^{10}\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 (384 \, c^{5} d^{5} e^{5} x^{4} - 105 \, c^{5} d^{9} e + 190 \, a c^{4} d^{7} e^{3} - 36 \, a^{2} c^{3} d^{5} e^{5} - 30 \, a^{3} c^{2} d^{3} e^{7} + 45 \, a^{4} c d e^{9} + 48 \, {\left (c^{5} d^{6} e^{4} + 11 \, a c^{4} d^{4} e^{6}\right )} x^{3} - 8 \, {\left (7 \, c^{5} d^{7} e^{3} - 12 \, a c^{4} d^{5} e^{5} - 3 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{2} + 2 \, {\left (35 \, c^{5} d^{8} e^{2} - 61 \, a c^{4} d^{6} e^{4} + 9 \, a^{2} c^{3} d^{4} e^{6} - 15 \, a^{3} c^{2} d^{2} e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{7680 \, c^{4} d^{4} e^{5}}, -\frac {15 \, {\left (7 \, c^{5} d^{10} - 15 \, a c^{4} d^{8} e^{2} + 6 \, a^{2} c^{3} d^{6} e^{4} + 2 \, a^{3} c^{2} d^{4} e^{6} + 3 \, a^{4} c d^{2} e^{8} - 3 \, a^{5} e^{10}\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 (384 \, c^{5} d^{5} e^{5} x^{4} - 105 \, c^{5} d^{9} e + 190 \, a c^{4} d^{7} e^{3} - 36 \, a^{2} c^{3} d^{5} e^{5} - 30 \, a^{3} c^{2} d^{3} e^{7} + 45 \, a^{4} c d e^{9} + 48 \, {\left (c^{5} d^{6} e^{4} + 11 \, a c^{4} d^{4} e^{6}\right )} x^{3} - 8 \, {\left (7 \, c^{5} d^{7} e^{3} - 12 \, a c^{4} d^{5} e^{5} - 3 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{2} + 2 \, {\left (35 \, c^{5} d^{8} e^{2} - 61 \, a c^{4} d^{6} e^{4} + 9 \, a^{2} c^{3} d^{4} e^{6} - 15 \, a^{3} c^{2} d^{2} e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{3840 \, c^{4} d^{4} e^{5}}\right ] \]

[In]

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

[Out]

[-1/7680*(15*(7*c^5*d^10 - 15*a*c^4*d^8*e^2 + 6*a^2*c^3*d^6*e^4 + 2*a^3*c^2*d^4*e^6 + 3*a^4*c*d^2*e^8 - 3*a^5*
e^10)*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*(384*c^5*d^5*e^5*x^4
- 105*c^5*d^9*e + 190*a*c^4*d^7*e^3 - 36*a^2*c^3*d^5*e^5 - 30*a^3*c^2*d^3*e^7 + 45*a^4*c*d*e^9 + 48*(c^5*d^6*e
^4 + 11*a*c^4*d^4*e^6)*x^3 - 8*(7*c^5*d^7*e^3 - 12*a*c^4*d^5*e^5 - 3*a^2*c^3*d^3*e^7)*x^2 + 2*(35*c^5*d^8*e^2
- 61*a*c^4*d^6*e^4 + 9*a^2*c^3*d^4*e^6 - 15*a^3*c^2*d^2*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(
c^4*d^4*e^5), -1/3840*(15*(7*c^5*d^10 - 15*a*c^4*d^8*e^2 + 6*a^2*c^3*d^6*e^4 + 2*a^3*c^2*d^4*e^6 + 3*a^4*c*d^2
*e^8 - 3*a^5*e^10)*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*(384*c^5*d^5*e^5*x^4 - 105*
c^5*d^9*e + 190*a*c^4*d^7*e^3 - 36*a^2*c^3*d^5*e^5 - 30*a^3*c^2*d^3*e^7 + 45*a^4*c*d*e^9 + 48*(c^5*d^6*e^4 + 1
1*a*c^4*d^4*e^6)*x^3 - 8*(7*c^5*d^7*e^3 - 12*a*c^4*d^5*e^5 - 3*a^2*c^3*d^3*e^7)*x^2 + 2*(35*c^5*d^8*e^2 - 61*a
*c^4*d^6*e^4 + 9*a^2*c^3*d^4*e^6 - 15*a^3*c^2*d^2*e^8)*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(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d),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 = 0.35 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.15 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {1}{1920} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, c d x + \frac {c^{5} d^{6} e^{3} + 11 \, a c^{4} d^{4} e^{5}}{c^{4} d^{4} e^{4}}\right )} x - \frac {7 \, c^{5} d^{7} e^{2} - 12 \, a c^{4} d^{5} e^{4} - 3 \, a^{2} c^{3} d^{3} e^{6}}{c^{4} d^{4} e^{4}}\right )} x + \frac {35 \, c^{5} d^{8} e - 61 \, a c^{4} d^{6} e^{3} + 9 \, a^{2} c^{3} d^{4} e^{5} - 15 \, a^{3} c^{2} d^{2} e^{7}}{c^{4} d^{4} e^{4}}\right )} x - \frac {105 \, c^{5} d^{9} - 190 \, a c^{4} d^{7} e^{2} + 36 \, a^{2} c^{3} d^{5} e^{4} + 30 \, a^{3} c^{2} d^{3} e^{6} - 45 \, a^{4} c d e^{8}}{c^{4} d^{4} e^{4}}\right )} - \frac {{\left (7 \, c^{5} d^{10} - 15 \, a c^{4} d^{8} e^{2} + 6 \, a^{2} c^{3} d^{6} e^{4} + 2 \, a^{3} c^{2} d^{4} e^{6} + 3 \, a^{4} c d^{2} e^{8} - 3 \, a^{5} e^{10}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{256 \, \sqrt {c d e} c^{3} d^{3} e^{4}} \]

[In]

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

[Out]

1/1920*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(6*(8*c*d*x + (c^5*d^6*e^3 + 11*a*c^4*d^4*e^5)/(c^4*d
^4*e^4))*x - (7*c^5*d^7*e^2 - 12*a*c^4*d^5*e^4 - 3*a^2*c^3*d^3*e^6)/(c^4*d^4*e^4))*x + (35*c^5*d^8*e - 61*a*c^
4*d^6*e^3 + 9*a^2*c^3*d^4*e^5 - 15*a^3*c^2*d^2*e^7)/(c^4*d^4*e^4))*x - (105*c^5*d^9 - 190*a*c^4*d^7*e^2 + 36*a
^2*c^3*d^5*e^4 + 30*a^3*c^2*d^3*e^6 - 45*a^4*c*d*e^8)/(c^4*d^4*e^4)) - 1/256*(7*c^5*d^10 - 15*a*c^4*d^8*e^2 +
6*a^2*c^3*d^6*e^4 + 2*a^3*c^2*d^4*e^6 + 3*a^4*c*d^2*e^8 - 3*a^5*e^10)*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. \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {x^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{d+e\,x} \,d x \]

[In]

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

[Out]

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