3.486 \(\int \frac{1}{x^4 (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=664 \[ -\frac{\left (33 a^2 c d^2 e^4-21 a^3 e^6-3 a c^2 d^4 e^2+7 c^3 d^6\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 d^3 e^2 x^3 \left (c d^2-a e^2\right )^3}-\frac{\left (-54 a^2 c^3 d^6 e^4-78 a^3 c^2 d^4 e^6+525 a^4 c d^2 e^8-315 a^5 e^{10}-55 a c^4 d^8 e^2+105 c^5 d^{10}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 a^4 d^5 e^4 x \left (c d^2-a e^2\right )^3}+\frac{\left (-18 a^2 c^2 d^4 e^4+168 a^3 c d^2 e^6-105 a^4 e^8-16 a c^3 d^6 e^2+35 c^4 d^8\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 a^3 d^4 e^3 x^2 \left (c d^2-a e^2\right )^3}+\frac{2 \left (c d e x \left (-9 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+13 a^2 c d^2 e^4-9 a^3 e^6+a c^2 d^4 e^2+3 c^3 d^6\right )}{3 a d^2 e x^3 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{5 \left (21 a^2 c d^2 e^4+21 a^3 e^6+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 a^{9/2} d^{11/2} e^{9/2}}-\frac{2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

(-2*e*(a*e + c*d*x))/(3*d*(c*d^2 - a*e^2)*x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (2*(3*c^3*d^6 +
a*c^2*d^4*e^2 + 13*a^2*c*d^2*e^4 - 9*a^3*e^6 + c*d*e*(3*c^2*d^4 + 14*a*c*d^2*e^2 - 9*a^2*e^4)*x))/(3*a*d^2*e*
(c*d^2 - a*e^2)^3*x^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - ((7*c^3*d^6 - 3*a*c^2*d^4*e^2 + 33*a^2*c*
d^2*e^4 - 21*a^3*e^6)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*a^2*d^3*e^2*(c*d^2 - a*e^2)^3*x^3) + ((3
5*c^4*d^8 - 16*a*c^3*d^6*e^2 - 18*a^2*c^2*d^4*e^4 + 168*a^3*c*d^2*e^6 - 105*a^4*e^8)*Sqrt[a*d*e + (c*d^2 + a*e
^2)*x + c*d*e*x^2])/(12*a^3*d^4*e^3*(c*d^2 - a*e^2)^3*x^2) - ((105*c^5*d^10 - 55*a*c^4*d^8*e^2 - 54*a^2*c^3*d^
6*e^4 - 78*a^3*c^2*d^4*e^6 + 525*a^4*c*d^2*e^8 - 315*a^5*e^10)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(2
4*a^4*d^5*e^4*(c*d^2 - a*e^2)^3*x) + (5*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*ArcTanh
[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(16*a
^(9/2)*d^(11/2)*e^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 1.16988, antiderivative size = 664, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {851, 822, 834, 806, 724, 206} \[ -\frac{\left (33 a^2 c d^2 e^4-21 a^3 e^6-3 a c^2 d^4 e^2+7 c^3 d^6\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 d^3 e^2 x^3 \left (c d^2-a e^2\right )^3}-\frac{\left (-54 a^2 c^3 d^6 e^4-78 a^3 c^2 d^4 e^6+525 a^4 c d^2 e^8-315 a^5 e^{10}-55 a c^4 d^8 e^2+105 c^5 d^{10}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 a^4 d^5 e^4 x \left (c d^2-a e^2\right )^3}+\frac{\left (-18 a^2 c^2 d^4 e^4+168 a^3 c d^2 e^6-105 a^4 e^8-16 a c^3 d^6 e^2+35 c^4 d^8\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 a^3 d^4 e^3 x^2 \left (c d^2-a e^2\right )^3}+\frac{2 \left (c d e x \left (-9 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+13 a^2 c d^2 e^4-9 a^3 e^6+a c^2 d^4 e^2+3 c^3 d^6\right )}{3 a d^2 e x^3 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{5 \left (21 a^2 c d^2 e^4+21 a^3 e^6+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 a^{9/2} d^{11/2} e^{9/2}}-\frac{2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

(-2*e*(a*e + c*d*x))/(3*d*(c*d^2 - a*e^2)*x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (2*(3*c^3*d^6 +
a*c^2*d^4*e^2 + 13*a^2*c*d^2*e^4 - 9*a^3*e^6 + c*d*e*(3*c^2*d^4 + 14*a*c*d^2*e^2 - 9*a^2*e^4)*x))/(3*a*d^2*e*
(c*d^2 - a*e^2)^3*x^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - ((7*c^3*d^6 - 3*a*c^2*d^4*e^2 + 33*a^2*c*
d^2*e^4 - 21*a^3*e^6)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*a^2*d^3*e^2*(c*d^2 - a*e^2)^3*x^3) + ((3
5*c^4*d^8 - 16*a*c^3*d^6*e^2 - 18*a^2*c^2*d^4*e^4 + 168*a^3*c*d^2*e^6 - 105*a^4*e^8)*Sqrt[a*d*e + (c*d^2 + a*e
^2)*x + c*d*e*x^2])/(12*a^3*d^4*e^3*(c*d^2 - a*e^2)^3*x^2) - ((105*c^5*d^10 - 55*a*c^4*d^8*e^2 - 54*a^2*c^3*d^
6*e^4 - 78*a^3*c^2*d^4*e^6 + 525*a^4*c*d^2*e^8 - 315*a^5*e^10)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(2
4*a^4*d^5*e^4*(c*d^2 - a*e^2)^3*x) + (5*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*ArcTanh
[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(16*a
^(9/2)*d^(11/2)*e^(9/2))

Rule 851

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])

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*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] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 724

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^4 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\int \frac{a e+c d x}{x^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\\ &=-\frac{2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{2 \int \frac{-\frac{3}{2} a e \left (c d^2-3 a e^2\right ) \left (c d^2-a e^2\right )+5 a c d e^2 \left (c d^2-a e^2\right ) x}{x^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 a d e \left (c d^2-a e^2\right )^2}\\ &=-\frac{2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{2 \left (3 c^3 d^6+a c^2 d^4 e^2+13 a^2 c d^2 e^4-9 a^3 e^6+c d e \left (3 c^2 d^4+14 a c d^2 e^2-9 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{4 \int \frac{\frac{3}{4} a e \left (c d^2-a e^2\right ) \left (7 c^3 d^6-3 a c^2 d^4 e^2+33 a^2 c d^2 e^4-21 a^3 e^6\right )+\frac{3}{2} a c d e^2 \left (c d^2-a e^2\right ) \left (3 c^2 d^4+14 a c d^2 e^2-9 a^2 e^4\right ) x}{x^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 a^2 d^2 e^2 \left (c d^2-a e^2\right )^4}\\ &=-\frac{2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{2 \left (3 c^3 d^6+a c^2 d^4 e^2+13 a^2 c d^2 e^4-9 a^3 e^6+c d e \left (3 c^2 d^4+14 a c d^2 e^2-9 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (7 c^3 d^6-3 a c^2 d^4 e^2+33 a^2 c d^2 e^4-21 a^3 e^6\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^3}-\frac{4 \int \frac{\frac{3}{8} a e \left (c d^2-a e^2\right ) \left (35 c^4 d^8-16 a c^3 d^6 e^2-18 a^2 c^2 d^4 e^4+168 a^3 c d^2 e^6-105 a^4 e^8\right )+\frac{3}{2} a c d e^2 \left (c d^2-a e^2\right ) \left (7 c^3 d^6-3 a c^2 d^4 e^2+33 a^2 c d^2 e^4-21 a^3 e^6\right ) x}{x^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{9 a^3 d^3 e^3 \left (c d^2-a e^2\right )^4}\\ &=-\frac{2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{2 \left (3 c^3 d^6+a c^2 d^4 e^2+13 a^2 c d^2 e^4-9 a^3 e^6+c d e \left (3 c^2 d^4+14 a c d^2 e^2-9 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (7 c^3 d^6-3 a c^2 d^4 e^2+33 a^2 c d^2 e^4-21 a^3 e^6\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^3}+\frac{\left (35 c^4 d^8-16 a c^3 d^6 e^2-18 a^2 c^2 d^4 e^4+168 a^3 c d^2 e^6-105 a^4 e^8\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 d^4 e^3 \left (c d^2-a e^2\right )^3 x^2}+\frac{2 \int \frac{\frac{3}{16} a e \left (c d^2-a e^2\right ) \left (105 c^5 d^{10}-55 a c^4 d^8 e^2-54 a^2 c^3 d^6 e^4-78 a^3 c^2 d^4 e^6+525 a^4 c d^2 e^8-315 a^5 e^{10}\right )+\frac{3}{8} a c d e^2 \left (c d^2-a e^2\right ) \left (35 c^4 d^8-16 a c^3 d^6 e^2-18 a^2 c^2 d^4 e^4+168 a^3 c d^2 e^6-105 a^4 e^8\right ) x}{x^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{9 a^4 d^4 e^4 \left (c d^2-a e^2\right )^4}\\ &=-\frac{2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{2 \left (3 c^3 d^6+a c^2 d^4 e^2+13 a^2 c d^2 e^4-9 a^3 e^6+c d e \left (3 c^2 d^4+14 a c d^2 e^2-9 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (7 c^3 d^6-3 a c^2 d^4 e^2+33 a^2 c d^2 e^4-21 a^3 e^6\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^3}+\frac{\left (35 c^4 d^8-16 a c^3 d^6 e^2-18 a^2 c^2 d^4 e^4+168 a^3 c d^2 e^6-105 a^4 e^8\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 d^4 e^3 \left (c d^2-a e^2\right )^3 x^2}-\frac{\left (105 c^5 d^{10}-55 a c^4 d^8 e^2-54 a^2 c^3 d^6 e^4-78 a^3 c^2 d^4 e^6+525 a^4 c d^2 e^8-315 a^5 e^{10}\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^4 d^5 e^4 \left (c d^2-a e^2\right )^3 x}-\frac{\left (5 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\right )\right ) \int \frac{1}{x \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 a^4 d^5 e^4}\\ &=-\frac{2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{2 \left (3 c^3 d^6+a c^2 d^4 e^2+13 a^2 c d^2 e^4-9 a^3 e^6+c d e \left (3 c^2 d^4+14 a c d^2 e^2-9 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (7 c^3 d^6-3 a c^2 d^4 e^2+33 a^2 c d^2 e^4-21 a^3 e^6\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^3}+\frac{\left (35 c^4 d^8-16 a c^3 d^6 e^2-18 a^2 c^2 d^4 e^4+168 a^3 c d^2 e^6-105 a^4 e^8\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 d^4 e^3 \left (c d^2-a e^2\right )^3 x^2}-\frac{\left (105 c^5 d^{10}-55 a c^4 d^8 e^2-54 a^2 c^3 d^6 e^4-78 a^3 c^2 d^4 e^6+525 a^4 c d^2 e^8-315 a^5 e^{10}\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^4 d^5 e^4 \left (c d^2-a e^2\right )^3 x}+\frac{\left (5 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a d e-x^2} \, dx,x,\frac{2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 a^4 d^5 e^4}\\ &=-\frac{2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{2 \left (3 c^3 d^6+a c^2 d^4 e^2+13 a^2 c d^2 e^4-9 a^3 e^6+c d e \left (3 c^2 d^4+14 a c d^2 e^2-9 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (7 c^3 d^6-3 a c^2 d^4 e^2+33 a^2 c d^2 e^4-21 a^3 e^6\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^3}+\frac{\left (35 c^4 d^8-16 a c^3 d^6 e^2-18 a^2 c^2 d^4 e^4+168 a^3 c d^2 e^6-105 a^4 e^8\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 d^4 e^3 \left (c d^2-a e^2\right )^3 x^2}-\frac{\left (105 c^5 d^{10}-55 a c^4 d^8 e^2-54 a^2 c^3 d^6 e^4-78 a^3 c^2 d^4 e^6+525 a^4 c d^2 e^8-315 a^5 e^{10}\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^4 d^5 e^4 \left (c d^2-a e^2\right )^3 x}+\frac{5 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\right ) \tanh ^{-1}\left (\frac{2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 a^{9/2} d^{11/2} e^{9/2}}\\ \end{align*}

Mathematica [A]  time = 1.44048, size = 593, normalized size = 0.89 \[ \frac{(a e+c d x) \left (x \left (x^2 \left (3 \sqrt{a} d^{3/2} \sqrt{e} \left (a e^2-c d^2\right ) \left (-42 a^2 c^2 d^4 e^5-84 a^3 c d^2 e^7+105 a^4 e^9-20 a c^3 d^6 e^3+105 c^4 d^8 e\right ) (a e+c d x)+(d+e x) \sqrt{a e+c d x} \left (3 \sqrt{a} \sqrt{d} \sqrt{e} \left (78 a^3 c^2 d^4 e^7+54 a^2 c^3 d^6 e^5-525 a^4 c d^2 e^9+315 a^5 e^{11}+55 a c^4 d^8 e^3-105 c^5 d^{10} e\right ) \sqrt{a e+c d x}+45 \sqrt{d+e x} \left (21 a^2 c d^2 e^4+21 a^3 e^6+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right )\right )+9 \sqrt{a} c d^{7/2} \sqrt{e} \left (c d^2-a e^2\right )^2 \left (3 a^2 c d^2 e^4+21 a^3 e^6-5 a c^2 d^4 e^2-35 c^3 d^6\right )\right )+3 a^{3/2} d^{5/2} e^{3/2} x \left (a e^2-c d^2\right )^3 \left (63 a^2 e^4+54 a c d^2 e^2+35 c^2 d^4\right )+6 a^{5/2} d^{7/2} e^{5/2} \left (c d^2-a e^2\right )^3 \left (9 a e^2+7 c d^2\right )\right )+24 a^{7/2} d^{9/2} e^{7/2} \left (a e^2-c d^2\right )^3\right )}{72 a^{9/2} d^{11/2} e^{9/2} x^3 \left (c d^2-a e^2\right )^3 ((d+e x) (a e+c d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

((a*e + c*d*x)*(24*a^(7/2)*d^(9/2)*e^(7/2)*(-(c*d^2) + a*e^2)^3 + x*(6*a^(5/2)*d^(7/2)*e^(5/2)*(c*d^2 - a*e^2)
^3*(7*c*d^2 + 9*a*e^2) + 3*a^(3/2)*d^(5/2)*e^(3/2)*(-(c*d^2) + a*e^2)^3*(35*c^2*d^4 + 54*a*c*d^2*e^2 + 63*a^2*
e^4)*x + x^2*(9*Sqrt[a]*c*d^(7/2)*Sqrt[e]*(c*d^2 - a*e^2)^2*(-35*c^3*d^6 - 5*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 +
21*a^3*e^6) + 3*Sqrt[a]*d^(3/2)*Sqrt[e]*(-(c*d^2) + a*e^2)*(105*c^4*d^8*e - 20*a*c^3*d^6*e^3 - 42*a^2*c^2*d^4
*e^5 - 84*a^3*c*d^2*e^7 + 105*a^4*e^9)*(a*e + c*d*x) + Sqrt[a*e + c*d*x]*(d + e*x)*(3*Sqrt[a]*Sqrt[d]*Sqrt[e]*
(-105*c^5*d^10*e + 55*a*c^4*d^8*e^3 + 54*a^2*c^3*d^6*e^5 + 78*a^3*c^2*d^4*e^7 - 525*a^4*c*d^2*e^9 + 315*a^5*e^
11)*Sqrt[a*e + c*d*x] + 45*(c*d^2 - a*e^2)^3*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*Sq
rt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])))))/(72*a^(9/2)*d^(11/2)*e^(
9/2)*(c*d^2 - a*e^2)^3*x^3*((a*e + c*d*x)*(d + e*x))^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.072, size = 1705, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)

[Out]

105/16*e^2/d^5/a/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2))/x)-35/16*d/a^4/e^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^3-89/24/d^4/a*e/x/(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2)-75/16/d/a^3/e^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^2-1/3/d^2/a/e/x^3/(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2)+105/16/d^3/a^2/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2))/x)*c+23/24*d/a^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)*c^3-35/24/a^3/e^3/x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^2-2/3*e^3/d^4/(a*e^2-c*d^2)/(d/e+x)/(c*d*e*
(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+8/3*e^4/d*c^2/(a*e^2-c*d^2)^3/(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(
1/2)-41/12*e^3/d^2/a/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^2+35/8*d^4/a
^4/e^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^5+25/12*d^2/a^3/e/(-a^2*e^
4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^4-105/16*e^2/d^5/a/(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(1/2)-105/16/d^3/a^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c+13/12/d^3/a/x^2/(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2)+105/16*e^6/d^5*a/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
+233/48*e^4/d^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c+7/12/d/a^2/e^2/x^2/
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c+105/8*e^5/d^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2)*x*c-43/24*e^2/d/a/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*
c^2-1/6/a^2*e/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^3+16/3*e^5/d^2*c^2/
(a*e^2-c*d^2)^3/(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+8/3*e^6/d^3*c/(a*e^2-c*d^2)^3/(c*d*e*(d/e+x)^2
+(a*e^2-c*d^2)*(d/e+x))^(1/2)*a+75/16/d/a^3/e^2/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^2+35/16*d/a^4/e^4/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e
)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^3+155/48*d^3/a^3/e^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a
*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^4+35/16*d^5/a^4/e^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2)*c^5-17/6/d^2/a^2/e/x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}{\left (e x + d\right )} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)*x^4), x)

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

[undef, undef, undef, 1]