3.5.86 \(\int \frac {1}{x^4 (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [486]

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

[Out]

-2/3*e*(c*d*x+a*e)/d/(-a*e^2+c*d^2)/x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+5/16*(21*a^3*e^6+21*a^2*c*d^2*
e^4+15*a*c^2*d^4*e^2+7*c^3*d^6)*arctanh(1/2*(2*a*d*e+(a*e^2+c*d^2)*x)/a^(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))/a^(9/2)/d^(11/2)/e^(9/2)+2/3*(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*(-9*a^2*e^4+14*a*c*d^2*e^2+3*c^2*d^4)*x)/a/d^2/e/(-a*e^2+c*d^2)^3/x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2
)-1/3*(-21*a^3*e^6+33*a^2*c*d^2*e^4-3*a*c^2*d^4*e^2+7*c^3*d^6)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^2/d^3
/e^2/(-a*e^2+c*d^2)^3/x^3+1/12*(-105*a^4*e^8+168*a^3*c*d^2*e^6-18*a^2*c^2*d^4*e^4-16*a*c^3*d^6*e^2+35*c^4*d^8)
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^3/d^4/e^3/(-a*e^2+c*d^2)^3/x^2-1/24*(-315*a^5*e^10+525*a^4*c*d^2*e^
8-78*a^3*c^2*d^4*e^6-54*a^2*c^3*d^6*e^4-55*a*c^4*d^8*e^2+105*c^5*d^10)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
/a^4/d^5/e^4/(-a*e^2+c*d^2)^3/x

________________________________________________________________________________________

Rubi [A]
time = 0.71, antiderivative size = 664, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {865, 836, 848, 820, 738, 212} \begin {gather*} \frac {2 \left (-9 a^3 e^6+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+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 {\left (-21 a^3 e^6+33 a^2 c d^2 e^4-3 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 {5 \left (21 a^3 e^6+21 a^2 c d^2 e^4+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 {\left (-105 a^4 e^8+168 a^3 c d^2 e^6-18 a^2 c^2 d^4 e^4-16 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 {\left (-315 a^5 e^{10}+525 a^4 c d^2 e^8-78 a^3 c^2 d^4 e^6-54 a^2 c^3 d^6 e^4-55 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 {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}} \end {gather*}

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 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 738

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 820

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)/(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[S
implify[m + 2*p + 3], 0]

Rule 836

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 848

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 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*} \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 ) \text {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*}

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Mathematica [A]
time = 1.02, size = 493, normalized size = 0.74 \begin {gather*} \frac {\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (-105 c^6 d^{11} x^3 (d+e x)^2-5 a c^5 d^9 e x^2 (7 d-11 e x) (d+e x)^2+a^2 c^4 d^7 e^2 x (d+e x)^2 \left (14 d^2+23 d e x+54 e^2 x^2\right )-2 a^3 c^3 d^5 e^3 (d+e x)^2 \left (4 d^3+4 d^2 e x-9 d e^2 x^2-39 e^3 x^3\right )+a^6 e^9 \left (8 d^4-18 d^3 e x+63 d^2 e^2 x^2+420 d e^3 x^3+315 e^4 x^4\right )+a^4 c^2 d^3 e^5 \left (24 d^5-12 d^4 e x+62 d^3 e^2 x^2+3 d^2 e^3 x^3-636 d e^4 x^4-525 e^5 x^5\right )-a^5 c d e^7 \left (24 d^5-40 d^4 e x+135 d^3 e^2 x^2+651 d^2 e^3 x^3+105 d e^4 x^4-315 e^5 x^5\right )\right )}{\left (c d^2-a e^2\right )^3 x^3 (d+e x)}+15 \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 ) \sqrt {a e+c d x} \sqrt {d+e x} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{24 a^{9/2} d^{11/2} e^{9/2} \sqrt {(a e+c d x) (d+e x)}} \end {gather*}

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]

((Sqrt[a]*Sqrt[d]*Sqrt[e]*(-105*c^6*d^11*x^3*(d + e*x)^2 - 5*a*c^5*d^9*e*x^2*(7*d - 11*e*x)*(d + e*x)^2 + a^2*
c^4*d^7*e^2*x*(d + e*x)^2*(14*d^2 + 23*d*e*x + 54*e^2*x^2) - 2*a^3*c^3*d^5*e^3*(d + e*x)^2*(4*d^3 + 4*d^2*e*x
- 9*d*e^2*x^2 - 39*e^3*x^3) + a^6*e^9*(8*d^4 - 18*d^3*e*x + 63*d^2*e^2*x^2 + 420*d*e^3*x^3 + 315*e^4*x^4) + a^
4*c^2*d^3*e^5*(24*d^5 - 12*d^4*e*x + 62*d^3*e^2*x^2 + 3*d^2*e^3*x^3 - 636*d*e^4*x^4 - 525*e^5*x^5) - a^5*c*d*e
^7*(24*d^5 - 40*d^4*e*x + 135*d^3*e^2*x^2 + 651*d^2*e^3*x^3 + 105*d*e^4*x^4 - 315*e^5*x^5)))/((c*d^2 - a*e^2)^
3*x^3*(d + e*x)) + 15*(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)*Sqrt[a*e + c*d*x]*Sqrt[d
+ e*x]*ArcTanh[(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])/(Sqrt[d]*Sqrt[a*e + c*d*x])])/(24*a^(9/2)*d^(11/2)*e^(9/2)*Sqrt
[(a*e + c*d*x)*(d + e*x)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2408\) vs. \(2(626)=1252\).
time = 0.10, size = 2409, normalized size = 3.63

method	result	size
default	\(\text {Expression too large to display}\)	\(2409\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1335 vs. \(2 (604) = 1208\).
time = 156.13, size = 2695, normalized size = 4.06 \begin {gather*} \text {Too large to display} \end {gather*}

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]

[1/96*(15*(7*c^7*d^15*x^4 - 21*a^7*x^5*e^15 - 21*a^7*d^2*x^3*e^13 - 21*(a^6*c*d*x^6 + 2*a^7*d*x^4)*e^14 + 21*(
2*a^5*c^2*d^3*x^6 + 3*a^6*c*d^3*x^4)*e^12 + 3*(23*a^5*c^2*d^4*x^5 + 14*a^6*c*d^4*x^3)*e^11 - 3*(5*a^4*c^3*d^5*
x^6 - 4*a^5*c^2*d^5*x^4)*e^10 - (34*a^4*c^3*d^6*x^5 + 15*a^5*c^2*d^6*x^3)*e^9 - (4*a^3*c^4*d^7*x^6 + 23*a^4*c^
3*d^7*x^4)*e^8 - (11*a^3*c^4*d^8*x^5 + 4*a^4*c^3*d^8*x^3)*e^7 - (3*a^2*c^5*d^9*x^6 + 10*a^3*c^4*d^9*x^4)*e^6 -
3*(4*a^2*c^5*d^10*x^5 + a^3*c^4*d^10*x^3)*e^5 - 3*(2*a*c^6*d^11*x^6 + 5*a^2*c^5*d^11*x^4)*e^4 - (5*a*c^6*d^12
*x^5 + 6*a^2*c^5*d^12*x^3)*e^3 + (7*c^7*d^13*x^6 + 8*a*c^6*d^13*x^4)*e^2 + 7*(2*c^7*d^14*x^5 + a*c^6*d^14*x^3)
*e)*sqrt(a*d)*e^(1/2)*log((c^2*d^4*x^2 + 8*a*c*d^3*x*e + a^2*x^2*e^4 + 8*a^2*d*x*e^3 + 4*(c*d^2*x + a*x*e^2 +
2*a*d*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(a*d)*e^(1/2) + 2*(3*a*c*d^2*x^2 + 4*a^2*d^2)*e^2)/x^
2) - 4*(105*a*c^6*d^14*x^3*e - 315*a^7*d*x^4*e^14 - 105*(3*a^6*c*d^2*x^5 + 4*a^7*d^2*x^3)*e^13 + 21*(5*a^6*c*d
^3*x^4 - 3*a^7*d^3*x^2)*e^12 + 3*(175*a^5*c^2*d^4*x^5 + 217*a^6*c*d^4*x^3 + 6*a^7*d^4*x)*e^11 + (636*a^5*c^2*d
^5*x^4 + 135*a^6*c*d^5*x^2 - 8*a^7*d^5)*e^10 - (78*a^4*c^3*d^6*x^5 + 3*a^5*c^2*d^6*x^3 + 40*a^6*c*d^6*x)*e^9 -
2*(87*a^4*c^3*d^7*x^4 + 31*a^5*c^2*d^7*x^2 - 12*a^6*c*d^7)*e^8 - 2*(27*a^3*c^4*d^8*x^5 + 53*a^4*c^3*d^8*x^3 -
6*a^5*c^2*d^8*x)*e^7 - (131*a^3*c^4*d^9*x^4 - 6*a^4*c^3*d^9*x^2 + 24*a^5*c^2*d^9)*e^6 - (55*a^2*c^5*d^10*x^5
+ 114*a^3*c^4*d^10*x^3 - 24*a^4*c^3*d^10*x)*e^5 - (75*a^2*c^5*d^11*x^4 + 51*a^3*c^4*d^11*x^2 - 8*a^4*c^3*d^11)
*e^4 + (105*a*c^6*d^12*x^5 + 15*a^2*c^5*d^12*x^3 - 14*a^3*c^4*d^12*x)*e^3 + 35*(6*a*c^6*d^13*x^4 + a^2*c^5*d^1
3*x^2)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))/(a^5*c^4*d^15*x^4*e^5 - a^9*d^6*x^5*e^14 - (a^8*c*d^7
*x^6 + 2*a^9*d^7*x^4)*e^13 + (a^8*c*d^8*x^5 - a^9*d^8*x^3)*e^12 + (3*a^7*c^2*d^9*x^6 + 5*a^8*c*d^9*x^4)*e^11 +
3*(a^7*c^2*d^10*x^5 + a^8*c*d^10*x^3)*e^10 - 3*(a^6*c^3*d^11*x^6 + a^7*c^2*d^11*x^4)*e^9 - (5*a^6*c^3*d^12*x^
5 + 3*a^7*c^2*d^12*x^3)*e^8 + (a^5*c^4*d^13*x^6 - a^6*c^3*d^13*x^4)*e^7 + (2*a^5*c^4*d^14*x^5 + a^6*c^3*d^14*x
^3)*e^6), -1/48*(15*(7*c^7*d^15*x^4 - 21*a^7*x^5*e^15 - 21*a^7*d^2*x^3*e^13 - 21*(a^6*c*d*x^6 + 2*a^7*d*x^4)*e
^14 + 21*(2*a^5*c^2*d^3*x^6 + 3*a^6*c*d^3*x^4)*e^12 + 3*(23*a^5*c^2*d^4*x^5 + 14*a^6*c*d^4*x^3)*e^11 - 3*(5*a^
4*c^3*d^5*x^6 - 4*a^5*c^2*d^5*x^4)*e^10 - (34*a^4*c^3*d^6*x^5 + 15*a^5*c^2*d^6*x^3)*e^9 - (4*a^3*c^4*d^7*x^6 +
23*a^4*c^3*d^7*x^4)*e^8 - (11*a^3*c^4*d^8*x^5 + 4*a^4*c^3*d^8*x^3)*e^7 - (3*a^2*c^5*d^9*x^6 + 10*a^3*c^4*d^9*
x^4)*e^6 - 3*(4*a^2*c^5*d^10*x^5 + a^3*c^4*d^10*x^3)*e^5 - 3*(2*a*c^6*d^11*x^6 + 5*a^2*c^5*d^11*x^4)*e^4 - (5*
a*c^6*d^12*x^5 + 6*a^2*c^5*d^12*x^3)*e^3 + (7*c^7*d^13*x^6 + 8*a*c^6*d^13*x^4)*e^2 + 7*(2*c^7*d^14*x^5 + a*c^6
*d^14*x^3)*e)*sqrt(-a*d*e)*arctan(1/2*(c*d^2*x + a*x*e^2 + 2*a*d*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e
)*sqrt(-a*d*e)/(a*c*d^3*x*e + a^2*d*x*e^3 + (a*c*d^2*x^2 + a^2*d^2)*e^2)) + 2*(105*a*c^6*d^14*x^3*e - 315*a^7*
d*x^4*e^14 - 105*(3*a^6*c*d^2*x^5 + 4*a^7*d^2*x^3)*e^13 + 21*(5*a^6*c*d^3*x^4 - 3*a^7*d^3*x^2)*e^12 + 3*(175*a
^5*c^2*d^4*x^5 + 217*a^6*c*d^4*x^3 + 6*a^7*d^4*x)*e^11 + (636*a^5*c^2*d^5*x^4 + 135*a^6*c*d^5*x^2 - 8*a^7*d^5)
*e^10 - (78*a^4*c^3*d^6*x^5 + 3*a^5*c^2*d^6*x^3 + 40*a^6*c*d^6*x)*e^9 - 2*(87*a^4*c^3*d^7*x^4 + 31*a^5*c^2*d^7
*x^2 - 12*a^6*c*d^7)*e^8 - 2*(27*a^3*c^4*d^8*x^5 + 53*a^4*c^3*d^8*x^3 - 6*a^5*c^2*d^8*x)*e^7 - (131*a^3*c^4*d^
9*x^4 - 6*a^4*c^3*d^9*x^2 + 24*a^5*c^2*d^9)*e^6 - (55*a^2*c^5*d^10*x^5 + 114*a^3*c^4*d^10*x^3 - 24*a^4*c^3*d^1
0*x)*e^5 - (75*a^2*c^5*d^11*x^4 + 51*a^3*c^4*d^11*x^2 - 8*a^4*c^3*d^11)*e^4 + (105*a*c^6*d^12*x^5 + 15*a^2*c^5
*d^12*x^3 - 14*a^3*c^4*d^12*x)*e^3 + 35*(6*a*c^6*d^13*x^4 + a^2*c^5*d^13*x^2)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c
*d*x^2 + a*d)*e))/(a^5*c^4*d^15*x^4*e^5 - a^9*d^6*x^5*e^14 - (a^8*c*d^7*x^6 + 2*a^9*d^7*x^4)*e^13 + (a^8*c*d^8
*x^5 - a^9*d^8*x^3)*e^12 + (3*a^7*c^2*d^9*x^6 + 5*a^8*c*d^9*x^4)*e^11 + 3*(a^7*c^2*d^10*x^5 + a^8*c*d^10*x^3)*
e^10 - 3*(a^6*c^3*d^11*x^6 + a^7*c^2*d^11*x^4)*e^9 - (5*a^6*c^3*d^12*x^5 + 3*a^7*c^2*d^12*x^3)*e^8 + (a^5*c^4*
d^13*x^6 - a^6*c^3*d^13*x^4)*e^7 + (2*a^5*c^4*d^14*x^5 + a^6*c^3*d^14*x^3)*e^6)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^4\,\left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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