\(\int \frac {1}{x^3 (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [485]

Optimal result

Integrand size = 40, antiderivative size = 522 \[ \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+11 a^2 c d^2 e^4-7 a^3 e^6+c d e \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (15 c^3 d^6-9 a c^2 d^4 e^2+61 a^2 c d^2 e^4-35 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^2}+\frac {\left (45 c^4 d^8-30 a c^3 d^6 e^2-36 a^2 c^2 d^4 e^4+190 a^3 c d^2 e^6-105 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 d^4 e^3 \left (c d^2-a e^2\right )^3 x}-\frac {5 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\right ) \text {arctanh}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 a^{7/2} d^{9/2} e^{7/2}} \]

[Out]

-2/3*e*(c*d*x+a*e)/d/(-a*e^2+c*d^2)/x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-5/8*(7*a^2*e^4+6*a*c*d^2*e^2+3
*c^2*d^4)*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^(7/2)/d^(9/2)/e^(7/2)+2/3*(3*c^3*d^6+a*c^2*d^4*e^2+11*a^2*c*d^2*e^4-7*a^3*e^6+c*d*e*(-7*a^2*e^4+12*a*c*d^
2*e^2+3*c^2*d^4)*x)/a/d^2/e/(-a*e^2+c*d^2)^3/x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/6*(-35*a^3*e^6+61*a
^2*c*d^2*e^4-9*a*c^2*d^4*e^2+15*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^2+1/12*(-105*a^4*e^8+190*a^3*c*d^2*e^6-36*a^2*c^2*d^4*e^4-30*a*c^3*d^6*e^2+45*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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {865, 836, 848, 820, 738, 212} \[ \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {5 \left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 a^{7/2} d^{9/2} e^{7/2}}+\frac {2 \left (-7 a^3 e^6+c d e x \left (-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right )+11 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{3 a d^2 e x^2 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {\left (-35 a^3 e^6+61 a^2 c d^2 e^4-9 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 a^2 d^3 e^2 x^2 \left (c d^2-a e^2\right )^3}+\frac {\left (-105 a^4 e^8+190 a^3 c d^2 e^6-36 a^2 c^2 d^4 e^4-30 a c^3 d^6 e^2+45 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 a^3 d^4 e^3 x \left (c d^2-a e^2\right )^3}-\frac {2 e (a e+c d x)}{3 d x^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[In]

Int[1/(x^3*(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^2*(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 + 11*a^2*c*d^2*e^4 - 7*a^3*e^6 + c*d*e*(3*c^2*d^4 + 12*a*c*d^2*e^2 - 7*a^2*e^4)*x))/(3*a*d^2*e*
(c*d^2 - a*e^2)^3*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - ((15*c^3*d^6 - 9*a*c^2*d^4*e^2 + 61*a^2*c
*d^2*e^4 - 35*a^3*e^6)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(6*a^2*d^3*e^2*(c*d^2 - a*e^2)^3*x^2) + ((
45*c^4*d^8 - 30*a*c^3*d^6*e^2 - 36*a^2*c^2*d^4*e^4 + 190*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) - (5*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*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])])/(8*a^
(7/2)*d^(9/2)*e^(7/2))

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*} \text {integral}& = \int \frac {a e+c d x}{x^3 \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^2 \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 {1}{2} a e \left (3 c d^2-7 a e^2\right ) \left (c d^2-a e^2\right )+4 a c d e^2 \left (c d^2-a e^2\right ) x}{x^3 \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^2 \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+11 a^2 c d^2 e^4-7 a^3 e^6+c d e \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {4 \int \frac {\frac {1}{4} a e \left (c d^2-a e^2\right ) \left (15 c^3 d^6-9 a c^2 d^4 e^2+61 a^2 c d^2 e^4-35 a^3 e^6\right )+a c d e^2 \left (c d^2-a e^2\right ) \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right ) x}{x^3 \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^2 \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+11 a^2 c d^2 e^4-7 a^3 e^6+c d e \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (15 c^3 d^6-9 a c^2 d^4 e^2+61 a^2 c d^2 e^4-35 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^2}-\frac {2 \int \frac {\frac {1}{8} a e \left (c d^2-a e^2\right ) \left (45 c^4 d^8-30 a c^3 d^6 e^2-36 a^2 c^2 d^4 e^4+190 a^3 c d^2 e^6-105 a^4 e^8\right )+\frac {1}{4} a c d e^2 \left (c d^2-a e^2\right ) \left (15 c^3 d^6-9 a c^2 d^4 e^2+61 a^2 c d^2 e^4-35 a^3 e^6\right ) x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 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^2 \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+11 a^2 c d^2 e^4-7 a^3 e^6+c d e \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (15 c^3 d^6-9 a c^2 d^4 e^2+61 a^2 c d^2 e^4-35 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^2}+\frac {\left (45 c^4 d^8-30 a c^3 d^6 e^2-36 a^2 c^2 d^4 e^4+190 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}+\frac {\left (5 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\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}{8 a^3 d^4 e^3} \\ & = -\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^2 \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+11 a^2 c d^2 e^4-7 a^3 e^6+c d e \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (15 c^3 d^6-9 a c^2 d^4 e^2+61 a^2 c d^2 e^4-35 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^2}+\frac {\left (45 c^4 d^8-30 a c^3 d^6 e^2-36 a^2 c^2 d^4 e^4+190 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}-\frac {\left (5 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\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 )}{4 a^3 d^4 e^3} \\ & = -\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^2 \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+11 a^2 c d^2 e^4-7 a^3 e^6+c d e \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (15 c^3 d^6-9 a c^2 d^4 e^2+61 a^2 c d^2 e^4-35 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^2}+\frac {\left (45 c^4 d^8-30 a c^3 d^6 e^2-36 a^2 c^2 d^4 e^4+190 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}-\frac {5 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\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 )}{8 a^{7/2} d^{9/2} e^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {a} \sqrt {d} \sqrt {e} (a e+c d x) \left (-45 c^5 d^9 x^2 (d+e x)^2-15 a c^4 d^7 e x (d-2 e x) (d+e x)^2+6 a^2 c^3 d^5 e^2 (d+e x)^2 \left (d^2+2 d e x+6 e^2 x^2\right )+a^5 e^8 \left (-6 d^3+21 d^2 e x+140 d e^2 x^2+105 e^3 x^3\right )-2 a^3 c^2 d^3 e^4 \left (9 d^4-9 d^3 e x-6 d^2 e^2 x^2+111 d e^3 x^3+95 e^4 x^4\right )+a^4 c d e^6 \left (18 d^4-48 d^3 e x-237 d^2 e^2 x^2-50 d e^3 x^3+105 e^4 x^4\right )\right )}{\left (-c d^2+a e^2\right )^3 x^2}-15 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\right ) (a e+c d x)^{3/2} (d+e x)^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{12 a^{7/2} d^{9/2} e^{7/2} ((a e+c d x) (d+e x))^{3/2}} \]

[In]

Integrate[1/(x^3*(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]*(a*e + c*d*x)*(-45*c^5*d^9*x^2*(d + e*x)^2 - 15*a*c^4*d^7*e*x*(d - 2*e*x)*(d + e*x)^
2 + 6*a^2*c^3*d^5*e^2*(d + e*x)^2*(d^2 + 2*d*e*x + 6*e^2*x^2) + a^5*e^8*(-6*d^3 + 21*d^2*e*x + 140*d*e^2*x^2 +
105*e^3*x^3) - 2*a^3*c^2*d^3*e^4*(9*d^4 - 9*d^3*e*x - 6*d^2*e^2*x^2 + 111*d*e^3*x^3 + 95*e^4*x^4) + a^4*c*d*e
^6*(18*d^4 - 48*d^3*e*x - 237*d^2*e^2*x^2 - 50*d*e^3*x^3 + 105*e^4*x^4)))/((-(c*d^2) + a*e^2)^3*x^2) - 15*(3*c
^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x]
)/(Sqrt[d]*Sqrt[a*e + c*d*x])])/(12*a^(7/2)*d^(9/2)*e^(7/2)*((a*e + c*d*x)*(d + e*x))^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1353\) vs. \(2(488)=976\).

Time = 0.77 (sec) , antiderivative size = 1354, normalized size of antiderivative = 2.59

method	result	size
default	\(\text {Expression too large to display}\)	\(1354\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1071 vs. \(2 (488) = 976\).

Time = 28.52 (sec) , antiderivative size = 2162, normalized size of antiderivative = 4.14 \[ \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/48*(15*((3*c^6*d^11*e^2 - 3*a*c^5*d^9*e^4 - 2*a^2*c^4*d^7*e^6 - 6*a^3*c^3*d^5*e^8 + 15*a^4*c^2*d^3*e^10 - 7
*a^5*c*d*e^12)*x^5 + (6*c^6*d^12*e - 3*a*c^5*d^10*e^3 - 7*a^2*c^4*d^8*e^5 - 14*a^3*c^3*d^6*e^7 + 24*a^4*c^2*d^
4*e^9 + a^5*c*d^2*e^11 - 7*a^6*e^13)*x^4 + (3*c^6*d^13 + 3*a*c^5*d^11*e^2 - 8*a^2*c^4*d^9*e^4 - 10*a^3*c^3*d^7
*e^6 + 3*a^4*c^2*d^5*e^8 + 23*a^5*c*d^3*e^10 - 14*a^6*d*e^12)*x^3 + (3*a*c^5*d^12*e - 3*a^2*c^4*d^10*e^3 - 2*a
^3*c^3*d^8*e^5 - 6*a^4*c^2*d^6*e^7 + 15*a^5*c*d^4*e^9 - 7*a^6*d^2*e^11)*x^2)*sqrt(a*d*e)*log((8*a^2*d^2*e^2 +
(c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2 + a*
e^2)*x)*sqrt(a*d*e) + 8*(a*c*d^3*e + a^2*d*e^3)*x)/x^2) - 4*(6*a^3*c^3*d^10*e^3 - 18*a^4*c^2*d^8*e^5 + 18*a^5*
c*d^6*e^7 - 6*a^6*d^4*e^9 - (45*a*c^5*d^10*e^3 - 30*a^2*c^4*d^8*e^5 - 36*a^3*c^3*d^6*e^7 + 190*a^4*c^2*d^4*e^9
- 105*a^5*c*d^2*e^11)*x^4 - (90*a*c^5*d^11*e^2 - 45*a^2*c^4*d^9*e^4 - 84*a^3*c^3*d^7*e^6 + 222*a^4*c^2*d^5*e^
8 + 50*a^5*c*d^3*e^10 - 105*a^6*d*e^12)*x^3 - (45*a*c^5*d^12*e - 66*a^3*c^3*d^8*e^5 - 12*a^4*c^2*d^6*e^7 + 237
*a^5*c*d^4*e^9 - 140*a^6*d^2*e^11)*x^2 - 3*(5*a^2*c^4*d^11*e^2 - 8*a^3*c^3*d^9*e^4 - 6*a^4*c^2*d^7*e^6 + 16*a^
5*c*d^5*e^8 - 7*a^6*d^3*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/((a^4*c^4*d^12*e^6 - 3*a^5*c^3*d
^10*e^8 + 3*a^6*c^2*d^8*e^10 - a^7*c*d^6*e^12)*x^5 + (2*a^4*c^4*d^13*e^5 - 5*a^5*c^3*d^11*e^7 + 3*a^6*c^2*d^9*
e^9 + a^7*c*d^7*e^11 - a^8*d^5*e^13)*x^4 + (a^4*c^4*d^14*e^4 - a^5*c^3*d^12*e^6 - 3*a^6*c^2*d^10*e^8 + 5*a^7*c
*d^8*e^10 - 2*a^8*d^6*e^12)*x^3 + (a^5*c^3*d^13*e^5 - 3*a^6*c^2*d^11*e^7 + 3*a^7*c*d^9*e^9 - a^8*d^7*e^11)*x^2
), 1/24*(15*((3*c^6*d^11*e^2 - 3*a*c^5*d^9*e^4 - 2*a^2*c^4*d^7*e^6 - 6*a^3*c^3*d^5*e^8 + 15*a^4*c^2*d^3*e^10 -
7*a^5*c*d*e^12)*x^5 + (6*c^6*d^12*e - 3*a*c^5*d^10*e^3 - 7*a^2*c^4*d^8*e^5 - 14*a^3*c^3*d^6*e^7 + 24*a^4*c^2*
d^4*e^9 + a^5*c*d^2*e^11 - 7*a^6*e^13)*x^4 + (3*c^6*d^13 + 3*a*c^5*d^11*e^2 - 8*a^2*c^4*d^9*e^4 - 10*a^3*c^3*d
^7*e^6 + 3*a^4*c^2*d^5*e^8 + 23*a^5*c*d^3*e^10 - 14*a^6*d*e^12)*x^3 + (3*a*c^5*d^12*e - 3*a^2*c^4*d^10*e^3 - 2
*a^3*c^3*d^8*e^5 - 6*a^4*c^2*d^6*e^7 + 15*a^5*c*d^4*e^9 - 7*a^6*d^2*e^11)*x^2)*sqrt(-a*d*e)*arctan(1/2*sqrt(c*
d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-a*d*e)/(a*c*d^2*e^2*x^2 + a^2*d^2*e^2
+ (a*c*d^3*e + a^2*d*e^3)*x)) - 2*(6*a^3*c^3*d^10*e^3 - 18*a^4*c^2*d^8*e^5 + 18*a^5*c*d^6*e^7 - 6*a^6*d^4*e^9
- (45*a*c^5*d^10*e^3 - 30*a^2*c^4*d^8*e^5 - 36*a^3*c^3*d^6*e^7 + 190*a^4*c^2*d^4*e^9 - 105*a^5*c*d^2*e^11)*x^
4 - (90*a*c^5*d^11*e^2 - 45*a^2*c^4*d^9*e^4 - 84*a^3*c^3*d^7*e^6 + 222*a^4*c^2*d^5*e^8 + 50*a^5*c*d^3*e^10 - 1
05*a^6*d*e^12)*x^3 - (45*a*c^5*d^12*e - 66*a^3*c^3*d^8*e^5 - 12*a^4*c^2*d^6*e^7 + 237*a^5*c*d^4*e^9 - 140*a^6*
d^2*e^11)*x^2 - 3*(5*a^2*c^4*d^11*e^2 - 8*a^3*c^3*d^9*e^4 - 6*a^4*c^2*d^7*e^6 + 16*a^5*c*d^5*e^8 - 7*a^6*d^3*e
^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/((a^4*c^4*d^12*e^6 - 3*a^5*c^3*d^10*e^8 + 3*a^6*c^2*d^8*e
^10 - a^7*c*d^6*e^12)*x^5 + (2*a^4*c^4*d^13*e^5 - 5*a^5*c^3*d^11*e^7 + 3*a^6*c^2*d^9*e^9 + a^7*c*d^7*e^11 - a^
8*d^5*e^13)*x^4 + (a^4*c^4*d^14*e^4 - a^5*c^3*d^12*e^6 - 3*a^6*c^2*d^10*e^8 + 5*a^7*c*d^8*e^10 - 2*a^8*d^6*e^1
2)*x^3 + (a^5*c^3*d^13*e^5 - 3*a^6*c^2*d^11*e^7 + 3*a^7*c*d^9*e^9 - a^8*d^7*e^11)*x^2)]

Sympy [F]

\[ \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]

[In]

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

Maxima [F]

\[ \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{3}} \,d x } \]

[In]

integrate(1/x^3/(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^3), x)

Giac [F]

\[ \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{3}} \,d x } \]

[In]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{x^3\,\left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \]

[In]

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