\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^8 (d+e x)} \, dx\) [468]

Optimal result

Integrand size = 40, antiderivative size = 500 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^8 (d+e x)} \, dx=\frac {\left (c d^2-a e^2\right )^3 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 a^4 d^5 e^4 x^2}-\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 a^3 d^4 e^3 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac {\left (c d^2-a e^2\right )^5 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \text {arctanh}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2048 a^{9/2} d^{11/2} e^{9/2}} \]

[Out]

-1/384*(-a*e^2+c*d^2)*(9*a^2*e^4+10*a*c*d^2*e^2+5*c^2*d^4)*(2*a*d*e+(a*e^2+c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(3/2)/a^3/d^4/e^3/x^4-1/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/d/x^7-1/84*(5*c/a/e-9*e/d^2)*(a*d*e
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^6+1/840*(-63*a^2*e^4+20*a*c*d^2*e^2+35*c^2*d^4)*(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(5/2)/a^2/d^3/e^2/x^5-1/2048*(-a*e^2+c*d^2)^5*(9*a^2*e^4+10*a*c*d^2*e^2+5*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^(9/2)/d^(11/2)/e^(9/2)+1
/1024*(-a*e^2+c*d^2)^3*(9*a^2*e^4+10*a*c*d^2*e^2+5*c^2*d^4)*(2*a*d*e+(a*e^2+c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)/a^4/d^5/e^4/x^2

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {863, 848, 820, 734, 738, 212} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^8 (d+e x)} \, dx=\frac {\left (-63 a^2 e^4+20 a c d^2 e^2+35 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac {\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 a^{9/2} d^{11/2} e^{9/2}}+\frac {\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 a^4 d^5 e^4 x^2}-\frac {\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 a^3 d^4 e^3 x^4}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{84 x^6} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^8*(d + e*x)),x]

[Out]

((c*d^2 - a*e^2)^3*(5*c^2*d^4 + 10*a*c*d^2*e^2 + 9*a^2*e^4)*(2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2])/(1024*a^4*d^5*e^4*x^2) - ((c*d^2 - a*e^2)*(5*c^2*d^4 + 10*a*c*d^2*e^2 + 9*a^2*e^4)*(2
*a*d*e + (c*d^2 + a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(384*a^3*d^4*e^3*x^4) - (a*d*e + (c
*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(7*d*x^7) - (((5*c)/(a*e) - (9*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(5/2))/(84*x^6) + ((35*c^2*d^4 + 20*a*c*d^2*e^2 - 63*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2
))/(840*a^2*d^3*e^2*x^5) - ((c*d^2 - a*e^2)^5*(5*c^2*d^4 + 10*a*c*d^2*e^2 + 9*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])])/(2048*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 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))
*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[p*((b^2
- 4*a*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m
+ 2*p + 2, 0] && GtQ[p, 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 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 863

Int[((x_)^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + c*(
x/e))*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b
*d*e + a*e^2, 0] &&  !IntegerQ[p] && ( !IntegerQ[n] ||  !IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2
]))

Rubi steps \begin{align*} \text {integral}& = \int \frac {(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^8} \, dx \\ & = -\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\int \frac {\left (-\frac {1}{2} a e \left (5 c d^2-9 a e^2\right )+2 a c d e^2 x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7} \, dx}{7 a d e} \\ & = -\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\int \frac {\left (-\frac {1}{4} a e \left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right )-\frac {1}{2} a c d e^2 \left (5 c d^2-9 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}}{x^6} \, dx}{42 a^2 d^2 e^2} \\ & = -\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}+\frac {\left (\left (c d^2-a e^2\right ) \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right )\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5} \, dx}{48 a^2 d^3 e^2} \\ & = -\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \left (2 a d e+\left (c d^2+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}}{384 a^3 d^4 e^3 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac {\left (\left (c d^2-a e^2\right )^3 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right )\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx}{256 a^3 d^4 e^3} \\ & = \frac {\left (c d^2-a e^2\right )^3 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 a^4 d^5 e^4 x^2}-\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \left (2 a d e+\left (c d^2+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}}{384 a^3 d^4 e^3 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}+\frac {\left (\left (c d^2-a e^2\right )^5 \left (5 c^2 d^4+10 a c d^2 e^2+9 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}{2048 a^4 d^5 e^4} \\ & = \frac {\left (c d^2-a e^2\right )^3 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 a^4 d^5 e^4 x^2}-\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \left (2 a d e+\left (c d^2+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}}{384 a^3 d^4 e^3 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac {\left (\left (c d^2-a e^2\right )^5 \left (5 c^2 d^4+10 a c d^2 e^2+9 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 )}{1024 a^4 d^5 e^4} \\ & = \frac {\left (c d^2-a e^2\right )^3 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 a^4 d^5 e^4 x^2}-\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \left (2 a d e+\left (c d^2+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}}{384 a^3 d^4 e^3 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac {\left (c d^2-a e^2\right )^5 \left (5 c^2 d^4+10 a c d^2 e^2+9 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 )}{2048 a^{9/2} d^{11/2} e^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.22 (sec) , antiderivative size = 497, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^8 (d+e x)} \, dx=\frac {\left (-c d^2+a e^2\right )^5 ((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (-525 c^6 d^{12} x^6+350 a c^5 d^{10} e x^5 (d+4 e x)-35 a^2 c^4 d^8 e^2 x^4 \left (8 d^2+26 d e x+15 e^2 x^2\right )+60 a^3 c^3 d^6 e^3 x^3 \left (4 d^3+12 d^2 e x+5 d e^2 x^2-10 e^3 x^3\right )+a^4 c^2 d^4 e^4 x^2 \left (23680 d^4+33520 d^3 e x+1824 d^2 e^2 x^2-2332 d e^3 x^3+3689 e^4 x^4\right )+2 a^5 c d^2 e^5 x \left (18560 d^5+24320 d^4 e x+744 d^3 e^2 x^2-872 d^2 e^3 x^3+1099 d e^4 x^4-1680 e^5 x^5\right )+3 a^6 e^6 \left (5120 d^6+6400 d^5 e x+128 d^4 e^2 x^2-144 d^3 e^3 x^3+168 d^2 e^4 x^4-210 d e^5 x^5+315 e^6 x^6\right )\right )}{\left (c d^2-a e^2\right )^5 x^7 (a e+c d x) (d+e x)}+\frac {105 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{107520 a^{9/2} d^{11/2} e^{9/2}} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^8*(d + e*x)),x]

[Out]

((-(c*d^2) + a*e^2)^5*((a*e + c*d*x)*(d + e*x))^(3/2)*((Sqrt[a]*Sqrt[d]*Sqrt[e]*(-525*c^6*d^12*x^6 + 350*a*c^5
*d^10*e*x^5*(d + 4*e*x) - 35*a^2*c^4*d^8*e^2*x^4*(8*d^2 + 26*d*e*x + 15*e^2*x^2) + 60*a^3*c^3*d^6*e^3*x^3*(4*d
^3 + 12*d^2*e*x + 5*d*e^2*x^2 - 10*e^3*x^3) + a^4*c^2*d^4*e^4*x^2*(23680*d^4 + 33520*d^3*e*x + 1824*d^2*e^2*x^
2 - 2332*d*e^3*x^3 + 3689*e^4*x^4) + 2*a^5*c*d^2*e^5*x*(18560*d^5 + 24320*d^4*e*x + 744*d^3*e^2*x^2 - 872*d^2*
e^3*x^3 + 1099*d*e^4*x^4 - 1680*e^5*x^5) + 3*a^6*e^6*(5120*d^6 + 6400*d^5*e*x + 128*d^4*e^2*x^2 - 144*d^3*e^3*
x^3 + 168*d^2*e^4*x^4 - 210*d*e^5*x^5 + 315*e^6*x^6)))/((c*d^2 - a*e^2)^5*x^7*(a*e + c*d*x)*(d + e*x)) + (105*
(5*c^2*d^4 + 10*a*c*d^2*e^2 + 9*a^2*e^4)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])
/((a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(107520*a^(9/2)*d^(11/2)*e^(9/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(45105\) vs. \(2(462)=924\).

Time = 3.19 (sec) , antiderivative size = 45106, normalized size of antiderivative = 90.21

method	result	size
default	\(\text {Expression too large to display}\)	\(45106\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^8/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [A] (verification not implemented)

none

Time = 78.30 (sec) , antiderivative size = 1300, normalized size of antiderivative = 2.60 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^8 (d+e x)} \, dx=\text {Too large to display} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^8/(e*x+d),x, algorithm="fricas")

[Out]

[-1/430080*(105*(5*c^7*d^14 - 15*a*c^6*d^12*e^2 + 9*a^2*c^5*d^10*e^4 + 5*a^3*c^4*d^8*e^6 + 15*a^4*c^3*d^6*e^8
- 45*a^5*c^2*d^4*e^10 + 35*a^6*c*d^2*e^12 - 9*a^7*e^14)*sqrt(a*d*e)*x^7*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*(15360*a^7*d^7*e^7 - (525*a*c^6*d^13*e - 1400*a^2*c^5*d^11*e^3 + 52
5*a^3*c^4*d^9*e^5 + 600*a^4*c^3*d^7*e^7 - 3689*a^5*c^2*d^5*e^9 + 3360*a^6*c*d^3*e^11 - 945*a^7*d*e^13)*x^6 + 2
*(175*a^2*c^5*d^12*e^2 - 455*a^3*c^4*d^10*e^4 + 150*a^4*c^3*d^8*e^6 - 1166*a^5*c^2*d^6*e^8 + 1099*a^6*c*d^4*e^
10 - 315*a^7*d^2*e^12)*x^5 - 8*(35*a^3*c^4*d^11*e^3 - 90*a^4*c^3*d^9*e^5 - 228*a^5*c^2*d^7*e^7 + 218*a^6*c*d^5
*e^9 - 63*a^7*d^3*e^11)*x^4 + 16*(15*a^4*c^3*d^10*e^4 + 2095*a^5*c^2*d^8*e^6 + 93*a^6*c*d^6*e^8 - 27*a^7*d^4*e
^10)*x^3 + 128*(185*a^5*c^2*d^9*e^5 + 380*a^6*c*d^7*e^7 + 3*a^7*d^5*e^9)*x^2 + 1280*(29*a^6*c*d^8*e^6 + 15*a^7
*d^6*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^5*d^6*e^5*x^7), 1/215040*(105*(5*c^7*d^14 - 15*a*
c^6*d^12*e^2 + 9*a^2*c^5*d^10*e^4 + 5*a^3*c^4*d^8*e^6 + 15*a^4*c^3*d^6*e^8 - 45*a^5*c^2*d^4*e^10 + 35*a^6*c*d^
2*e^12 - 9*a^7*e^14)*sqrt(-a*d*e)*x^7*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*(15360*a^7*d^7*e^7
- (525*a*c^6*d^13*e - 1400*a^2*c^5*d^11*e^3 + 525*a^3*c^4*d^9*e^5 + 600*a^4*c^3*d^7*e^7 - 3689*a^5*c^2*d^5*e^9
+ 3360*a^6*c*d^3*e^11 - 945*a^7*d*e^13)*x^6 + 2*(175*a^2*c^5*d^12*e^2 - 455*a^3*c^4*d^10*e^4 + 150*a^4*c^3*d^
8*e^6 - 1166*a^5*c^2*d^6*e^8 + 1099*a^6*c*d^4*e^10 - 315*a^7*d^2*e^12)*x^5 - 8*(35*a^3*c^4*d^11*e^3 - 90*a^4*c
^3*d^9*e^5 - 228*a^5*c^2*d^7*e^7 + 218*a^6*c*d^5*e^9 - 63*a^7*d^3*e^11)*x^4 + 16*(15*a^4*c^3*d^10*e^4 + 2095*a
^5*c^2*d^8*e^6 + 93*a^6*c*d^6*e^8 - 27*a^7*d^4*e^10)*x^3 + 128*(185*a^5*c^2*d^9*e^5 + 380*a^6*c*d^7*e^7 + 3*a^
7*d^5*e^9)*x^2 + 1280*(29*a^6*c*d^8*e^6 + 15*a^7*d^6*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^5
*d^6*e^5*x^7)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^8 (d+e x)} \, dx=\text {Timed out} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/x**8/(e*x+d),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^8 (d+e x)} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )} x^{8}} \,d x } \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^8/(e*x+d),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)*x^8), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4452 vs. \(2 (462) = 924\).

Time = 0.78 (sec) , antiderivative size = 4452, normalized size of antiderivative = 8.90 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^8 (d+e x)} \, dx=\text {Too large to display} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^8/(e*x+d),x, algorithm="giac")

[Out]

1/1024*(5*c^7*d^14 - 15*a*c^6*d^12*e^2 + 9*a^2*c^5*d^10*e^4 + 5*a^3*c^4*d^8*e^6 + 15*a^4*c^3*d^6*e^8 - 45*a^5*
c^2*d^4*e^10 + 35*a^6*c*d^2*e^12 - 9*a^7*e^14)*arctan(-(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a
*d*e))/sqrt(-a*d*e))/(sqrt(-a*d*e)*a^4*d^5*e^4) - 1/107520*(525*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*
e^2*x + a*d*e))*a^6*c^7*d^20*e^6 - 1575*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^7*c^6*
d^18*e^8 + 945*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^8*c^5*d^16*e^10 + 215565*(sqrt(
c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^9*c^4*d^14*e^12 + 431655*(sqrt(c*d*e)*x - sqrt(c*d*e
*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^10*c^3*d^12*e^14 + 210315*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e
^2*x + a*d*e))*a^11*c^2*d^10*e^16 + 3675*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^12*c*
d^8*e^18 - 945*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^13*d^6*e^20 + 30720*sqrt(c*d*e)
*a^10*c^3*d^13*e^13 + 43008*sqrt(c*d*e)*a^11*c^2*d^11*e^15 - 3500*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x +
a*e^2*x + a*d*e))^3*a^5*c^7*d^19*e^5 + 10500*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a
^6*c^6*d^17*e^7 + 1068900*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^7*c^5*d^15*e^9 + 4
655700*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^8*c^4*d^13*e^11 + 6225660*(sqrt(c*d*e
)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^9*c^3*d^11*e^13 + 2827020*(sqrt(c*d*e)*x - sqrt(c*d*e*x
^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^10*c^2*d^9*e^15 + 405580*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^
2*x + a*d*e))^3*a^11*c*d^7*e^17 + 6300*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^12*d^
5*e^19 + 645120*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))^2*a^8*c^4*d^14*e^10
+ 1935360*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))^2*a^9*c^3*d^12*e^12 + 1634
304*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))^2*a^10*c^2*d^10*e^14 + 430080*sq
rt(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))^2*a^11*c*d^8*e^16 + 9905*(sqrt(c*d*e)*
x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^4*c^7*d^18*e^4 + 615405*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 +
c*d^2*x + a*e^2*x + a*d*e))^5*a^5*c^6*d^16*e^6 + 5823909*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x
+ a*d*e))^5*a^6*c^5*d^14*e^8 + 17499825*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^7*c^
4*d^12*e^10 + 19168275*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^8*c^3*d^10*e^12 + 765
2295*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^9*c^2*d^8*e^14 + 929495*(sqrt(c*d*e)*x
- sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^10*c*d^6*e^16 + 25179*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d
^2*x + a*e^2*x + a*d*e))^5*a^11*d^4*e^18 + 1075200*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))^4*a^6*c^5*d^15*e^7 + 6666240*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))^4*a^7*c^4*d^13*e^9 + 13332480*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)
)^4*a^8*c^3*d^11*e^11 + 9719808*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))^4*a^
9*c^2*d^9*e^13 + 2365440*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))^4*a^10*c*d^
7*e^15 + 215040*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))^4*a^11*d^5*e^17 + 15
360*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^3*c^7*d^17*e^3 + 752640*(sqrt(c*d*e)*x -
sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^4*c^6*d^15*e^5 + 6730752*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c
*d^2*x + a*e^2*x + a*d*e))^7*a^5*c^5*d^13*e^7 + 18170880*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x +
a*d*e))^7*a^6*c^4*d^11*e^9 + 16665600*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^7*c^3
*d^9*e^11 + 4623360*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^8*c^2*d^7*e^13 + 107520*
(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^9*c*d^5*e^15 - 27648*(sqrt(c*d*e)*x - sqrt(c
*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^10*d^3*e^17 + 215040*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))^6*a^4*c^6*d^16*e^4 + 2938880*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))^6*a^5*c^5*d^14*e^6 + 14192640*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))^6*a^6*c^4*d^12*e^8 + 23654400*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))^6*a^7*c^3*d^10*e^10 + 14120960*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))^6*a^8*c^2*d^8*e^12 + 2795520*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))^6*a
^9*c*d^6*e^14 - 9905*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^2*c^7*d^16*e^2 + 29715*
(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^3*c^6*d^14*e^4 + 1745499*(sqrt(c*d*e)*x - sq
rt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^4*c^5*d^12*e^6 + 5294415*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^
2*x + a*e^2*x + a*d*e))^9*a^5*c^4*d^10*e^8 + 3625965*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d
*e))^9*a^6*c^3*d^8*e^10 + 89145*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^7*c^2*d^6*e^
12 - 69335*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^8*c*d^4*e^14 + 17829*(sqrt(c*d*e)
*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^9*d^2*e^16 + 1576960*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))^8*a^4*c^5*d^13*e^5 + 8171520*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))^8*a^5*c^4*d^11*e^7 + 12257280*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))^8*a^6*c^3*d^9*e^9 + 5232640*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))^8*a^7*c^2*d^7*e^11 + 3500*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^11*a
*c^7*d^15*e - 10500*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^11*a^2*c^6*d^13*e^3 + 6300*(
sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^11*a^3*c^5*d^11*e^5 + 290220*(sqrt(c*d*e)*x - sqr
t(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^11*a^4*c^4*d^9*e^7 + 10500*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x
+ a*e^2*x + a*d*e))^11*a^5*c^3*d^7*e^9 - 31500*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^
11*a^6*c^2*d^5*e^11 + 24500*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^11*a^7*c*d^3*e^13 -
6300*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^11*a^8*d*e^15 + 1075200*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))^10*a^4*c^4*d^10*e^6 + 1505280*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))^10*a^5*c^3*d^8*e^8 - 525*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*
d^2*x + a*e^2*x + a*d*e))^13*c^7*d^14 + 1575*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^13*
a*c^6*d^12*e^2 - 945*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^13*a^2*c^5*d^10*e^4 - 525*(
sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^13*a^3*c^4*d^8*e^6 - 1575*(sqrt(c*d*e)*x - sqrt(c
*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^13*a^4*c^3*d^6*e^8 + 4725*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a
*e^2*x + a*d*e))^13*a^5*c^2*d^4*e^10 - 3675*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^13*a
^6*c*d^2*e^12 + 945*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^13*a^7*e^14)/((a*d*e - (sqrt
(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2)^7*a^4*d^5*e^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^8 (d+e x)} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^8\,\left (d+e\,x\right )} \,d x \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^8*(d + e*x)),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^8*(d + e*x)), x)