3.456 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{x^7 (d+e x)} \, dx\)

Optimal. Leaf size=498 \[ -\frac{\left (21 a^2 c d^2 e^4-105 a^3 e^6+33 a c^2 d^4 e^2+35 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}+\frac{\left (-21 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}+\frac{\left (6 a^2 c^2 d^4 e^4-21 a^4 e^8+8 a c^3 d^6 e^2+7 c^4 d^8\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 a^4 d^5 e^4 x^2}-\frac{\left (21 a^2 c d^2 e^4+21 a^3 e^6+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 a^{9/2} d^{11/2} e^{9/2}}-\frac{\left (\frac{c}{a e}-\frac{3 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{20 x^5}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 d x^6} \]

[Out]

((7*c^4*d^8 + 8*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 21*a^4*e^8)*(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])/(512*a^4*d^5*e^4*x^2) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(6*d*x^6) -
((c/(a*e) - (3*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(20*x^5) + ((7*c^2*d^4 + 6*a*c*d^2*e^2
- 21*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(160*a^2*d^3*e^2*x^4) - ((35*c^3*d^6 + 33*a*c^2*d
^4*e^2 + 21*a^2*c*d^2*e^4 - 105*a^3*e^6)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(960*a^3*d^4*e^3*x^3)
- ((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)*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])])/(1024*a^(9/2)*d^(11/2)*e
^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.724514, antiderivative size = 498, 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 = {849, 834, 806, 720, 724, 206} \[ -\frac{\left (21 a^2 c d^2 e^4-105 a^3 e^6+33 a c^2 d^4 e^2+35 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}+\frac{\left (-21 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}+\frac{\left (6 a^2 c^2 d^4 e^4-21 a^4 e^8+8 a c^3 d^6 e^2+7 c^4 d^8\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 a^4 d^5 e^4 x^2}-\frac{\left (21 a^2 c d^2 e^4+21 a^3 e^6+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 a^{9/2} d^{11/2} e^{9/2}}-\frac{\left (\frac{c}{a e}-\frac{3 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{20 x^5}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 d x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^7*(d + e*x)),x]

[Out]

((7*c^4*d^8 + 8*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 21*a^4*e^8)*(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])/(512*a^4*d^5*e^4*x^2) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(6*d*x^6) -
((c/(a*e) - (3*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(20*x^5) + ((7*c^2*d^4 + 6*a*c*d^2*e^2
- 21*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(160*a^2*d^3*e^2*x^4) - ((35*c^3*d^6 + 33*a*c^2*d
^4*e^2 + 21*a^2*c*d^2*e^4 - 105*a^3*e^6)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(960*a^3*d^4*e^3*x^3)
- ((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)*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])])/(1024*a^(9/2)*d^(11/2)*e
^(9/2))

Rule 849

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
]))

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 720

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] /; FreeQ[
{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 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{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx &=\int \frac{(a e+c d x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^7} \, dx\\ &=-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6}-\frac{\int \frac{\left (-\frac{3}{2} a e \left (c d^2-3 a e^2\right )+3 a c d e^2 x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^6} \, dx}{6 a d e}\\ &=-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6}-\frac{\left (\frac{c}{a e}-\frac{3 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 x^5}+\frac{\int \frac{\left (-\frac{3}{4} a e \left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right )-3 a c d e^2 \left (c d^2-3 a e^2\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5} \, dx}{30 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 )^{3/2}}{6 d x^6}-\frac{\left (\frac{c}{a e}-\frac{3 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 x^5}+\frac{\left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}-\frac{\int \frac{\left (-\frac{3}{8} a e \left (35 c^3 d^6+33 a c^2 d^4 e^2+21 a^2 c d^2 e^4-105 a^3 e^6\right )-\frac{3}{4} a c d e^2 \left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4} \, dx}{120 a^3 d^3 e^3}\\ &=-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6}-\frac{\left (\frac{c}{a e}-\frac{3 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 x^5}+\frac{\left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}-\frac{\left (35 c^3 d^6+33 a c^2 d^4 e^2+21 a^2 c d^2 e^4-105 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}-\frac{\left (7 c^4 d^8+8 a c^3 d^6 e^2+6 a^2 c^2 d^4 e^4-21 a^4 e^8\right ) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx}{128 a^3 d^4 e^3}\\ &=\frac{\left (7 c^4 d^8+8 a c^3 d^6 e^2+6 a^2 c^2 d^4 e^4-21 a^4 e^8\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}}{512 a^4 d^5 e^4 x^2}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6}-\frac{\left (\frac{c}{a e}-\frac{3 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 x^5}+\frac{\left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}-\frac{\left (35 c^3 d^6+33 a c^2 d^4 e^2+21 a^2 c d^2 e^4-105 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}+\frac{\left (\left (c d^2-a e^2\right )^3 \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}{1024 a^4 d^5 e^4}\\ &=\frac{\left (7 c^4 d^8+8 a c^3 d^6 e^2+6 a^2 c^2 d^4 e^4-21 a^4 e^8\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}}{512 a^4 d^5 e^4 x^2}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6}-\frac{\left (\frac{c}{a e}-\frac{3 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 x^5}+\frac{\left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}-\frac{\left (35 c^3 d^6+33 a c^2 d^4 e^2+21 a^2 c d^2 e^4-105 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}-\frac{\left (\left (c d^2-a e^2\right )^3 \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 )}{512 a^4 d^5 e^4}\\ &=\frac{\left (7 c^4 d^8+8 a c^3 d^6 e^2+6 a^2 c^2 d^4 e^4-21 a^4 e^8\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}}{512 a^4 d^5 e^4 x^2}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6}-\frac{\left (\frac{c}{a e}-\frac{3 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 x^5}+\frac{\left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}-\frac{\left (35 c^3 d^6+33 a c^2 d^4 e^2+21 a^2 c d^2 e^4-105 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}-\frac{\left (c d^2-a e^2\right )^3 \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 )}{1024 a^{9/2} d^{11/2} e^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.88486, size = 380, normalized size = 0.76 \[ -\frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{16 x^2 (d+e x) \left (63 a^2 e^4+54 a c d^2 e^2+35 c^2 d^4\right ) (a e+c d x)^2}{a^2 d^2 e^2}+\frac{5 x^3 \left (21 a^2 c d^2 e^4+21 a^3 e^6+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \left (\sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (a^2 e^2 \left (-8 d^2-2 d e x+3 e^2 x^2\right )-2 a c d^2 e x (7 d+4 e x)-3 c^2 d^4 x^2\right )+3 x^3 \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right )\right )}{a^{7/2} d^{9/2} e^{7/2} \sqrt{d+e x} \sqrt{a e+c d x}}-\frac{128 x (d+e x) \left (9 a e^2+7 c d^2\right ) (a e+c d x)^2}{a d e}+1280 (d+e x) (a e+c d x)^2\right )}{7680 a d e x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^7*(d + e*x)),x]

[Out]

-(Sqrt[(a*e + c*d*x)*(d + e*x)]*(1280*(a*e + c*d*x)^2*(d + e*x) - (128*(7*c*d^2 + 9*a*e^2)*x*(a*e + c*d*x)^2*(
d + e*x))/(a*d*e) + (16*(35*c^2*d^4 + 54*a*c*d^2*e^2 + 63*a^2*e^4)*x^2*(a*e + c*d*x)^2*(d + e*x))/(a^2*d^2*e^2
) + (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)*x^3*(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e +
c*d*x]*Sqrt[d + e*x]*(-3*c^2*d^4*x^2 - 2*a*c*d^2*e*x*(7*d + 4*e*x) + a^2*e^2*(-8*d^2 - 2*d*e*x + 3*e^2*x^2)) +
3*(c*d^2 - a*e^2)^3*x^3*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])]))/(a^(7/2)*d^(9/
2)*e^(7/2)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(7680*a*d*e*x^6)

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Maple [B]  time = 0.108, size = 3387, normalized size = 6.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^7/(e*x+d),x)

[Out]

-533/1536*e^6/d^7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1/3*e^6/d^7*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^
(3/2)+7/1536*d^5/a^6/e^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^6+7/512*d^6/a^5/e^5*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2)*c^6-21/512*e^7/d^6*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/8*e^5/d^4*c*(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(1/2)-23/96/d/a^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^3+1/64*e/a^2*(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2)*c^3+19/60/d^3/a/x^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+1/8*e^5/d^4*c*(c*d*e*(d/e+x)^2+
(a*e^2-c*d^2)*(d/e+x))^(1/2)-1/6/d^2/a/e/x^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-7/96/a^3/e^3/x^4*(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^2-1/4*e^8/d^7*a*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-1/8*e^9/d^8*
a^2/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-3/16*e^9/d^6*a^2*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d
*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)+1/4*e^6/d^5*c*(c*d*e*(d/e+x)^2+(a*e^2
-c*d^2)*(d/e+x))^(1/2)*x-1/16*e^5/d^2*c^2*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^
2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)+107/192*e^2/d^5/a/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-15
/512*e^3/d^2/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^2-235/384*e^4/d^5/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(3/2)*c+1045/1536*e^4/d^7/a/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-703/1536*e^2/d^3/a^2*(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(3/2)*c^2-491/768*e^3/d^6/a/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+15/1024*e^4/d/(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+257/768/d^3/
a^3/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^2+1/4*e^8/d^7*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+1/8*
e^9/d^8*a^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+3/16*e^9/d^6*a^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c
)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)-149/512*e^6/d^5*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(1/2)*x+1/16*e^5/d^2*c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(1/2))/(d*e*c)^(1/2)+21/1024*e^8/d^5*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)-43/96*e/d^4/a/x^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+13/512/e*d^2/a^3*(
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^4-131/1536/e^2*d/a^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^4-5/38
4/e^4*d^3/a^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^5+1/64/e^3*d^4/a^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)*c^5+65/192/d^3/a^2/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c+3/1024*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^4-109/768/e/a^4*c^4*(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-1/12/e^3/a^4/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^3+7/256*d/a^3*(a*d*e
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^4+1/16*e^11/d^8*a^3/c*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/
2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)+3/16*e^7/d^4*a*c*ln((1/2*a*e^2-1/2*c*d^2+(d/e+
x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)-21/512*e^4/d^3/a*(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^2+877/1536*e^2/d^5/a^2/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c+3/256*e
^2/d/a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^3-21/512*e^6/d^3*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)*c-43/96*e/d^4/a^2/x^2*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(5/2)*c-1045/1536*e^5/d^6/a*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-877/1536*e^3/d^4/a^2*c^2*(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-257/768*e/d^2/a^3*c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+1/256*e
^2*d/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)*c
^3-1/16*e^11/d^8*a^3/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))
/(d*e*c)^(1/2)-3/16*e^7/d^4*a*c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(1/2))/(d*e*c)^(1/2)+7/512*d^5/a^5/e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^6-7/1536*d^3/a^6/e^6/x*(
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^5+7/60/d/a^2/e^2/x^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c-7/768*
d^2/a^5/e^5/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^4+7/1536*d^4/a^6/e^5*c^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(3/2)*x-7/1024*d^7/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^6+7/192*d/a^4/e^4/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^3-41/1536/e^3*d^2/
a^5*c^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+3/512/e^2*d^5/a^3/(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^5+29/192/e^2/d/a^3/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(5/2)*c^2+109/768/e^2/d/a^4/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^3+15/512/e^2*d^3/a^4*(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^5+41/1536/e^4*d/a^5/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^4-11/48/e/
d^2/a^2/x^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c-91/384/e/d^2/a^3/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
5/2)*c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^7/(e*x+d),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)*x^7), x)

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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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^7/(e*x+d),x, algorithm="fricas")

[Out]

Timed out

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/x**7/(e*x+d),x)

[Out]

Timed out

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Giac [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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^7/(e*x+d),x, algorithm="giac")

[Out]

Timed out