3.455 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{x^6 (d+e x)} \, dx\)

Optimal. Leaf size=395 \[ -\frac{\left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 a^3 d^4 e^3 x^2}+\frac{\left (-35 a^2 e^4+12 a c d^2 e^2+15 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}+\frac{\left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 a^{7/2} d^{9/2} e^{7/2}}-\frac{\left (\frac{3 c}{a e}-\frac{7 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{40 x^4}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 d x^5} \]

[Out]

-((c*d^2 - a*e^2)*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*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])/(128*a^3*d^4*e^3*x^2) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(5*d*x^5) - (((3*
c)/(a*e) - (7*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(40*x^4) + ((15*c^2*d^4 + 12*a*c*d^2*e^2
- 35*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(240*a^2*d^3*e^2*x^3) + ((c*d^2 - a*e^2)^3*(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])])/(256*a^(7/2)*d^(9/2)*e^(7/2))

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Rubi [A]  time = 0.513995, antiderivative size = 395, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{\left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 a^3 d^4 e^3 x^2}+\frac{\left (-35 a^2 e^4+12 a c d^2 e^2+15 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}+\frac{\left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 a^{7/2} d^{9/2} e^{7/2}}-\frac{\left (\frac{3 c}{a e}-\frac{7 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{40 x^4}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 d x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)),x]

[Out]

-((c*d^2 - a*e^2)*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*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])/(128*a^3*d^4*e^3*x^2) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(5*d*x^5) - (((3*
c)/(a*e) - (7*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(40*x^4) + ((15*c^2*d^4 + 12*a*c*d^2*e^2
- 35*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(240*a^2*d^3*e^2*x^3) + ((c*d^2 - a*e^2)^3*(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])])/(256*a^(7/2)*d^(9/2)*e^(7/2))

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^6 (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^6} \, dx\\ &=-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac{\int \frac{\left (-\frac{1}{2} a e \left (3 c d^2-7 a e^2\right )+2 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^5} \, dx}{5 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}}{5 d x^5}-\frac{\left (\frac{3 c}{a e}-\frac{7 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}}{40 x^4}+\frac{\int \frac{\left (-\frac{1}{4} a e \left (15 c^2 d^4+12 a c d^2 e^2-35 a^2 e^4\right )-\frac{1}{2} a c d e^2 \left (3 c d^2-7 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^4} \, dx}{20 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}}{5 d x^5}-\frac{\left (\frac{3 c}{a e}-\frac{7 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}}{40 x^4}+\frac{\left (15 c^2 d^4+12 a c d^2 e^2-35 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}}{240 a^2 d^3 e^2 x^3}+\frac{\left (\left (c d^2-a e^2\right ) \left (3 c^2 d^4+6 a c d^2 e^2+7 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}{32 a^2 d^3 e^2}\\ &=-\frac{\left (c d^2-a e^2\right ) \left (3 c^2 d^4+6 a c d^2 e^2+7 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}}{128 a^3 d^4 e^3 x^2}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac{\left (\frac{3 c}{a e}-\frac{7 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}}{40 x^4}+\frac{\left (15 c^2 d^4+12 a c d^2 e^2-35 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}}{240 a^2 d^3 e^2 x^3}-\frac{\left (\left (c d^2-a e^2\right )^3 \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}{256 a^3 d^4 e^3}\\ &=-\frac{\left (c d^2-a e^2\right ) \left (3 c^2 d^4+6 a c d^2 e^2+7 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}}{128 a^3 d^4 e^3 x^2}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac{\left (\frac{3 c}{a e}-\frac{7 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}}{40 x^4}+\frac{\left (15 c^2 d^4+12 a c d^2 e^2-35 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}}{240 a^2 d^3 e^2 x^3}+\frac{\left (\left (c d^2-a e^2\right )^3 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\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 )}{128 a^3 d^4 e^3}\\ &=-\frac{\left (c d^2-a e^2\right ) \left (3 c^2 d^4+6 a c d^2 e^2+7 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}}{128 a^3 d^4 e^3 x^2}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac{\left (\frac{3 c}{a e}-\frac{7 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}}{40 x^4}+\frac{\left (15 c^2 d^4+12 a c d^2 e^2-35 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}}{240 a^2 d^3 e^2 x^3}+\frac{\left (c d^2-a e^2\right )^3 \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 )}{256 a^{7/2} d^{9/2} e^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.523731, size = 310, normalized size = 0.78 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{5 x^2 \left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \left (\sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (a^2 e^2 \left (-8 d^2-2 d e x+3 e^2 x^2\right )-2 a c d^2 e x (7 d+4 e x)-3 c^2 d^4 x^2\right )+3 x^3 \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right )\right )}{a^{5/2} d^{7/2} e^{5/2} \sqrt{d+e x} \sqrt{a e+c d x}}+\frac{48 x (d+e x) \left (7 a e^2+5 c d^2\right ) (a e+c d x)^2}{a d e}-384 (d+e x) (a e+c d x)^2\right )}{1920 a d e x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-384*(a*e + c*d*x)^2*(d + e*x) + (48*(5*c*d^2 + 7*a*e^2)*x*(a*e + c*d*x)^2*(d
+ e*x))/(a*d*e) + (5*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*x^2*(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sq
rt[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^(5/2)*d^(7/2)*e^(5/2
)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(1920*a*d*e*x^5)

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Maple [B]  time = 0.102, size = 2888, normalized size = 7.3 \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^6/(e*x+d),x)

[Out]

3/16*e^8/d^5*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^5/d^4*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+1/16*e^4/d*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)-1/8*e^8/
d^7*a^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-3/16*e^8/d^5*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)+39/128*e^5/d^4*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2)*x-1/16*e^4/d*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)-7/256*e^7/d^4*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)-1/4*e^7/d^6*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-263/384*e^3/d^6/a/x*(a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+1/32*e^2/d/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^2+19/48*e/d^2/a^2*(a*d*e
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^2+121/192*e^2/d^5/a/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+227/384*e^
3/d^4/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c-25/48*e/d^4/a/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-1/
32*e/a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^3+73/192/d^3/a^2/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5
/2)*c+23/96/d/a^3*c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-1/16/a^3/e^3/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(5/2)*c^2-3/128*d^3/a^3/e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^4-1/5/d^2/a/e/x^5*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(5/2)-3/128*d^5/a^4/e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^5+3/128*d^2/a^4/e^3*(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^4-1/128*d^4/a^5/e^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^5-3/64/a^4/e^
3/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^3+7/128*e^6/d^5*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+15/128
*e^4/d^3*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/8*e^4/d^3*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)
+3/8/d^3/a/x^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-1/32*d/a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^3+
7/48/e/a^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^3-3/128*e^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^2+1/4*e^7/d^6*a*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e
+x))^(1/2)*x+1/8*e^8/d^7*a^2/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+1/3*e^5/d^6*(c*d*e*(d/e+x)^2+(a*e
^2-c*d^2)*(d/e+x))^(3/2)+45/128*e^5/d^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1/4/d^2/a^2/e/x^3*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(5/2)*c-3/256*d^4/a^2/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)*c^4+1/8/d/a^2/e^2/x^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c+9/64/d/a
^3/e^2/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^2+3/64*d/a^4/e^2*c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3
/2)*x-23/96/e/d^2/a^3/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^2+3/64*e^3/d^2/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(1/2)*x*c^2+15/256*e^5/d^2*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+263/384*e^4/d^5/a*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+103/192*e^2/d^3/a^
2*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-103/192*e/d^4/a^2/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c-
1/128*e*d^2/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^10/d^7*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)-3/64/e*d^2/a^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^4-3/16*e^6/d^3*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)+1/16*e^10/d^7*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^6/d^3*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)-3/128*d^4/a^4/e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^5+3/256*d^6/a^3/e^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+1/128
*d^2/a^5/e^5/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^4+1/64*d/a^4/e^4/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(5/2)*c^3-1/128*d^3/a^5/e^4*c^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x

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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^{6}}\,{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^6/(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^6), 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^6/(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**6/(e*x+d),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \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^6/(e*x+d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError