3.465 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^5 (d+e x)} \, dx\)

Optimal. Leaf size=404 \[ -\frac{\left (x \left (11 a^2 c d^2 e^4-3 a^3 e^6+83 a c^2 d^4 e^2+5 c^3 d^6\right )+2 a d e \left (5 c d^2-a e^2\right ) \left (3 a e^2+c d^2\right )\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 a d^2 e x^2}+\frac{\left (-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8-60 a c^3 d^6 e^2+5 c^4 d^8\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 a^{3/2} d^{5/2} e^{3/2}}+c^{5/2} d^{5/2} e^{3/2} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )-\frac{\left (x \left (3 a e^2+11 c d^2\right )+6 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 d x^4} \]

[Out]

-((2*a*d*e*(5*c*d^2 - a*e^2)*(c*d^2 + 3*a*e^2) + (5*c^3*d^6 + 83*a*c^2*d^4*e^2 + 11*a^2*c*d^2*e^4 - 3*a^3*e^6)
*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*a*d^2*e*x^2) - ((6*a*d*e + (11*c*d^2 + 3*a*e^2)*x)*(a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24*d*x^4) + c^(5/2)*d^(5/2)*e^(3/2)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e
*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])] + ((5*c^4*d^8 - 60*a*c^3*d^6*e^2
- 90*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 3*a^4*e^8)*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])])/(128*a^(3/2)*d^(5/2)*e^(3/2))

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Rubi [A]  time = 0.45556, antiderivative size = 404, 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, 810, 843, 621, 206, 724} \[ -\frac{\left (x \left (11 a^2 c d^2 e^4-3 a^3 e^6+83 a c^2 d^4 e^2+5 c^3 d^6\right )+2 a d e \left (5 c d^2-a e^2\right ) \left (3 a e^2+c d^2\right )\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 a d^2 e x^2}+\frac{\left (-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8-60 a c^3 d^6 e^2+5 c^4 d^8\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 a^{3/2} d^{5/2} e^{3/2}}+c^{5/2} d^{5/2} e^{3/2} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )-\frac{\left (x \left (3 a e^2+11 c d^2\right )+6 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 d x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2*a*d*e*(5*c*d^2 - a*e^2)*(c*d^2 + 3*a*e^2) + (5*c^3*d^6 + 83*a*c^2*d^4*e^2 + 11*a^2*c*d^2*e^4 - 3*a^3*e^6)
*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*a*d^2*e*x^2) - ((6*a*d*e + (11*c*d^2 + 3*a*e^2)*x)*(a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24*d*x^4) + c^(5/2)*d^(5/2)*e^(3/2)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e
*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])] + ((5*c^4*d^8 - 60*a*c^3*d^6*e^2
- 90*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 3*a^4*e^8)*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])])/(128*a^(3/2)*d^(5/2)*e^(3/2))

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 810

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*
c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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])

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]

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 )^{5/2}}{x^5 (d+e x)} \, dx &=\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^5} \, dx\\ &=-\frac{\left (6 a d e+\left (11 c d^2+3 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}}{24 d x^4}-\frac{\int \frac{\left (-\frac{1}{2} a e \left (5 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )-8 a c^2 d^3 e^2 x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx}{8 a d e}\\ &=-\frac{\left (2 a d e \left (5 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (5 c^3 d^6+83 a c^2 d^4 e^2+11 a^2 c d^2 e^4-3 a^3 e^6\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a d^2 e x^2}-\frac{\left (6 a d e+\left (11 c d^2+3 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}}{24 d x^4}+\frac{\int \frac{-\frac{1}{4} a e \left (5 c^4 d^8-60 a c^3 d^6 e^2-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8\right )+32 a^2 c^3 d^5 e^4 x}{x \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{32 a^2 d^2 e^2}\\ &=-\frac{\left (2 a d e \left (5 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (5 c^3 d^6+83 a c^2 d^4 e^2+11 a^2 c d^2 e^4-3 a^3 e^6\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a d^2 e x^2}-\frac{\left (6 a d e+\left (11 c d^2+3 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}}{24 d x^4}+\left (c^3 d^3 e^2\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx-\frac{\left (5 c^4 d^8-60 a c^3 d^6 e^2-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8\right ) \int \frac{1}{x \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 a d^2 e}\\ &=-\frac{\left (2 a d e \left (5 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (5 c^3 d^6+83 a c^2 d^4 e^2+11 a^2 c d^2 e^4-3 a^3 e^6\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a d^2 e x^2}-\frac{\left (6 a d e+\left (11 c d^2+3 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}}{24 d x^4}+\left (2 c^3 d^3 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d e-x^2} \, dx,x,\frac{c d^2+a e^2+2 c d e x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )+\frac{\left (5 c^4 d^8-60 a c^3 d^6 e^2-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8\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 )}{64 a d^2 e}\\ &=-\frac{\left (2 a d e \left (5 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (5 c^3 d^6+83 a c^2 d^4 e^2+11 a^2 c d^2 e^4-3 a^3 e^6\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a d^2 e x^2}-\frac{\left (6 a d e+\left (11 c d^2+3 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}}{24 d x^4}+c^{5/2} d^{5/2} e^{3/2} \tanh ^{-1}\left (\frac{c d^2+a e^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )+\frac{\left (5 c^4 d^8-60 a c^3 d^6 e^2-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8\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 )}{128 a^{3/2} d^{5/2} e^{3/2}}\\ \end{align*}

Mathematica [A]  time = 3.72771, size = 404, normalized size = 1. \[ \frac{\sqrt{a e+c d x} \left (-\frac{\sqrt{d} \sqrt{e} (d+e x) \sqrt{a e+c d x} \left (a^2 c d^2 e^2 x \left (136 d^2+244 d e x+57 e^2 x^2\right )+3 a^3 e^3 \left (24 d^2 e x+16 d^3+2 d e^2 x^2-3 e^3 x^3\right )+a c^2 d^4 e x^2 (118 d+337 e x)+15 c^3 d^6 x^3\right )}{a x^4}+\frac{3 \sqrt{d+e x} \left (-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8-60 a c^3 d^6 e^2+5 c^4 d^8\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right )}{a^{3/2}}+384 c^{3/2} d^4 e^3 \sqrt{c d} \sqrt{c d^2-a e^2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right )\right )}{192 d^{5/2} e^{3/2} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a*e + c*d*x]*(-((Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*(d + e*x)*(15*c^3*d^6*x^3 + a*c^2*d^4*e*x^2*(118*d +
337*e*x) + a^2*c*d^2*e^2*x*(136*d^2 + 244*d*e*x + 57*e^2*x^2) + 3*a^3*e^3*(16*d^3 + 24*d^2*e*x + 2*d*e^2*x^2 -
3*e^3*x^3)))/(a*x^4)) + 384*c^(3/2)*d^4*Sqrt[c*d]*e^3*Sqrt[c*d^2 - a*e^2]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2
)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])] + (3*(5*c^4*d^8 - 60*a
*c^3*d^6*e^2 - 90*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 3*a^4*e^8)*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*
d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/a^(3/2)))/(192*d^(5/2)*e^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [B]  time = 0.087, size = 3646, normalized size = 9. \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)^(5/2)/x^5/(e*x+d),x)

[Out]

3/64*e^8/d^5*a^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/16*e^7/d^6*a^2/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(
d/e+x))^(3/2)+1/8*e^6/d^5*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-7/64*e^4/d^3*c*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(3/2)*x-9/64*e^7/d^4*a^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+3/128*e^10/d^7*a^4/c^2*(c*d*
e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+3/64*e^4/d*a*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-13/32*e/
d^4/a/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+19/96/e^2*d/a^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^3+
25/64*e^2/d^5/a/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)-15/32*e^2/d^3/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5
/2)*c-15/32*e*d^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^3+35/96/e*d^2/a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^3+85/192*d/a^2*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(3/2)*x*c^3+43/192/e/a^3*c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+5/32*e^5*a^2/(a*d*e)^(1/2)*l
n((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+31/64/d^3/a^2/x*(a*d*
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c-17/64*e^4/d*a*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-3/128*e^7/d^2*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)+9/64*e^
7/d^4*a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-3/128*e^10/d^7*a^4/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)+15/128*e^8/d^3*a^3*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)+253/256*e^2*d^3*c^3*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)+1/16*e^7/d^6*a^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1/4/d^2/a/e/x^4*(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)-5/192*d^4/a^3/e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^4-1/64*d^3/a^4/e^
4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^4-5/64*d^5/a^2/e^2*c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/9
6/a^3/e^3/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^2-15/128*e^8/d^3*a^3*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)+3/256*e^2*d^3*c^3*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^6
/d^5*a*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x-3/64*e^8/d^5*a^3/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+
x))^(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))^(3/2)*x-1/5*e^4/d^5*(c*d*e*(d/e+x)^2+(a*e^2-c*
d^2)*(d/e+x))^(5/2)-61/320*e^4/d^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+3/8/d^3/a/x^3*(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(7/2)+25/32*d^3/a*c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/96/d/a^2*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(5/2)*c^2+35/96*e/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^2+1/64*e^5/d^4*a*(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(3/2)+3/64*e^6/d^3*a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-19/96*e^3/d^2*c*(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(3/2)+127/128*e^2*d*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/8*e^3*c^2*(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(1/2)*x-3/64*e^3*c^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+1/16*e^3/d^2*c*(c*d*e*(d/
e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)-3/128*e^2*d*c^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-15/256*e^10/
d^5*a^4/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
)+15/256*e^4*d*a*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)-3/64*e^9/d^6*a^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-3/32*e^5/d^2*a*c*(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2)*x+3/256*e^12/d^7*a^5/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)-15/128*e^6/d*a^2*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)-43/192/e^2/d/a^3/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^
2-25/64*e^3/d^4/a*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x-31/64*e/d^2/a^2*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(5/2)*x-13/48/e/d^2/a^2/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c-45/64*e^3*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^2-35/192*e^2/d/a*(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*c^2+15/256*e^10/d^5*a^4/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^9/d^6*a^3/c*(c*d*e*(d/e+x)^2+(a*e^2-c*
d^2)*(d/e+x))^(1/2)*x+9/64*e^5/d^2*a*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-3/256*e^12/d^7*a^5/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)+15/128*e^6/d*a^2*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)-15/256*e^4*d*a*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)+45/64*e*d^2/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^3-
1/64*d^2/a^4/e^3*c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+1/24/d/a^2/e^2/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(7/2)*c-5/192*d^3/a^3/e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*c^4-5/64*d^4/a^2/e*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2)*x*c^4+5/128*d^6/a/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+1/64*d/a^4/e^4/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^3

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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{5}{2}}}{{\left (e x + d\right )} x^{5}}\,{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)^(5/2)/x^5/(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^5), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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)^(5/2)/x^5/(e*x+d),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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)**(5/2)/x**5/(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: TypeError} \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)^(5/2)/x^5/(e*x+d),x, algorithm="giac")

[Out]

Exception raised: TypeError