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

Optimal. Leaf size=389 \[ -\frac{\left (25 a^2 c d^2 e^4-105 a^3 e^6+17 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 a^3 d^4 e^3 x}+\frac{\left (-35 a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{96 a^2 d^3 e^2 x^2}+\frac{\left (c d^2-a e^2\right ) \left (15 a^2 c d^2 e^4+35 a^3 e^6+9 a c^2 d^4 e^2+5 c^3 d^6\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 a^{7/2} d^{9/2} e^{7/2}}-\frac{\left (\frac{c}{a e}-\frac{7 e}{d^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 x^3}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^4} \]

[Out]

-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(4*d*x^4) - ((c/(a*e) - (7*e)/d^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(24*x^3) + ((5*c^2*d^4 + 6*a*c*d^2*e^2 - 35*a^2*e^4)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
])/(96*a^2*d^3*e^2*x^2) - ((15*c^3*d^6 + 17*a*c^2*d^4*e^2 + 25*a^2*c*d^2*e^4 - 105*a^3*e^6)*Sqrt[a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2])/(192*a^3*d^4*e^3*x) + ((c*d^2 - a*e^2)*(5*c^3*d^6 + 9*a*c^2*d^4*e^2 + 15*a^2*c*d^2*
e^4 + 35*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])])/(128*a^(7/2)*d^(9/2)*e^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.593513, antiderivative size = 389, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {849, 834, 806, 724, 206} \[ -\frac{\left (25 a^2 c d^2 e^4-105 a^3 e^6+17 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 a^3 d^4 e^3 x}+\frac{\left (-35 a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{96 a^2 d^3 e^2 x^2}+\frac{\left (c d^2-a e^2\right ) \left (15 a^2 c d^2 e^4+35 a^3 e^6+9 a c^2 d^4 e^2+5 c^3 d^6\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 a^{7/2} d^{9/2} e^{7/2}}-\frac{\left (\frac{c}{a e}-\frac{7 e}{d^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 x^3}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(4*d*x^4) - ((c/(a*e) - (7*e)/d^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(24*x^3) + ((5*c^2*d^4 + 6*a*c*d^2*e^2 - 35*a^2*e^4)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
])/(96*a^2*d^3*e^2*x^2) - ((15*c^3*d^6 + 17*a*c^2*d^4*e^2 + 25*a^2*c*d^2*e^4 - 105*a^3*e^6)*Sqrt[a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2])/(192*a^3*d^4*e^3*x) + ((c*d^2 - a*e^2)*(5*c^3*d^6 + 9*a*c^2*d^4*e^2 + 15*a^2*c*d^2*
e^4 + 35*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])])/(128*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 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{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5 (d+e x)} \, dx &=\int \frac{a e+c d x}{x^5 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac{\int \frac{-\frac{1}{2} a e \left (c d^2-7 a e^2\right )+3 a c d e^2 x}{x^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 a d e}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac{\left (\frac{c}{a e}-\frac{7 e}{d^2}\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac{\int \frac{-\frac{1}{4} a e \left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right )-a c d e^2 \left (c d^2-7 a e^2\right ) x}{x^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{12 a^2 d^2 e^2}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac{\left (\frac{c}{a e}-\frac{7 e}{d^2}\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac{\left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac{\int \frac{-\frac{1}{8} a e \left (15 c^3 d^6+17 a c^2 d^4 e^2+25 a^2 c d^2 e^4-105 a^3 e^6\right )-\frac{1}{4} a c d e^2 \left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) x}{x^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{24 a^3 d^3 e^3}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac{\left (\frac{c}{a e}-\frac{7 e}{d^2}\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac{\left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac{\left (15 c^3 d^6+17 a c^2 d^4 e^2+25 a^2 c d^2 e^4-105 a^3 e^6\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 a^3 d^4 e^3 x}-\frac{\left (\left (c d^2-a e^2\right ) \left (5 c^3 d^6+9 a c^2 d^4 e^2+15 a^2 c d^2 e^4+35 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}{128 a^3 d^4 e^3}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac{\left (\frac{c}{a e}-\frac{7 e}{d^2}\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac{\left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac{\left (15 c^3 d^6+17 a c^2 d^4 e^2+25 a^2 c d^2 e^4-105 a^3 e^6\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 a^3 d^4 e^3 x}+\frac{\left (\left (c d^2-a e^2\right ) \left (5 c^3 d^6+9 a c^2 d^4 e^2+15 a^2 c d^2 e^4+35 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 )}{64 a^3 d^4 e^3}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac{\left (\frac{c}{a e}-\frac{7 e}{d^2}\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac{\left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac{\left (15 c^3 d^6+17 a c^2 d^4 e^2+25 a^2 c d^2 e^4-105 a^3 e^6\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 a^3 d^4 e^3 x}+\frac{\left (c d^2-a e^2\right ) \left (5 c^3 d^6+9 a c^2 d^4 e^2+15 a^2 c d^2 e^4+35 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 )}{128 a^{7/2} d^{9/2} e^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.377932, size = 273, normalized size = 0.7 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{\sqrt{a} \sqrt{d} \sqrt{e} \left (a^2 c d^2 e^2 x \left (-8 d^2+12 d e x-25 e^2 x^2\right )+a^3 e^3 \left (56 d^2 e x-48 d^3-70 d e^2 x^2+105 e^3 x^3\right )+a c^2 d^4 e x^2 (10 d-17 e x)-15 c^3 d^6 x^3\right )}{x^4}+\frac{3 \left (6 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-35 a^4 e^8+4 a c^3 d^6 e^2+5 c^4 d^8\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right )}{\sqrt{d+e x} \sqrt{a e+c d x}}\right )}{192 a^{7/2} d^{9/2} e^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[a]*Sqrt[d]*Sqrt[e]*(-15*c^3*d^6*x^3 + a*c^2*d^4*e*x^2*(10*d - 17*e*x) +
a^2*c*d^2*e^2*x*(-8*d^2 + 12*d*e*x - 25*e^2*x^2) + a^3*e^3*(-48*d^3 + 56*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3)
))/x^4 + (3*(5*c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 35*a^4*e^8)*ArcTanh[(Sqrt[d]
*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(192*a^(7/2)*d^(9/2)
*e^(7/2))

________________________________________________________________________________________

Maple [B]  time = 0.073, size = 1494, normalized size = 3.8 \begin{align*} \text{result 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)^(1/2)/x^5/(e*x+d),x)

[Out]

13/24/d^3/a/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-17/32/d/a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^
2-1/2*e^4/d^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)*c-35/128*e^5/d^4*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)+93/64*e^2/d^5/a/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+43/64/d^3/a^2/x*(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(3/2)*c-39/32*e^2/d^3/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c-29/32*e/d^4/a/x^2*(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(3/2)-7/32/e^2*d/a^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^3+5/32*e^3/d^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+3/64*e/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^2-19/64/e/a^
3*c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-e^4/d^5*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-29/64*e^
4/d^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*e^6/d^5*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)*a+1/2*e^4/d^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)*c-1/4/d^2/a/e/x^4*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(3/2)-5/64*d^3/a^4/e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^4-5/32/a^3/e^3/x^2*(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^2+1/2*e^6/d^5*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)*a-7/16/e/d^2/a^2/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c+19/
64/e^2/d/a^3/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^2+1/32/e*d^2/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^3-93/64*e^3/d^4/a*c*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(1/2)*x-43/64*e/d^2/a^2*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+5/24/d/a^2/e^2/x^3*(a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c-5/64*d^2/a^4/e^3*c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+5/64*d/a^4/e^4
/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^3+5/128*d^4/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^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{{\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)^(1/2)/x^5/(e*x+d),x, algorithm="maxima")

[Out]

integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/((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)^(1/2)/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)**(1/2)/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: 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)^(1/2)/x^5/(e*x+d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError