3.476 \(\int \frac{1}{x^3 (d+e x) \sqrt{a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\)

Optimal. Leaf size=329 \[ -\frac{3 \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 a^{5/2} d^{7/2} e^{5/2}}+\frac{\left (3 c d^2-5 a e^2\right ) \left (3 a e^2+c d^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a^2 d^3 e^2 x \left (c d^2-a e^2\right )}-\frac{\left (c d^2-5 a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d^2 e x^2 \left (c d^2-a e^2\right )}-\frac{2 e (a e+c d x)}{d x^2 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

(-2*e*(a*e + c*d*x))/(d*(c*d^2 - a*e^2)*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - ((c*d^2 - 5*a*e^2)*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(2*a*d^2*e*(c*d^2 - a*e^2)*x^2) + ((3*c*d^2 - 5*a*e^2)*(c*d^2 + 3
*a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*a^2*d^3*e^2*(c*d^2 - a*e^2)*x) - (3*(c^2*d^4 + 2*a*c*d
^2*e^2 + 5*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])])/(8*a^(5/2)*d^(7/2)*e^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.507791, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {851, 822, 834, 806, 724, 206} \[ -\frac{3 \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 a^{5/2} d^{7/2} e^{5/2}}+\frac{\left (3 c d^2-5 a e^2\right ) \left (3 a e^2+c d^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a^2 d^3 e^2 x \left (c d^2-a e^2\right )}-\frac{\left (c d^2-5 a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d^2 e x^2 \left (c d^2-a e^2\right )}-\frac{2 e (a e+c d x)}{d x^2 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

(-2*e*(a*e + c*d*x))/(d*(c*d^2 - a*e^2)*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - ((c*d^2 - 5*a*e^2)*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(2*a*d^2*e*(c*d^2 - a*e^2)*x^2) + ((3*c*d^2 - 5*a*e^2)*(c*d^2 + 3
*a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*a^2*d^3*e^2*(c*d^2 - a*e^2)*x) - (3*(c^2*d^4 + 2*a*c*d
^2*e^2 + 5*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])])/(8*a^(5/2)*d^(7/2)*e^(5/2))

Rule 851

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Int[((f + g*x)^n*(a + b*x + c*x^2)^(m + p))/(a/d + (c*x)/e)^m, x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] &&
NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[m, 0] && In
tegerQ[n] && (LtQ[n, 0] || GtQ[p, 0])

Rule 822

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

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{1}{x^3 (d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\int \frac{a e+c d x}{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\\ &=-\frac{2 e (a e+c d x)}{d \left (c d^2-a e^2\right ) x^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{2 \int \frac{-\frac{1}{2} a e \left (c d^2-5 a e^2\right ) \left (c d^2-a e^2\right )+2 a c d e^2 \left (c d^2-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}{a d e \left (c d^2-a e^2\right )^2}\\ &=-\frac{2 e (a e+c d x)}{d \left (c d^2-a e^2\right ) x^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (c d^2-5 a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 a d^2 e \left (c d^2-a e^2\right ) x^2}+\frac{\int \frac{-\frac{1}{4} a e \left (3 c d^2-5 a e^2\right ) \left (c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )-\frac{1}{2} a c d e^2 \left (c d^2-5 a e^2\right ) \left (c d^2-a e^2\right ) x}{x^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{a^2 d^2 e^2 \left (c d^2-a e^2\right )^2}\\ &=-\frac{2 e (a e+c d x)}{d \left (c d^2-a e^2\right ) x^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (c d^2-5 a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 a d^2 e \left (c d^2-a e^2\right ) x^2}+\frac{\left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 a^2 d^3 e^2 \left (c d^2-a e^2\right ) x}+\frac{\left (3 \left (c^2 d^4+2 a c d^2 e^2+5 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}{8 a^2 d^3 e^2}\\ &=-\frac{2 e (a e+c d x)}{d \left (c d^2-a e^2\right ) x^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (c d^2-5 a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 a d^2 e \left (c d^2-a e^2\right ) x^2}+\frac{\left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 a^2 d^3 e^2 \left (c d^2-a e^2\right ) x}-\frac{\left (3 \left (c^2 d^4+2 a c d^2 e^2+5 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 )}{4 a^2 d^3 e^2}\\ &=-\frac{2 e (a e+c d x)}{d \left (c d^2-a e^2\right ) x^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (c d^2-5 a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 a d^2 e \left (c d^2-a e^2\right ) x^2}+\frac{\left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 a^2 d^3 e^2 \left (c d^2-a e^2\right ) x}-\frac{3 \left (c^2 d^4+2 a c d^2 e^2+5 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 )}{8 a^{5/2} d^{7/2} e^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.205207, size = 283, normalized size = 0.86 \[ \frac{\sqrt{a} \sqrt{d} \sqrt{e} \left (-a^2 c d e^2 \left (-4 d^2 e x+2 d^3+d e^2 x^2+15 e^3 x^3\right )+a^3 e^4 \left (2 d^2-5 d e x-15 e^2 x^2\right )+a c^2 d^3 e x \left (d^2+5 d e x+4 e^2 x^2\right )+3 c^3 d^5 x^2 (d+e x)\right )-3 x^2 \sqrt{d+e x} \left (3 a^2 c d^2 e^4-5 a^3 e^6+a c^2 d^4 e^2+c^3 d^6\right ) \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right )}{4 a^{5/2} d^{7/2} e^{5/2} x^2 \left (c d^2-a e^2\right ) \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

(Sqrt[a]*Sqrt[d]*Sqrt[e]*(3*c^3*d^5*x^2*(d + e*x) + a^3*e^4*(2*d^2 - 5*d*e*x - 15*e^2*x^2) + a*c^2*d^3*e*x*(d^
2 + 5*d*e*x + 4*e^2*x^2) - a^2*c*d*e^2*(2*d^3 - 4*d^2*e*x + d*e^2*x^2 + 15*e^3*x^3)) - 3*(c^3*d^6 + a*c^2*d^4*
e^2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*x^2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sq
rt[a]*Sqrt[e]*Sqrt[d + e*x])])/(4*a^(5/2)*d^(7/2)*e^(5/2)*(c*d^2 - a*e^2)*x^2*Sqrt[(a*e + c*d*x)*(d + e*x)])

________________________________________________________________________________________

Maple [A]  time = 0.066, size = 414, normalized size = 1.3 \begin{align*} -{\frac{15\,{e}^{2}}{8\,{d}^{3}}\ln \left ({\frac{1}{x} \left ( 2\,ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+2\,\sqrt{ade}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ) } \right ){\frac{1}{\sqrt{ade}}}}+2\,{\frac{{e}^{2}}{{d}^{3} \left ( a{e}^{2}-c{d}^{2} \right ) }\sqrt{cde \left ({\frac{d}{e}}+x \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-1}}-{\frac{1}{2\,a{d}^{2}e{x}^{2}}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}+{\frac{7}{4\,{d}^{3}ax}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}+{\frac{3\,c}{4\,{a}^{2}d{e}^{2}x}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}-{\frac{3\,c}{4\,ad}\ln \left ({\frac{1}{x} \left ( 2\,ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+2\,\sqrt{ade}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ) } \right ){\frac{1}{\sqrt{ade}}}}-{\frac{3\,{c}^{2}d}{8\,{a}^{2}{e}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+2\,\sqrt{ade}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ) } \right ){\frac{1}{\sqrt{ade}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

-15/8*e^2/d^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)+2*e^2/d^3/(a*e^2-c*d^2)/(d/e+x)*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-1/2/d^2/a/e/x^2*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+7/4/d^3/a/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+3/4/d/a^2/e^2/x*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(1/2)*c-3/4/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-3/8*d/a^2/e^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^2

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (e x + d\right )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)*x^3), x)

________________________________________________________________________________________

Fricas [A]  time = 27.7265, size = 1593, normalized size = 4.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

[1/16*(3*((c^3*d^6*e + a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - 5*a^3*e^7)*x^3 + (c^3*d^7 + a*c^2*d^5*e^2 + 3*a^2*c*d
^3*e^4 - 5*a^3*d*e^6)*x^2)*sqrt(a*d*e)*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*(2*a^2*c*d^5*e^2 - 2*a^3*d^3*e^4 - (3*a*c^2*d^5*e^2 + 4*a^2*c*d^3*e^4 - 15*a^3*d*e^6)*x^2 - (3*a*c^2
*d^6*e + 2*a^2*c*d^4*e^3 - 5*a^3*d^2*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/((a^3*c*d^6*e^4 - a^
4*d^4*e^6)*x^3 + (a^3*c*d^7*e^3 - a^4*d^5*e^5)*x^2), 1/8*(3*((c^3*d^6*e + a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - 5*
a^3*e^7)*x^3 + (c^3*d^7 + a*c^2*d^5*e^2 + 3*a^2*c*d^3*e^4 - 5*a^3*d*e^6)*x^2)*sqrt(-a*d*e)*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*(2*a^2*c*d^5*e^2 - 2*a^3*d^3*e^4 - (3*a*c^2*d^5*e^2 + 4*a^2*c*d^3*e^4 - 15*a
^3*d*e^6)*x^2 - (3*a*c^2*d^6*e + 2*a^2*c*d^4*e^3 - 5*a^3*d^2*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*
x))/((a^3*c*d^6*e^4 - a^4*d^4*e^6)*x^3 + (a^3*c*d^7*e^3 - a^4*d^5*e^5)*x^2)]

________________________________________________________________________________________

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(1/x**3/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

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(1/x^3/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError