3.2389 \(\int \frac{1}{(d+e x)^3 (a+b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=371 \[ -\frac{e \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{4 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{e \sqrt{a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{3 e^2 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2*Sqrt[a +
b*x + c*x^2]) - (e*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*Sqrt[a + b*x + c*x^2])/(2*(b^2 - 4*a*c)*(c
*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) - (e*(2*c*d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(2*b*d + 13*a*e))*Sqrt
[a + b*x + c*x^2])/(4*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (3*e^2*(16*c^2*d^2 + 5*b^2*e^2 - 4*
c*e*(4*b*d + a*e))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2
])])/(8*(c*d^2 - b*d*e + a*e^2)^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.475524, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {740, 834, 806, 724, 206} \[ -\frac{e \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{4 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{e \sqrt{a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{3 e^2 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2*Sqrt[a +
b*x + c*x^2]) - (e*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*Sqrt[a + b*x + c*x^2])/(2*(b^2 - 4*a*c)*(c
*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) - (e*(2*c*d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(2*b*d + 13*a*e))*Sqrt
[a + b*x + c*x^2])/(4*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (3*e^2*(16*c^2*d^2 + 5*b^2*e^2 - 4*
c*e*(4*b*d + a*e))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2
])])/(8*(c*d^2 - b*d*e + a*e^2)^(7/2))

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*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*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

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}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{\frac{1}{2} e \left (4 b c d-5 b^2 e+12 a c e\right )+2 c e (2 c d-b e) x}{(d+e x)^3 \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \sqrt{a+b x+c x^2}}-\frac{e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt{a+b x+c x^2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{\int \frac{\frac{1}{4} e \left (28 b^2 c d e-80 a c^2 d e-15 b^3 e^2-4 b c \left (2 c d^2-13 a e^2\right )\right )-\frac{1}{2} c e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \sqrt{a+b x+c x^2}}-\frac{e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt{a+b x+c x^2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac{e (2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt{a+b x+c x^2}}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac{\left (3 e^2 \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{8 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \sqrt{a+b x+c x^2}}-\frac{e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt{a+b x+c x^2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac{e (2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt{a+b x+c x^2}}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{\left (3 e^2 \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x}{\sqrt{a+b x+c x^2}}\right )}{4 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \sqrt{a+b x+c x^2}}-\frac{e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt{a+b x+c x^2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac{e (2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt{a+b x+c x^2}}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac{3 e^2 \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 1.41918, size = 353, normalized size = 0.95 \[ \frac{2 \left (-\frac{e \sqrt{a+x (b+c x)} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{4 (d+e x)^2 \left (e (a e-b d)+c d^2\right )}+\frac{1}{16} e \left (-\frac{2 \sqrt{a+x (b+c x)} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{(d+e x) \left (e (a e-b d)+c d^2\right )^2}-\frac{3 e \left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{\left (e (a e-b d)+c d^2\right )^{5/2}}\right )+\frac{-2 c (a e+c d x)+b^2 e+b c (e x-d)}{(d+e x)^2 \sqrt{a+x (b+c x)}}\right )}{\left (b^2-4 a c\right ) \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-(e*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*Sqrt[a + x*(b + c*x)])/(4*(c*d^2 + e*(-(b*d) + a*e))*(
d + e*x)^2) + (b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/((d + e*x)^2*Sqrt[a + x*(b + c*x)]) + (e*((-2*(2*c*
d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(2*b*d + 13*a*e))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))^
2*(d + e*x)) - (3*(b^2 - 4*a*c)*e*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c
*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(5/2)))/16
))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))

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Maple [B]  time = 0.236, size = 2380, normalized size = 6.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15/8*e^3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*
b^4-45*e/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x
*b*c^3*d^2+45/2*e^2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e
^2)^(1/2)*x*b^2*c^2*d-45/2*e/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e
+c*d^2)/e^2)^(1/2)*b^2*c^2*d^2+15/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-
b*d*e+c*d^2)/e^2)^(1/2)*b*c^3*d^3+15/2*e^2/(a*e^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-
b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a
*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b*c*d+3/2*e*c/(a*e^2-b*d*e+c*d^2)^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln(
(2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*
(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))-1/2/e/(a*e^2-b*d*e+c*d^2)/(d/e+x)^2/((d/e+x)^2*c+(b*e-2*c*d)/
e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+15/8*e^3/(a*e^2-b*d*e+c*d^2)^3/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*
e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2-3/2*e*c/(a*e^2-b*d*e+c*d^2)^2/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+
c*d^2)/e^2)^(1/2)+5/4*e/(a*e^2-b*d*e+c*d^2)^2/(d/e+x)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e
^2)^(1/2)*b+15/2*e/(a*e^2-b*d*e+c*d^2)^3/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c^2
*d^2-13/(a*e^2-b*d*e+c*d^2)^2*c^2/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2
)*b*d+13*e/(a*e^2-b*d*e+c*d^2)^2*c^2/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(
1/2)*x*b-26/(a*e^2-b*d*e+c*d^2)^2*c^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^
(1/2)*x*d-15/4*e^3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^
2)^(1/2)*x*b^3*c+30/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e
^2)^(1/2)*x*c^4*d^3+45/4*e^2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e
+c*d^2)/e^2)^(1/2)*b^3*c*d-15/8*e^3/(a*e^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c
*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*
d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^2-5/2/(a*e^2-b*d*e+c*d^2)^2/(d/e+x)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^
2-b*d*e+c*d^2)/e^2)^(1/2)*c*d-15/2*e^2/(a*e^2-b*d*e+c*d^2)^3/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c
*d^2)/e^2)^(1/2)*b*c*d-15/2*e/(a*e^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/
e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*
d^2)/e^2)^(1/2))/(d/e+x))*c^2*d^2+13/2*e/(a*e^2-b*d*e+c*d^2)^2*c/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x
)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 92.8938, size = 11780, normalized size = 31.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(3*(16*(a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - 16*(a*b^3*c - 4*a^2*b*c^2)*d^3*e^3 + (5*a*b^4 - 24*a^2*b^2*c +
16*a^3*c^2)*d^2*e^4 + (16*(b^2*c^3 - 4*a*c^4)*d^2*e^4 - 16*(b^3*c^2 - 4*a*b*c^3)*d*e^5 + (5*b^4*c - 24*a*b^2*
c^2 + 16*a^2*c^3)*e^6)*x^4 + (32*(b^2*c^3 - 4*a*c^4)*d^3*e^3 - 16*(b^3*c^2 - 4*a*b*c^3)*d^2*e^4 - 2*(3*b^4*c -
8*a*b^2*c^2 - 16*a^2*c^3)*d*e^5 + (5*b^5 - 24*a*b^3*c + 16*a^2*b*c^2)*e^6)*x^3 + (16*(b^2*c^3 - 4*a*c^4)*d^4*
e^2 + 16*(b^3*c^2 - 4*a*b*c^3)*d^3*e^3 - 3*(9*b^4*c - 40*a*b^2*c^2 + 16*a^2*c^3)*d^2*e^4 + 2*(5*b^5 - 32*a*b^3
*c + 48*a^2*b*c^2)*d*e^5 + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*e^6)*x^2 + (16*(b^3*c^2 - 4*a*b*c^3)*d^4*e^2
- 16*(b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*d^3*e^3 + (5*b^5 - 56*a*b^3*c + 144*a^2*b*c^2)*d^2*e^4 + 2*(5*a*b^4 - 2
4*a^2*b^2*c + 16*a^3*c^2)*d*e^5)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2
- (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d
- 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) + 4
*(8*b*c^4*d^7 - 16*(2*b^2*c^3 - 3*a*c^4)*d^6*e + 16*(3*b^3*c^2 - 7*a*b*c^3)*d^5*e^2 - 32*(b^4*c - 3*a*b^2*c^2
+ a^2*c^3)*d^4*e^3 + (8*b^5 - 33*a*b^3*c + 44*a^2*b*c^2)*d^3*e^4 + (a*b^4 + 14*a^2*b^2*c - 88*a^3*c^2)*d^2*e^5
- 11*(a^2*b^3 - 4*a^3*b*c)*d*e^6 + 2*(a^3*b^2 - 4*a^4*c)*e^7 + (16*c^5*d^5*e^2 - 40*b*c^4*d^4*e^3 + 2*(31*b^2
*c^3 - 44*a*c^4)*d^3*e^4 - (53*b^3*c^2 - 132*a*b*c^3)*d^2*e^5 + (15*b^4*c - 14*a*b^2*c^2 - 104*a^2*c^3)*d*e^6
- (15*a*b^3*c - 52*a^2*b*c^2)*e^7)*x^3 + (32*c^5*d^6*e - 72*b*c^4*d^5*e^2 + 80*(b^2*c^3 - a*c^4)*d^4*e^3 - (27
*b^3*c^2 - 28*a*b*c^3)*d^3*e^4 - 2*(14*b^4*c - 73*a*b^2*c^2 + 68*a^2*c^3)*d^2*e^5 + (15*b^5 - 49*a*b^3*c - 20*
a^2*b*c^2)*d*e^6 - (15*a*b^4 - 62*a^2*b^2*c + 24*a^3*c^2)*e^7)*x^2 + (16*c^5*d^7 - 24*b*c^4*d^6*e - 16*(b^2*c^
3 - 4*a*c^4)*d^5*e^2 + 80*(b^3*c^2 - 3*a*b*c^3)*d^4*e^3 - (81*b^4*c - 274*a*b^2*c^2 + 40*a^2*c^3)*d^3*e^4 + (2
5*b^5 - 63*a*b^3*c - 76*a^2*b*c^2)*d^2*e^5 - 2*(10*a*b^4 - 47*a^2*b^2*c + 44*a^3*c^2)*d*e^6 - 5*(a^2*b^3 - 4*a
^3*b*c)*e^7)*x)*sqrt(c*x^2 + b*x + a))/((a*b^2*c^4 - 4*a^2*c^5)*d^10 - 4*(a*b^3*c^3 - 4*a^2*b*c^4)*d^9*e + 2*(
3*a*b^4*c^2 - 10*a^2*b^2*c^3 - 8*a^3*c^4)*d^8*e^2 - 4*(a*b^5*c - a^2*b^3*c^2 - 12*a^3*b*c^3)*d^7*e^3 + (a*b^6
+ 8*a^2*b^4*c - 42*a^3*b^2*c^2 - 24*a^4*c^3)*d^6*e^4 - 4*(a^2*b^5 - a^3*b^3*c - 12*a^4*b*c^2)*d^5*e^5 + 2*(3*a
^3*b^4 - 10*a^4*b^2*c - 8*a^5*c^2)*d^4*e^6 - 4*(a^4*b^3 - 4*a^5*b*c)*d^3*e^7 + (a^5*b^2 - 4*a^6*c)*d^2*e^8 + (
(b^2*c^5 - 4*a*c^6)*d^8*e^2 - 4*(b^3*c^4 - 4*a*b*c^5)*d^7*e^3 + 2*(3*b^4*c^3 - 10*a*b^2*c^4 - 8*a^2*c^5)*d^6*e
^4 - 4*(b^5*c^2 - a*b^3*c^3 - 12*a^2*b*c^4)*d^5*e^5 + (b^6*c + 8*a*b^4*c^2 - 42*a^2*b^2*c^3 - 24*a^3*c^4)*d^4*
e^6 - 4*(a*b^5*c - a^2*b^3*c^2 - 12*a^3*b*c^3)*d^3*e^7 + 2*(3*a^2*b^4*c - 10*a^3*b^2*c^2 - 8*a^4*c^3)*d^2*e^8
- 4*(a^3*b^3*c - 4*a^4*b*c^2)*d*e^9 + (a^4*b^2*c - 4*a^5*c^2)*e^10)*x^4 + (2*(b^2*c^5 - 4*a*c^6)*d^9*e - 7*(b^
3*c^4 - 4*a*b*c^5)*d^8*e^2 + 8*(b^4*c^3 - 3*a*b^2*c^4 - 4*a^2*c^5)*d^7*e^3 - 2*(b^5*c^2 + 6*a*b^3*c^3 - 40*a^2
*b*c^4)*d^6*e^4 - 2*(b^6*c - 10*a*b^4*c^2 + 18*a^2*b^2*c^3 + 24*a^3*c^4)*d^5*e^5 + (b^7 - 34*a^2*b^3*c^2 + 72*
a^3*b*c^3)*d^4*e^6 - 4*(a*b^6 - 4*a^2*b^4*c - 2*a^3*b^2*c^2 + 8*a^4*c^3)*d^3*e^7 + 2*(3*a^2*b^5 - 14*a^3*b^3*c
+ 8*a^4*b*c^2)*d^2*e^8 - 2*(2*a^3*b^4 - 9*a^4*b^2*c + 4*a^5*c^2)*d*e^9 + (a^4*b^3 - 4*a^5*b*c)*e^10)*x^3 + ((
b^2*c^5 - 4*a*c^6)*d^10 - 2*(b^3*c^4 - 4*a*b*c^5)*d^9*e - (2*b^4*c^3 - 13*a*b^2*c^4 + 20*a^2*c^5)*d^8*e^2 + 8*
(b^5*c^2 - 5*a*b^3*c^3 + 4*a^2*b*c^4)*d^7*e^3 - (7*b^6*c - 22*a*b^4*c^2 - 34*a^2*b^2*c^3 + 40*a^3*c^4)*d^6*e^4
+ 2*(b^7 + 4*a*b^5*c - 38*a^2*b^3*c^2 + 24*a^3*b*c^3)*d^5*e^5 - (7*a*b^6 - 22*a^2*b^4*c - 34*a^3*b^2*c^2 + 40
*a^4*c^3)*d^4*e^6 + 8*(a^2*b^5 - 5*a^3*b^3*c + 4*a^4*b*c^2)*d^3*e^7 - (2*a^3*b^4 - 13*a^4*b^2*c + 20*a^5*c^2)*
d^2*e^8 - 2*(a^4*b^3 - 4*a^5*b*c)*d*e^9 + (a^5*b^2 - 4*a^6*c)*e^10)*x^2 + ((b^3*c^4 - 4*a*b*c^5)*d^10 - 2*(2*b
^4*c^3 - 9*a*b^2*c^4 + 4*a^2*c^5)*d^9*e + 2*(3*b^5*c^2 - 14*a*b^3*c^3 + 8*a^2*b*c^4)*d^8*e^2 - 4*(b^6*c - 4*a*
b^4*c^2 - 2*a^2*b^2*c^3 + 8*a^3*c^4)*d^7*e^3 + (b^7 - 34*a^2*b^3*c^2 + 72*a^3*b*c^3)*d^6*e^4 - 2*(a*b^6 - 10*a
^2*b^4*c + 18*a^3*b^2*c^2 + 24*a^4*c^3)*d^5*e^5 - 2*(a^2*b^5 + 6*a^3*b^3*c - 40*a^4*b*c^2)*d^4*e^6 + 8*(a^3*b^
4 - 3*a^4*b^2*c - 4*a^5*c^2)*d^3*e^7 - 7*(a^4*b^3 - 4*a^5*b*c)*d^2*e^8 + 2*(a^5*b^2 - 4*a^6*c)*d*e^9)*x), 1/8*
(3*(16*(a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - 16*(a*b^3*c - 4*a^2*b*c^2)*d^3*e^3 + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3
*c^2)*d^2*e^4 + (16*(b^2*c^3 - 4*a*c^4)*d^2*e^4 - 16*(b^3*c^2 - 4*a*b*c^3)*d*e^5 + (5*b^4*c - 24*a*b^2*c^2 + 1
6*a^2*c^3)*e^6)*x^4 + (32*(b^2*c^3 - 4*a*c^4)*d^3*e^3 - 16*(b^3*c^2 - 4*a*b*c^3)*d^2*e^4 - 2*(3*b^4*c - 8*a*b^
2*c^2 - 16*a^2*c^3)*d*e^5 + (5*b^5 - 24*a*b^3*c + 16*a^2*b*c^2)*e^6)*x^3 + (16*(b^2*c^3 - 4*a*c^4)*d^4*e^2 + 1
6*(b^3*c^2 - 4*a*b*c^3)*d^3*e^3 - 3*(9*b^4*c - 40*a*b^2*c^2 + 16*a^2*c^3)*d^2*e^4 + 2*(5*b^5 - 32*a*b^3*c + 48
*a^2*b*c^2)*d*e^5 + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*e^6)*x^2 + (16*(b^3*c^2 - 4*a*b*c^3)*d^4*e^2 - 16*(b
^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*d^3*e^3 + (5*b^5 - 56*a*b^3*c + 144*a^2*b*c^2)*d^2*e^4 + 2*(5*a*b^4 - 24*a^2*b
^2*c + 16*a^3*c^2)*d*e^5)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2
+ b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 +
(b*c*d^2 - b^2*d*e + a*b*e^2)*x)) - 2*(8*b*c^4*d^7 - 16*(2*b^2*c^3 - 3*a*c^4)*d^6*e + 16*(3*b^3*c^2 - 7*a*b*c^
3)*d^5*e^2 - 32*(b^4*c - 3*a*b^2*c^2 + a^2*c^3)*d^4*e^3 + (8*b^5 - 33*a*b^3*c + 44*a^2*b*c^2)*d^3*e^4 + (a*b^4
+ 14*a^2*b^2*c - 88*a^3*c^2)*d^2*e^5 - 11*(a^2*b^3 - 4*a^3*b*c)*d*e^6 + 2*(a^3*b^2 - 4*a^4*c)*e^7 + (16*c^5*d
^5*e^2 - 40*b*c^4*d^4*e^3 + 2*(31*b^2*c^3 - 44*a*c^4)*d^3*e^4 - (53*b^3*c^2 - 132*a*b*c^3)*d^2*e^5 + (15*b^4*c
- 14*a*b^2*c^2 - 104*a^2*c^3)*d*e^6 - (15*a*b^3*c - 52*a^2*b*c^2)*e^7)*x^3 + (32*c^5*d^6*e - 72*b*c^4*d^5*e^2
+ 80*(b^2*c^3 - a*c^4)*d^4*e^3 - (27*b^3*c^2 - 28*a*b*c^3)*d^3*e^4 - 2*(14*b^4*c - 73*a*b^2*c^2 + 68*a^2*c^3)
*d^2*e^5 + (15*b^5 - 49*a*b^3*c - 20*a^2*b*c^2)*d*e^6 - (15*a*b^4 - 62*a^2*b^2*c + 24*a^3*c^2)*e^7)*x^2 + (16*
c^5*d^7 - 24*b*c^4*d^6*e - 16*(b^2*c^3 - 4*a*c^4)*d^5*e^2 + 80*(b^3*c^2 - 3*a*b*c^3)*d^4*e^3 - (81*b^4*c - 274
*a*b^2*c^2 + 40*a^2*c^3)*d^3*e^4 + (25*b^5 - 63*a*b^3*c - 76*a^2*b*c^2)*d^2*e^5 - 2*(10*a*b^4 - 47*a^2*b^2*c +
44*a^3*c^2)*d*e^6 - 5*(a^2*b^3 - 4*a^3*b*c)*e^7)*x)*sqrt(c*x^2 + b*x + a))/((a*b^2*c^4 - 4*a^2*c^5)*d^10 - 4*
(a*b^3*c^3 - 4*a^2*b*c^4)*d^9*e + 2*(3*a*b^4*c^2 - 10*a^2*b^2*c^3 - 8*a^3*c^4)*d^8*e^2 - 4*(a*b^5*c - a^2*b^3*
c^2 - 12*a^3*b*c^3)*d^7*e^3 + (a*b^6 + 8*a^2*b^4*c - 42*a^3*b^2*c^2 - 24*a^4*c^3)*d^6*e^4 - 4*(a^2*b^5 - a^3*b
^3*c - 12*a^4*b*c^2)*d^5*e^5 + 2*(3*a^3*b^4 - 10*a^4*b^2*c - 8*a^5*c^2)*d^4*e^6 - 4*(a^4*b^3 - 4*a^5*b*c)*d^3*
e^7 + (a^5*b^2 - 4*a^6*c)*d^2*e^8 + ((b^2*c^5 - 4*a*c^6)*d^8*e^2 - 4*(b^3*c^4 - 4*a*b*c^5)*d^7*e^3 + 2*(3*b^4*
c^3 - 10*a*b^2*c^4 - 8*a^2*c^5)*d^6*e^4 - 4*(b^5*c^2 - a*b^3*c^3 - 12*a^2*b*c^4)*d^5*e^5 + (b^6*c + 8*a*b^4*c^
2 - 42*a^2*b^2*c^3 - 24*a^3*c^4)*d^4*e^6 - 4*(a*b^5*c - a^2*b^3*c^2 - 12*a^3*b*c^3)*d^3*e^7 + 2*(3*a^2*b^4*c -
10*a^3*b^2*c^2 - 8*a^4*c^3)*d^2*e^8 - 4*(a^3*b^3*c - 4*a^4*b*c^2)*d*e^9 + (a^4*b^2*c - 4*a^5*c^2)*e^10)*x^4 +
(2*(b^2*c^5 - 4*a*c^6)*d^9*e - 7*(b^3*c^4 - 4*a*b*c^5)*d^8*e^2 + 8*(b^4*c^3 - 3*a*b^2*c^4 - 4*a^2*c^5)*d^7*e^
3 - 2*(b^5*c^2 + 6*a*b^3*c^3 - 40*a^2*b*c^4)*d^6*e^4 - 2*(b^6*c - 10*a*b^4*c^2 + 18*a^2*b^2*c^3 + 24*a^3*c^4)*
d^5*e^5 + (b^7 - 34*a^2*b^3*c^2 + 72*a^3*b*c^3)*d^4*e^6 - 4*(a*b^6 - 4*a^2*b^4*c - 2*a^3*b^2*c^2 + 8*a^4*c^3)*
d^3*e^7 + 2*(3*a^2*b^5 - 14*a^3*b^3*c + 8*a^4*b*c^2)*d^2*e^8 - 2*(2*a^3*b^4 - 9*a^4*b^2*c + 4*a^5*c^2)*d*e^9 +
(a^4*b^3 - 4*a^5*b*c)*e^10)*x^3 + ((b^2*c^5 - 4*a*c^6)*d^10 - 2*(b^3*c^4 - 4*a*b*c^5)*d^9*e - (2*b^4*c^3 - 13
*a*b^2*c^4 + 20*a^2*c^5)*d^8*e^2 + 8*(b^5*c^2 - 5*a*b^3*c^3 + 4*a^2*b*c^4)*d^7*e^3 - (7*b^6*c - 22*a*b^4*c^2 -
34*a^2*b^2*c^3 + 40*a^3*c^4)*d^6*e^4 + 2*(b^7 + 4*a*b^5*c - 38*a^2*b^3*c^2 + 24*a^3*b*c^3)*d^5*e^5 - (7*a*b^6
- 22*a^2*b^4*c - 34*a^3*b^2*c^2 + 40*a^4*c^3)*d^4*e^6 + 8*(a^2*b^5 - 5*a^3*b^3*c + 4*a^4*b*c^2)*d^3*e^7 - (2*
a^3*b^4 - 13*a^4*b^2*c + 20*a^5*c^2)*d^2*e^8 - 2*(a^4*b^3 - 4*a^5*b*c)*d*e^9 + (a^5*b^2 - 4*a^6*c)*e^10)*x^2 +
((b^3*c^4 - 4*a*b*c^5)*d^10 - 2*(2*b^4*c^3 - 9*a*b^2*c^4 + 4*a^2*c^5)*d^9*e + 2*(3*b^5*c^2 - 14*a*b^3*c^3 + 8
*a^2*b*c^4)*d^8*e^2 - 4*(b^6*c - 4*a*b^4*c^2 - 2*a^2*b^2*c^3 + 8*a^3*c^4)*d^7*e^3 + (b^7 - 34*a^2*b^3*c^2 + 72
*a^3*b*c^3)*d^6*e^4 - 2*(a*b^6 - 10*a^2*b^4*c + 18*a^3*b^2*c^2 + 24*a^4*c^3)*d^5*e^5 - 2*(a^2*b^5 + 6*a^3*b^3*
c - 40*a^4*b*c^2)*d^4*e^6 + 8*(a^3*b^4 - 3*a^4*b^2*c - 4*a^5*c^2)*d^3*e^7 - 7*(a^4*b^3 - 4*a^5*b*c)*d^2*e^8 +
2*(a^5*b^2 - 4*a^6*c)*d*e^9)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.45875, size = 3475, normalized size = 9.37 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*((2*c^7*d^9 - 9*b*c^6*d^8*e + 18*b^2*c^5*d^7*e^2 - 21*b^3*c^4*d^6*e^3 + 15*b^4*c^3*d^5*e^4 + 6*a*b^2*c^4*d^
5*e^4 - 12*a^2*c^5*d^5*e^4 - 6*b^5*c^2*d^4*e^5 - 15*a*b^3*c^3*d^4*e^5 + 30*a^2*b*c^4*d^4*e^5 + b^6*c*d^3*e^6 +
12*a*b^4*c^2*d^3*e^6 - 18*a^2*b^2*c^3*d^3*e^6 - 16*a^3*c^4*d^3*e^6 - 3*a*b^5*c*d^2*e^7 - 3*a^2*b^3*c^2*d^2*e^
7 + 24*a^3*b*c^3*d^2*e^7 + 3*a^2*b^4*c*d*e^8 - 6*a^3*b^2*c^2*d*e^8 - 6*a^4*c^3*d*e^8 - a^3*b^3*c*e^9 + 3*a^4*b
*c^2*e^9)*x/(b^2*c^6*d^12 - 4*a*c^7*d^12 - 6*b^3*c^5*d^11*e + 24*a*b*c^6*d^11*e + 15*b^4*c^4*d^10*e^2 - 54*a*b
^2*c^5*d^10*e^2 - 24*a^2*c^6*d^10*e^2 - 20*b^5*c^3*d^9*e^3 + 50*a*b^3*c^4*d^9*e^3 + 120*a^2*b*c^5*d^9*e^3 + 15
*b^6*c^2*d^8*e^4 - 225*a^2*b^2*c^4*d^8*e^4 - 60*a^3*c^5*d^8*e^4 - 6*b^7*c*d^7*e^5 - 36*a*b^5*c^2*d^7*e^5 + 180
*a^2*b^3*c^3*d^7*e^5 + 240*a^3*b*c^4*d^7*e^5 + b^8*d^6*e^6 + 26*a*b^6*c*d^6*e^6 - 30*a^2*b^4*c^2*d^6*e^6 - 340
*a^3*b^2*c^3*d^6*e^6 - 80*a^4*c^4*d^6*e^6 - 6*a*b^7*d^5*e^7 - 36*a^2*b^5*c*d^5*e^7 + 180*a^3*b^3*c^2*d^5*e^7 +
240*a^4*b*c^3*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 225*a^4*b^2*c^2*d^4*e^8 - 60*a^5*c^3*d^4*e^8 - 20*a^3*b^5*d^3*e^
9 + 50*a^4*b^3*c*d^3*e^9 + 120*a^5*b*c^2*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 54*a^5*b^2*c*d^2*e^10 - 24*a^6*c^2*d^
2*e^10 - 6*a^5*b^3*d*e^11 + 24*a^6*b*c*d*e^11 + a^6*b^2*e^12 - 4*a^7*c*e^12) + (b*c^6*d^9 - 6*b^2*c^5*d^8*e +
6*a*c^6*d^8*e + 15*b^3*c^4*d^7*e^2 - 24*a*b*c^5*d^7*e^2 - 20*b^4*c^3*d^6*e^3 + 34*a*b^2*c^4*d^6*e^3 + 16*a^2*c
^5*d^6*e^3 + 15*b^5*c^2*d^5*e^4 - 15*a*b^3*c^3*d^5*e^4 - 54*a^2*b*c^4*d^5*e^4 - 6*b^6*c*d^4*e^5 - 9*a*b^4*c^2*
d^4*e^5 + 66*a^2*b^2*c^3*d^4*e^5 + 12*a^3*c^4*d^4*e^5 + b^7*d^3*e^6 + 11*a*b^5*c*d^3*e^6 - 31*a^2*b^3*c^2*d^3*
e^6 - 32*a^3*b*c^3*d^3*e^6 - 3*a*b^6*d^2*e^7 + 30*a^3*b^2*c^2*d^2*e^7 + 3*a^2*b^5*d*e^8 - 9*a^3*b^3*c*d*e^8 -
3*a^4*b*c^2*d*e^8 - a^3*b^4*e^9 + 4*a^4*b^2*c*e^9 - 2*a^5*c^2*e^9)/(b^2*c^6*d^12 - 4*a*c^7*d^12 - 6*b^3*c^5*d^
11*e + 24*a*b*c^6*d^11*e + 15*b^4*c^4*d^10*e^2 - 54*a*b^2*c^5*d^10*e^2 - 24*a^2*c^6*d^10*e^2 - 20*b^5*c^3*d^9*
e^3 + 50*a*b^3*c^4*d^9*e^3 + 120*a^2*b*c^5*d^9*e^3 + 15*b^6*c^2*d^8*e^4 - 225*a^2*b^2*c^4*d^8*e^4 - 60*a^3*c^5
*d^8*e^4 - 6*b^7*c*d^7*e^5 - 36*a*b^5*c^2*d^7*e^5 + 180*a^2*b^3*c^3*d^7*e^5 + 240*a^3*b*c^4*d^7*e^5 + b^8*d^6*
e^6 + 26*a*b^6*c*d^6*e^6 - 30*a^2*b^4*c^2*d^6*e^6 - 340*a^3*b^2*c^3*d^6*e^6 - 80*a^4*c^4*d^6*e^6 - 6*a*b^7*d^5
*e^7 - 36*a^2*b^5*c*d^5*e^7 + 180*a^3*b^3*c^2*d^5*e^7 + 240*a^4*b*c^3*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 225*a^4*b
^2*c^2*d^4*e^8 - 60*a^5*c^3*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 50*a^4*b^3*c*d^3*e^9 + 120*a^5*b*c^2*d^3*e^9 + 15*a
^4*b^4*d^2*e^10 - 54*a^5*b^2*c*d^2*e^10 - 24*a^6*c^2*d^2*e^10 - 6*a^5*b^3*d*e^11 + 24*a^6*b*c*d*e^11 + a^6*b^2
*e^12 - 4*a^7*c*e^12))/sqrt(c*x^2 + b*x + a) + 3/4*(16*c^2*d^2*e^2 - 16*b*c*d*e^3 + 5*b^2*e^4 - 4*a*c*e^4)*arc
tan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^3*d^6 - 3*b*c^2*d^5
*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3
*a^2*b*d*e^5 + a^3*e^6)*sqrt(-c*d^2 + b*d*e - a*e^2)) - 1/4*(56*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(5/2)*
d^3*e^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^2*d^2*e^3 + 56*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^2*
d^3*e^2 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(3/2)*d^2*e^3 + 14*b^2*c^(3/2)*d^3*e^2 - 24*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*b*c*d*e^4 - 44*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c*d^2*e^3 - 88*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*a*c^2*d^2*e^3 + 13*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*sqrt(c)*d*e^4 - 28*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(3/2)*d*e^4 - 7*b^3*sqrt(c)*d^2*e^3 - 44*a*b*c^(3/2)*d^2*e^3 + 7*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*b^2*e^5 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c*e^5 + 9*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))*b^3*d*e^4 + 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c*d*e^4 + 8*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^2*a*b*sqrt(c)*e^5 + 23*a*b^2*sqrt(c)*d*e^4 + 28*a^2*c^(3/2)*d*e^4 - 9*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*a*b^2*e^5 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c*e^5 - 16*a^2*b*sqrt(c)*e^5)/((c^3*d^6 - 3*b*c^2*d
^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 -
3*a^2*b*d*e^5 + a^3*e^6)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqr
t(c)*d + b*d - a*e)^2)