3.2398 \(\int \frac{1}{(d+e x)^2 (a+b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=473 \[ \frac{e \sqrt{a+b x+c x^2} \left (4 c^2 e^2 \left (-32 a^2 e^2-36 a b d e+3 b^2 d^2\right )+20 b^2 c e^3 (5 a e+b d)-16 c^3 d^2 e (4 b d-9 a e)-15 b^4 e^4+32 c^4 d^4\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 c e (b d-8 a e)-5 b^2 e^2+8 c^2 d^2\right )+6 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac{5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{7/2}} \]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(a + b*x
+ c*x^2)^(3/2)) - (2*(6*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(b*d
- 8*a*e)) - c*(2*c*d - b*e)*(8*c^2*d^2 - 5*b^2*e^2 - 4*c*e*(2*b*d - 7*a*e))*x))/(3*(b^2 - 4*a*c)^2*(c*d^2 - b
*d*e + a*e^2)^2*(d + e*x)*Sqrt[a + b*x + c*x^2]) + (e*(32*c^4*d^4 - 15*b^4*e^4 - 16*c^3*d^2*e*(4*b*d - 9*a*e)
+ 20*b^2*c*e^3*(b*d + 5*a*e) + 4*c^2*e^2*(3*b^2*d^2 - 36*a*b*d*e - 32*a^2*e^2))*Sqrt[a + b*x + c*x^2])/(3*(b^2
- 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (5*e^4*(2*c*d - b*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])])/(2*(c*d^2 - b*d*e + a*e^2)^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.614158, antiderivative size = 473, 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, 822, 806, 724, 206} \[ \frac{e \sqrt{a+b x+c x^2} \left (4 c^2 e^2 \left (-32 a^2 e^2-36 a b d e+3 b^2 d^2\right )+20 b^2 c e^3 (5 a e+b d)-16 c^3 d^2 e (4 b d-9 a e)-15 b^4 e^4+32 c^4 d^4\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 c e (b d-8 a e)-5 b^2 e^2+8 c^2 d^2\right )+6 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac{5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(a + b*x
+ c*x^2)^(3/2)) - (2*(6*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(b*d
- 8*a*e)) - c*(2*c*d - b*e)*(8*c^2*d^2 - 5*b^2*e^2 - 4*c*e*(2*b*d - 7*a*e))*x))/(3*(b^2 - 4*a*c)^2*(c*d^2 - b
*d*e + a*e^2)^2*(d + e*x)*Sqrt[a + b*x + c*x^2]) + (e*(32*c^4*d^4 - 15*b^4*e^4 - 16*c^3*d^2*e*(4*b*d - 9*a*e)
+ 20*b^2*c*e^3*(b*d + 5*a*e) + 4*c^2*e^2*(3*b^2*d^2 - 36*a*b*d*e - 32*a^2*e^2))*Sqrt[a + b*x + c*x^2])/(3*(b^2
- 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (5*e^4*(2*c*d - b*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])])/(2*(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 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 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)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \int \frac{\frac{1}{2} \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )+3 c e (2 c d-b e) x}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \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 )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (6 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt{a+b x+c x^2}}+\frac{4 \int \frac{-\frac{1}{4} e \left (10 b^3 c d e^2-15 b^4 e^3+32 a c^2 e \left (c d^2-4 a e^2\right )-8 b c^2 d \left (2 c d^2+11 a e^2\right )+4 b^2 c e \left (4 c d^2+25 a e^2\right )\right )+\frac{1}{2} c e (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x}{(d+e x)^2 \sqrt{a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \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 )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (6 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt{a+b x+c x^2}}+\frac{e \left (32 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (4 b d-9 a e)+20 b^2 c e^3 (b d+5 a e)+4 c^2 e^2 \left (3 b^2 d^2-36 a b d e-32 a^2 e^2\right )\right ) \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac{\left (5 e^4 (2 c d-b e)\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{2 \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 )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (6 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt{a+b x+c x^2}}+\frac{e \left (32 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (4 b d-9 a e)+20 b^2 c e^3 (b d+5 a e)+4 c^2 e^2 \left (3 b^2 d^2-36 a b d e-32 a^2 e^2\right )\right ) \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{\left (5 e^4 (2 c d-b e)\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 )}{\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 )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (6 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt{a+b x+c x^2}}+\frac{e \left (32 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (4 b d-9 a e)+20 b^2 c e^3 (b d+5 a e)+4 c^2 e^2 \left (3 b^2 d^2-36 a b d e-32 a^2 e^2\right )\right ) \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac{5 e^4 (2 c d-b e) \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 )}{2 \left (c d^2-b d e+a e^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 1.93774, size = 482, normalized size = 1.02 \[ \frac{2 \left (\frac{e \sqrt{a+x (b+c x)} \left (-4 c^2 e^2 \left (32 a^2 e^2+36 a b d e-3 b^2 d^2\right )+20 b^2 c e^3 (5 a e+b d)-16 c^3 d^2 e (4 b d-9 a e)-15 b^4 e^4+32 c^4 d^4\right )}{2 \left (b^2-4 a c\right ) (d+e x) \left (e (a e-b d)+c d^2\right )^2}+\frac{-8 c^2 \left (4 a^2 e^3-a c d e (d-7 e x)+2 c^2 d^3 x\right )+2 b^2 c e \left (16 a e^2+c d (5 d+e x)\right )-4 b c^2 \left (a e^2 (9 d-7 e x)+2 c d^2 (d-3 e x)\right )+b^3 c e^2 (3 d-5 e x)-5 b^4 e^3}{\left (b^2-4 a c\right ) (d+e x) \sqrt{a+x (b+c x)} \left (e (b d-a e)-c d^2\right )}+\frac{15 e^4 \left (b^2-4 a c\right ) (b e-2 c d) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{4 \left (e (a e-b d)+c d^2\right )^{5/2}}+\frac{-2 c (a e+c d x)+b^2 e+b c (e x-d)}{(d+e x) (a+x (b+c x))^{3/2}}\right )}{3 \left (b^2-4 a c\right ) \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(2*((e*(32*c^4*d^4 - 15*b^4*e^4 - 16*c^3*d^2*e*(4*b*d - 9*a*e) + 20*b^2*c*e^3*(b*d + 5*a*e) - 4*c^2*e^2*(-3*b^
2*d^2 + 36*a*b*d*e + 32*a^2*e^2))*Sqrt[a + x*(b + c*x)])/(2*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*
x)) + (b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/((d + e*x)*(a + x*(b + c*x))^(3/2)) + (-5*b^4*e^3 + b^3*c*e
^2*(3*d - 5*e*x) - 8*c^2*(4*a^2*e^3 + 2*c^2*d^3*x - a*c*d*e*(d - 7*e*x)) - 4*b*c^2*(a*e^2*(9*d - 7*e*x) + 2*c*
d^2*(d - 3*e*x)) + 2*b^2*c*e*(16*a*e^2 + c*d*(5*d + e*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(d + e*x)
*Sqrt[a + x*(b + c*x)]) + (15*(b^2 - 4*a*c)*e^4*(-2*c*d + b*e)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*S
qrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(4*(c*d^2 + e*(-(b*d) + a*e))^(5/2))))/(3*(b^2 - 4*a*c)
*(c*d^2 + e*(-(b*d) + a*e)))

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Maple [B]  time = 0.239, size = 2765, normalized size = 5.9 \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)^2/(c*x^2+b*x+a)^(5/2),x)

[Out]

-5/2*e^4/(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-5/6*e^2/(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)^(3/2)*b+5/3*e/(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)^(3/2)*c*d-20/3*e/(a*e^2-b*d*e+c*d^2)^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)^(3/2)*x*c^2*b*d-20*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*c^2*d-160/3*e/(a*e^2-b*d*e+
c*d^2)^2*c^3/(4*a*c-b^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)*x*b*d-1/(a*e^2-b*
d*e+c*d^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)^(3/2)+5/6*e^2/(a*e^2-b*d*e+c*d^
2)^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)^(3/2)*b^3+5*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)*c*d-16/3*c^2/(a*e^2-b*d*e+c*d^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)^(3/2)*x-8/3*c/(a*e^2-b*d*e+c*d^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)^(3/2)*b+5/2*e^4/(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+5/2*e^4/(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-128/3*c^3/(a*e^2-b*d*e+c*d^2)/(4
*a*c-b^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)*x-64/3*c^2/(a*e^2-b*d*e+c*d^2)/(
4*a*c-b^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)*b+20/3*e^2/(a*e^2-b*d*e+c*d^2)^
2*c/(4*a*c-b^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)*b^3-5*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))*c*d+80/3/(a*e^2-b*d*
e+c*d^2)^2*c^3/(4*a*c-b^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)*b*d^2+20/3/(a*e
^2-b*d*e+c*d^2)^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)^(3/2)*x*c^3*d^2+10/3
/(a*e^2-b*d*e+c*d^2)^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)^(3/2)*b*c^2*d^2
+160/3/(a*e^2-b*d*e+c*d^2)^2*c^4/(4*a*c-b^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)*x*d^2-10*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^2*c*d+10*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*c^2*d^2+5/3*e^2/(a*e^2-b*d*e+c*d^2)^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)^(3/2)*x*c*b^2-10/3*e/(a*e^2-b*d*e+c*d^2)^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)^(3/2)*b^2*c*d+40/3*e^2/(a*e^2-b*d*e+c*d^2)^2*c^2/(4*a*c-b^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)*x*b^2-80/3*e/(a*e^2-b*d*e+c*d^2)^2*c^2/(4*a*c-b^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)*b^2*d+5*e^4/(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+20*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*c^3*d^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)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 102.847, size = 18303, normalized size = 38.7 \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)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/12*(15*(2*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d^2*e^4 - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d*e^5
+ (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d*e^5 - (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^6)*x^5 + (2*(b^4*c^
3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^2*e^4 + 3*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^5 - 2*(b^6*c - 8*a*b^4*c^
2 + 16*a^2*b^2*c^3)*e^6)*x^4 + (4*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e^4 + 4*(a*b^4*c^2 - 8*a^2*b^2*c^
3 + 16*a^3*c^4)*d*e^5 - (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*e^6)*x^3 + (2*(b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d^2*
e^4 - (b^7 - 10*a*b^5*c + 32*a^2*b^3*c^2 - 32*a^3*b*c^3)*d*e^5 - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e^6)
*x^2 + (4*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d^2*e^4 - 2*(a*b^6 - 9*a^2*b^4*c + 24*a^3*b^2*c^2 - 16*a^4*
c^3)*d*e^5 - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e^6)*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)*sqr
t(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*(2*(b^3*c^4 - 12*a*b*c^5)*d^7 - 8*(b^4*c^3 - 11*a*b^2*c^4 + 4*a^2*c^5)*d^6*e + 4*(3*b
^5*c^2 - 26*a*b^3*c^3 - 4*a^2*b*c^4)*d^5*e^2 - 8*(b^6*c - 3*a*b^4*c^2 - 32*a^2*b^2*c^3 + 32*a^3*c^4)*d^4*e^3 +
2*(b^7 + 16*a*b^5*c - 155*a^2*b^3*c^2 + 180*a^3*b*c^3)*d^3*e^4 - (16*a*b^6 - 91*a^2*b^4*c + 16*a^3*b^2*c^2 +
176*a^4*c^3)*d^2*e^5 + (11*a^2*b^5 - 76*a^3*b^3*c + 112*a^4*b*c^2)*d*e^6 + 3*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c
^2)*e^7 - (32*c^7*d^6*e - 96*b*c^6*d^5*e^2 + 4*(19*b^2*c^5 + 44*a*c^6)*d^4*e^3 + 8*(b^3*c^4 - 44*a*b*c^5)*d^3*
e^4 - (35*b^4*c^3 - 256*a*b^2*c^4 - 16*a^2*c^5)*d^2*e^5 + (15*b^5*c^2 - 80*a*b^3*c^3 - 16*a^2*b*c^4)*d*e^6 - (
15*a*b^4*c^2 - 100*a^2*b^2*c^3 + 128*a^3*c^4)*e^7)*x^4 - 2*(16*c^7*d^7 - 24*b*c^6*d^6*e - 2*(17*b^2*c^5 - 44*a
*c^6)*d^5*e^2 + (61*b^3*c^4 - 44*a*b*c^5)*d^4*e^3 - 4*(b^4*c^3 + 49*a*b^2*c^4 - 32*a^2*c^5)*d^3*e^4 - 2*(15*b^
5*c^2 - 121*a*b^3*c^3 + 88*a^2*b*c^4)*d^2*e^5 + (15*b^6*c - 90*a*b^4*c^2 + 38*a^2*b^2*c^3 + 56*a^3*c^4)*d*e^6
- 3*(5*a*b^5*c - 35*a^2*b^3*c^2 + 52*a^3*b*c^3)*e^7)*x^3 - 3*(16*b*c^6*d^7 - 4*(11*b^2*c^5 - 4*a*c^6)*d^6*e +
2*(13*b^3*c^4 + 20*a*b*c^5)*d^5*e^2 + 8*(2*b^4*c^3 - 17*a*b^2*c^4 + 16*a^2*c^5)*d^4*e^3 - 2*(7*b^5*c^2 - 34*a*
b^3*c^3 + 64*a^2*b*c^4)*d^3*e^4 - (5*b^6*c - 42*a*b^4*c^2 + 28*a^2*b^2*c^3 - 48*a^3*c^4)*d^2*e^5 + (5*b^7 - 30
*a*b^5*c + 18*a^2*b^3*c^2 + 8*a^3*b*c^3)*d*e^6 - (5*a*b^6 - 30*a^2*b^4*c + 16*a^3*b^2*c^2 + 64*a^4*c^3)*e^7)*x
^2 - 2*(6*(b^2*c^5 + 4*a*c^6)*d^7 - (19*b^3*c^4 + 60*a*b*c^5)*d^6*e + 16*(b^4*c^3 + 4*a*b^2*c^4 + 7*a^2*c^5)*d
^5*e^2 + 2*(3*b^5*c^2 - 46*a*b^3*c^3 - 44*a^2*b*c^4)*d^4*e^3 - 2*(7*b^6*c - 51*a*b^4*c^2 + 61*a^2*b^2*c^3 - 76
*a^3*c^4)*d^3*e^4 + (5*b^7 - 43*a*b^5*c + 125*a^2*b^3*c^2 - 156*a^3*b*c^3)*d^2*e^5 + (5*a*b^6 - 32*a^2*b^4*c +
20*a^3*b^2*c^2 + 64*a^4*c^3)*d*e^6 - 2*(5*a^2*b^5 - 37*a^3*b^3*c + 64*a^4*b*c^2)*e^7)*x)*sqrt(c*x^2 + b*x + a
))/((a^2*b^4*c^4 - 8*a^3*b^2*c^5 + 16*a^4*c^6)*d^9 - 4*(a^2*b^5*c^3 - 8*a^3*b^3*c^4 + 16*a^4*b*c^5)*d^8*e + 2*
(3*a^2*b^6*c^2 - 22*a^3*b^4*c^3 + 32*a^4*b^2*c^4 + 32*a^5*c^5)*d^7*e^2 - 4*(a^2*b^7*c - 5*a^3*b^5*c^2 - 8*a^4*
b^3*c^3 + 48*a^5*b*c^4)*d^6*e^3 + (a^2*b^8 + 4*a^3*b^6*c - 74*a^4*b^4*c^2 + 144*a^5*b^2*c^3 + 96*a^6*c^4)*d^5*
e^4 - 4*(a^3*b^7 - 5*a^4*b^5*c - 8*a^5*b^3*c^2 + 48*a^6*b*c^3)*d^4*e^5 + 2*(3*a^4*b^6 - 22*a^5*b^4*c + 32*a^6*
b^2*c^2 + 32*a^7*c^3)*d^3*e^6 - 4*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*d^2*e^7 + (a^6*b^4 - 8*a^7*b^2*c + 16
*a^8*c^2)*d*e^8 + ((b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^8*e - 4*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^7*e
^2 + 2*(3*b^6*c^4 - 22*a*b^4*c^5 + 32*a^2*b^2*c^6 + 32*a^3*c^7)*d^6*e^3 - 4*(b^7*c^3 - 5*a*b^5*c^4 - 8*a^2*b^3
*c^5 + 48*a^3*b*c^6)*d^5*e^4 + (b^8*c^2 + 4*a*b^6*c^3 - 74*a^2*b^4*c^4 + 144*a^3*b^2*c^5 + 96*a^4*c^6)*d^4*e^5
- 4*(a*b^7*c^2 - 5*a^2*b^5*c^3 - 8*a^3*b^3*c^4 + 48*a^4*b*c^5)*d^3*e^6 + 2*(3*a^2*b^6*c^2 - 22*a^3*b^4*c^3 +
32*a^4*b^2*c^4 + 32*a^5*c^5)*d^2*e^7 - 4*(a^3*b^5*c^2 - 8*a^4*b^3*c^3 + 16*a^5*b*c^4)*d*e^8 + (a^4*b^4*c^2 - 8
*a^5*b^2*c^3 + 16*a^6*c^4)*e^9)*x^5 + ((b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^9 - 2*(b^5*c^5 - 8*a*b^3*c^6 + 1
6*a^2*b*c^7)*d^8*e - 2*(b^6*c^4 - 10*a*b^4*c^5 + 32*a^2*b^2*c^6 - 32*a^3*c^7)*d^7*e^2 + 4*(2*b^7*c^3 - 17*a*b^
5*c^4 + 40*a^2*b^3*c^5 - 16*a^3*b*c^6)*d^6*e^3 - (7*b^8*c^2 - 44*a*b^6*c^3 + 10*a^2*b^4*c^4 + 240*a^3*b^2*c^5
- 96*a^4*c^6)*d^5*e^4 + 2*(b^9*c + 2*a*b^7*c^2 - 64*a^2*b^5*c^3 + 160*a^3*b^3*c^4)*d^4*e^5 - 2*(4*a*b^8*c - 23
*a^2*b^6*c^2 - 10*a^3*b^4*c^3 + 160*a^4*b^2*c^4 - 32*a^5*c^5)*d^3*e^6 + 4*(3*a^2*b^7*c - 23*a^3*b^5*c^2 + 40*a
^4*b^3*c^3 + 16*a^5*b*c^4)*d^2*e^7 - (8*a^3*b^6*c - 65*a^4*b^4*c^2 + 136*a^5*b^2*c^3 - 16*a^6*c^4)*d*e^8 + 2*(
a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*e^9)*x^4 + (2*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^9 - (7*b^6*c^
4 - 58*a*b^4*c^5 + 128*a^2*b^2*c^6 - 32*a^3*c^7)*d^8*e + 8*(b^7*c^3 - 8*a*b^5*c^4 + 16*a^2*b^3*c^5)*d^7*e^2 -
2*(b^8*c^2 - 4*a*b^6*c^3 - 20*a^2*b^4*c^4 + 96*a^3*b^2*c^5 - 64*a^4*c^6)*d^6*e^3 - 2*(b^9*c - 10*a*b^7*c^2 + 3
8*a^2*b^5*c^3 - 80*a^3*b^3*c^4 + 96*a^4*b*c^5)*d^5*e^4 + (b^10 - 2*a*b^8*c - 26*a^2*b^6*c^2 + 60*a^3*b^4*c^3 +
192*a^5*c^5)*d^4*e^5 - 4*(a*b^9 - 6*a^2*b^7*c + 4*a^3*b^5*c^2 + 64*a^5*b*c^4)*d^3*e^6 + 2*(3*a^2*b^8 - 20*a^3
*b^6*c + 20*a^4*b^4*c^2 + 32*a^5*b^2*c^3 + 64*a^6*c^4)*d^2*e^7 - 2*(2*a^3*b^7 - 13*a^4*b^5*c + 8*a^5*b^3*c^2 +
48*a^6*b*c^3)*d*e^8 + (a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*e^9)*x^3 + ((b^6*c^4 - 6*a*b^4*c^5 + 32*a^3*c^7)*d
^9 - 2*(2*b^7*c^3 - 13*a*b^5*c^4 + 8*a^2*b^3*c^5 + 48*a^3*b*c^6)*d^8*e + 2*(3*b^8*c^2 - 20*a*b^6*c^3 + 20*a^2*
b^4*c^4 + 32*a^3*b^2*c^5 + 64*a^4*c^6)*d^7*e^2 - 4*(b^9*c - 6*a*b^7*c^2 + 4*a^2*b^5*c^3 + 64*a^4*b*c^5)*d^6*e^
3 + (b^10 - 2*a*b^8*c - 26*a^2*b^6*c^2 + 60*a^3*b^4*c^3 + 192*a^5*c^5)*d^5*e^4 - 2*(a*b^9 - 10*a^2*b^7*c + 38*
a^3*b^5*c^2 - 80*a^4*b^3*c^3 + 96*a^5*b*c^4)*d^4*e^5 - 2*(a^2*b^8 - 4*a^3*b^6*c - 20*a^4*b^4*c^2 + 96*a^5*b^2*
c^3 - 64*a^6*c^4)*d^3*e^6 + 8*(a^3*b^7 - 8*a^4*b^5*c + 16*a^5*b^3*c^2)*d^2*e^7 - (7*a^4*b^6 - 58*a^5*b^4*c + 1
28*a^6*b^2*c^2 - 32*a^7*c^3)*d*e^8 + 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*e^9)*x^2 + (2*(a*b^5*c^4 - 8*a^2
*b^3*c^5 + 16*a^3*b*c^6)*d^9 - (8*a*b^6*c^3 - 65*a^2*b^4*c^4 + 136*a^3*b^2*c^5 - 16*a^4*c^6)*d^8*e + 4*(3*a*b^
7*c^2 - 23*a^2*b^5*c^3 + 40*a^3*b^3*c^4 + 16*a^4*b*c^5)*d^7*e^2 - 2*(4*a*b^8*c - 23*a^2*b^6*c^2 - 10*a^3*b^4*c
^3 + 160*a^4*b^2*c^4 - 32*a^5*c^5)*d^6*e^3 + 2*(a*b^9 + 2*a^2*b^7*c - 64*a^3*b^5*c^2 + 160*a^4*b^3*c^3)*d^5*e^
4 - (7*a^2*b^8 - 44*a^3*b^6*c + 10*a^4*b^4*c^2 + 240*a^5*b^2*c^3 - 96*a^6*c^4)*d^4*e^5 + 4*(2*a^3*b^7 - 17*a^4
*b^5*c + 40*a^5*b^3*c^2 - 16*a^6*b*c^3)*d^3*e^6 - 2*(a^4*b^6 - 10*a^5*b^4*c + 32*a^6*b^2*c^2 - 32*a^7*c^3)*d^2
*e^7 - 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*d*e^8 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*e^9)*x), 1/6*(15*
(2*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d^2*e^4 - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d*e^5 + (2*(b^4*c
^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d*e^5 - (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^6)*x^5 + (2*(b^4*c^3 - 8*a*b^2
*c^4 + 16*a^2*c^5)*d^2*e^4 + 3*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^5 - 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*
b^2*c^3)*e^6)*x^4 + (4*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e^4 + 4*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*
c^4)*d*e^5 - (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*e^6)*x^3 + (2*(b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d^2*e^4 - (b^7
- 10*a*b^5*c + 32*a^2*b^3*c^2 - 32*a^3*b*c^3)*d*e^5 - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e^6)*x^2 + (4*(
a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d^2*e^4 - 2*(a*b^6 - 9*a^2*b^4*c + 24*a^3*b^2*c^2 - 16*a^4*c^3)*d*e^5
- (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e^6)*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*(2*(b^3*c^4 - 12*a*b*c^5)*d^7 - 8*(b^4*c^3 - 11*a*b^
2*c^4 + 4*a^2*c^5)*d^6*e + 4*(3*b^5*c^2 - 26*a*b^3*c^3 - 4*a^2*b*c^4)*d^5*e^2 - 8*(b^6*c - 3*a*b^4*c^2 - 32*a^
2*b^2*c^3 + 32*a^3*c^4)*d^4*e^3 + 2*(b^7 + 16*a*b^5*c - 155*a^2*b^3*c^2 + 180*a^3*b*c^3)*d^3*e^4 - (16*a*b^6 -
91*a^2*b^4*c + 16*a^3*b^2*c^2 + 176*a^4*c^3)*d^2*e^5 + (11*a^2*b^5 - 76*a^3*b^3*c + 112*a^4*b*c^2)*d*e^6 + 3*
(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*e^7 - (32*c^7*d^6*e - 96*b*c^6*d^5*e^2 + 4*(19*b^2*c^5 + 44*a*c^6)*d^4*e^
3 + 8*(b^3*c^4 - 44*a*b*c^5)*d^3*e^4 - (35*b^4*c^3 - 256*a*b^2*c^4 - 16*a^2*c^5)*d^2*e^5 + (15*b^5*c^2 - 80*a*
b^3*c^3 - 16*a^2*b*c^4)*d*e^6 - (15*a*b^4*c^2 - 100*a^2*b^2*c^3 + 128*a^3*c^4)*e^7)*x^4 - 2*(16*c^7*d^7 - 24*b
*c^6*d^6*e - 2*(17*b^2*c^5 - 44*a*c^6)*d^5*e^2 + (61*b^3*c^4 - 44*a*b*c^5)*d^4*e^3 - 4*(b^4*c^3 + 49*a*b^2*c^4
- 32*a^2*c^5)*d^3*e^4 - 2*(15*b^5*c^2 - 121*a*b^3*c^3 + 88*a^2*b*c^4)*d^2*e^5 + (15*b^6*c - 90*a*b^4*c^2 + 38
*a^2*b^2*c^3 + 56*a^3*c^4)*d*e^6 - 3*(5*a*b^5*c - 35*a^2*b^3*c^2 + 52*a^3*b*c^3)*e^7)*x^3 - 3*(16*b*c^6*d^7 -
4*(11*b^2*c^5 - 4*a*c^6)*d^6*e + 2*(13*b^3*c^4 + 20*a*b*c^5)*d^5*e^2 + 8*(2*b^4*c^3 - 17*a*b^2*c^4 + 16*a^2*c^
5)*d^4*e^3 - 2*(7*b^5*c^2 - 34*a*b^3*c^3 + 64*a^2*b*c^4)*d^3*e^4 - (5*b^6*c - 42*a*b^4*c^2 + 28*a^2*b^2*c^3 -
48*a^3*c^4)*d^2*e^5 + (5*b^7 - 30*a*b^5*c + 18*a^2*b^3*c^2 + 8*a^3*b*c^3)*d*e^6 - (5*a*b^6 - 30*a^2*b^4*c + 16
*a^3*b^2*c^2 + 64*a^4*c^3)*e^7)*x^2 - 2*(6*(b^2*c^5 + 4*a*c^6)*d^7 - (19*b^3*c^4 + 60*a*b*c^5)*d^6*e + 16*(b^4
*c^3 + 4*a*b^2*c^4 + 7*a^2*c^5)*d^5*e^2 + 2*(3*b^5*c^2 - 46*a*b^3*c^3 - 44*a^2*b*c^4)*d^4*e^3 - 2*(7*b^6*c - 5
1*a*b^4*c^2 + 61*a^2*b^2*c^3 - 76*a^3*c^4)*d^3*e^4 + (5*b^7 - 43*a*b^5*c + 125*a^2*b^3*c^2 - 156*a^3*b*c^3)*d^
2*e^5 + (5*a*b^6 - 32*a^2*b^4*c + 20*a^3*b^2*c^2 + 64*a^4*c^3)*d*e^6 - 2*(5*a^2*b^5 - 37*a^3*b^3*c + 64*a^4*b*
c^2)*e^7)*x)*sqrt(c*x^2 + b*x + a))/((a^2*b^4*c^4 - 8*a^3*b^2*c^5 + 16*a^4*c^6)*d^9 - 4*(a^2*b^5*c^3 - 8*a^3*b
^3*c^4 + 16*a^4*b*c^5)*d^8*e + 2*(3*a^2*b^6*c^2 - 22*a^3*b^4*c^3 + 32*a^4*b^2*c^4 + 32*a^5*c^5)*d^7*e^2 - 4*(a
^2*b^7*c - 5*a^3*b^5*c^2 - 8*a^4*b^3*c^3 + 48*a^5*b*c^4)*d^6*e^3 + (a^2*b^8 + 4*a^3*b^6*c - 74*a^4*b^4*c^2 + 1
44*a^5*b^2*c^3 + 96*a^6*c^4)*d^5*e^4 - 4*(a^3*b^7 - 5*a^4*b^5*c - 8*a^5*b^3*c^2 + 48*a^6*b*c^3)*d^4*e^5 + 2*(3
*a^4*b^6 - 22*a^5*b^4*c + 32*a^6*b^2*c^2 + 32*a^7*c^3)*d^3*e^6 - 4*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*d^2*
e^7 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*d*e^8 + ((b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^8*e - 4*(b^5*c^5 -
8*a*b^3*c^6 + 16*a^2*b*c^7)*d^7*e^2 + 2*(3*b^6*c^4 - 22*a*b^4*c^5 + 32*a^2*b^2*c^6 + 32*a^3*c^7)*d^6*e^3 - 4*(
b^7*c^3 - 5*a*b^5*c^4 - 8*a^2*b^3*c^5 + 48*a^3*b*c^6)*d^5*e^4 + (b^8*c^2 + 4*a*b^6*c^3 - 74*a^2*b^4*c^4 + 144*
a^3*b^2*c^5 + 96*a^4*c^6)*d^4*e^5 - 4*(a*b^7*c^2 - 5*a^2*b^5*c^3 - 8*a^3*b^3*c^4 + 48*a^4*b*c^5)*d^3*e^6 + 2*(
3*a^2*b^6*c^2 - 22*a^3*b^4*c^3 + 32*a^4*b^2*c^4 + 32*a^5*c^5)*d^2*e^7 - 4*(a^3*b^5*c^2 - 8*a^4*b^3*c^3 + 16*a^
5*b*c^4)*d*e^8 + (a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*e^9)*x^5 + ((b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d
^9 - 2*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^8*e - 2*(b^6*c^4 - 10*a*b^4*c^5 + 32*a^2*b^2*c^6 - 32*a^3*c^7)
*d^7*e^2 + 4*(2*b^7*c^3 - 17*a*b^5*c^4 + 40*a^2*b^3*c^5 - 16*a^3*b*c^6)*d^6*e^3 - (7*b^8*c^2 - 44*a*b^6*c^3 +
10*a^2*b^4*c^4 + 240*a^3*b^2*c^5 - 96*a^4*c^6)*d^5*e^4 + 2*(b^9*c + 2*a*b^7*c^2 - 64*a^2*b^5*c^3 + 160*a^3*b^3
*c^4)*d^4*e^5 - 2*(4*a*b^8*c - 23*a^2*b^6*c^2 - 10*a^3*b^4*c^3 + 160*a^4*b^2*c^4 - 32*a^5*c^5)*d^3*e^6 + 4*(3*
a^2*b^7*c - 23*a^3*b^5*c^2 + 40*a^4*b^3*c^3 + 16*a^5*b*c^4)*d^2*e^7 - (8*a^3*b^6*c - 65*a^4*b^4*c^2 + 136*a^5*
b^2*c^3 - 16*a^6*c^4)*d*e^8 + 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*e^9)*x^4 + (2*(b^5*c^5 - 8*a*b^3*c^
6 + 16*a^2*b*c^7)*d^9 - (7*b^6*c^4 - 58*a*b^4*c^5 + 128*a^2*b^2*c^6 - 32*a^3*c^7)*d^8*e + 8*(b^7*c^3 - 8*a*b^5
*c^4 + 16*a^2*b^3*c^5)*d^7*e^2 - 2*(b^8*c^2 - 4*a*b^6*c^3 - 20*a^2*b^4*c^4 + 96*a^3*b^2*c^5 - 64*a^4*c^6)*d^6*
e^3 - 2*(b^9*c - 10*a*b^7*c^2 + 38*a^2*b^5*c^3 - 80*a^3*b^3*c^4 + 96*a^4*b*c^5)*d^5*e^4 + (b^10 - 2*a*b^8*c -
26*a^2*b^6*c^2 + 60*a^3*b^4*c^3 + 192*a^5*c^5)*d^4*e^5 - 4*(a*b^9 - 6*a^2*b^7*c + 4*a^3*b^5*c^2 + 64*a^5*b*c^4
)*d^3*e^6 + 2*(3*a^2*b^8 - 20*a^3*b^6*c + 20*a^4*b^4*c^2 + 32*a^5*b^2*c^3 + 64*a^6*c^4)*d^2*e^7 - 2*(2*a^3*b^7
- 13*a^4*b^5*c + 8*a^5*b^3*c^2 + 48*a^6*b*c^3)*d*e^8 + (a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*e^9)*x^3 + ((b^6*
c^4 - 6*a*b^4*c^5 + 32*a^3*c^7)*d^9 - 2*(2*b^7*c^3 - 13*a*b^5*c^4 + 8*a^2*b^3*c^5 + 48*a^3*b*c^6)*d^8*e + 2*(3
*b^8*c^2 - 20*a*b^6*c^3 + 20*a^2*b^4*c^4 + 32*a^3*b^2*c^5 + 64*a^4*c^6)*d^7*e^2 - 4*(b^9*c - 6*a*b^7*c^2 + 4*a
^2*b^5*c^3 + 64*a^4*b*c^5)*d^6*e^3 + (b^10 - 2*a*b^8*c - 26*a^2*b^6*c^2 + 60*a^3*b^4*c^3 + 192*a^5*c^5)*d^5*e^
4 - 2*(a*b^9 - 10*a^2*b^7*c + 38*a^3*b^5*c^2 - 80*a^4*b^3*c^3 + 96*a^5*b*c^4)*d^4*e^5 - 2*(a^2*b^8 - 4*a^3*b^6
*c - 20*a^4*b^4*c^2 + 96*a^5*b^2*c^3 - 64*a^6*c^4)*d^3*e^6 + 8*(a^3*b^7 - 8*a^4*b^5*c + 16*a^5*b^3*c^2)*d^2*e^
7 - (7*a^4*b^6 - 58*a^5*b^4*c + 128*a^6*b^2*c^2 - 32*a^7*c^3)*d*e^8 + 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)
*e^9)*x^2 + (2*(a*b^5*c^4 - 8*a^2*b^3*c^5 + 16*a^3*b*c^6)*d^9 - (8*a*b^6*c^3 - 65*a^2*b^4*c^4 + 136*a^3*b^2*c^
5 - 16*a^4*c^6)*d^8*e + 4*(3*a*b^7*c^2 - 23*a^2*b^5*c^3 + 40*a^3*b^3*c^4 + 16*a^4*b*c^5)*d^7*e^2 - 2*(4*a*b^8*
c - 23*a^2*b^6*c^2 - 10*a^3*b^4*c^3 + 160*a^4*b^2*c^4 - 32*a^5*c^5)*d^6*e^3 + 2*(a*b^9 + 2*a^2*b^7*c - 64*a^3*
b^5*c^2 + 160*a^4*b^3*c^3)*d^5*e^4 - (7*a^2*b^8 - 44*a^3*b^6*c + 10*a^4*b^4*c^2 + 240*a^5*b^2*c^3 - 96*a^6*c^4
)*d^4*e^5 + 4*(2*a^3*b^7 - 17*a^4*b^5*c + 40*a^5*b^3*c^2 - 16*a^6*b*c^3)*d^3*e^6 - 2*(a^4*b^6 - 10*a^5*b^4*c +
32*a^6*b^2*c^2 - 32*a^7*c^3)*d^2*e^7 - 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*d*e^8 + (a^6*b^4 - 8*a^7*b^2*
c + 16*a^8*c^2)*e^9)*x)]

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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(1/(e*x+d)**2/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Giac [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/(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

Timed out