3.1594 \(\int \frac{b+2 c x}{(d+e x)^2 (a+b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=423 \[ \frac{e^2 \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{2 e \left (-c x \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )-4 b c \left (c d^2-4 a e^2\right )-20 a c^2 d e+9 b^2 c d e-5 b^3 e^2\right )}{3 \left (b^2-4 a c\right ) (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{e^3 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.571644, antiderivative size = 423, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {822, 806, 724, 206} \[ \frac{e^2 \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{2 e \left (-c x \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )-4 b c \left (c d^2-4 a e^2\right )-20 a c^2 d e+9 b^2 c d e-5 b^3 e^2\right )}{3 \left (b^2-4 a c\right ) (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{e^3 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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{b+2 c x}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) 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 (b^2-4 a c\right ) e (4 c d-5 b e)-3 c \left (b^2-4 a c\right ) e^2 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 (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) 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 e \left (9 b^2 c d e-20 a c^2 d e-5 b^3 e^2-4 b c \left (c d^2-4 a e^2\right )-c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \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} \left (b^2-4 a c\right ) e^2 \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 \left (b^2-4 a c\right ) e^2 \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}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) 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 e \left (9 b^2 c d e-20 a c^2 d e-5 b^3 e^2-4 b c \left (c d^2-4 a e^2\right )-c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt{a+b x+c x^2}}+\frac{e^2 (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}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{\left (e^3 \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}{2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) 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 e \left (9 b^2 c d e-20 a c^2 d e-5 b^3 e^2-4 b c \left (c d^2-4 a e^2\right )-c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt{a+b x+c x^2}}+\frac{e^2 (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}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac{\left (e^3 \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 )}{\left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) 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 e \left (9 b^2 c d e-20 a c^2 d e-5 b^3 e^2-4 b c \left (c d^2-4 a e^2\right )-c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt{a+b x+c x^2}}+\frac{e^2 (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}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{e^3 \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 )}{2 \left (c d^2-b d e+a e^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 1.58167, size = 384, normalized size = 0.91 \[ \frac{2 \left (\frac{e \left (4 b c \left (c d (d-2 e x)-4 a e^2\right )+4 c^2 \left (a e (5 d-3 e x)+2 c d^2 x\right )+b^2 c e (5 e x-9 d)+5 b^3 e^2\right )}{(d+e x) \sqrt{a+x (b+c x)} \left (e (a e-b d)+c d^2\right )}+\frac{1}{4} e^2 \left (\frac{2 \sqrt{a+x (b+c x)} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{(d+e x) \left (e (a e-b d)+c d^2\right )^2}+\frac{3 e \left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{\left (e (a e-b d)+c d^2\right )^{5/2}}\right )+\frac{\left (b^2-4 a c\right ) (b e-c d+c e x)}{(d+e x) (a+x (b+c x))^{3/2}}\right )}{3 \left (b^2-4 a c\right ) \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(((b^2 - 4*a*c)*(-(c*d) + b*e + c*e*x))/((d + e*x)*(a + x*(b + c*x))^(3/2)) + (e*(5*b^3*e^2 + b^2*c*e*(-9*d
+ 5*e*x) + 4*c^2*(2*c*d^2*x + a*e*(5*d - 3*e*x)) + 4*b*c*(-4*a*e^2 + c*d*(d - 2*e*x))))/((c*d^2 + e*(-(b*d) +
a*e))*(d + e*x)*Sqrt[a + x*(b + c*x)]) + (e^2*((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)))/4))/(3*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))

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Maple [B]  time = 0.021, size = 4855, normalized size = 11.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10/3*e/(a*e^2-b*d*e+c*d^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b*c*d+10*e^3/(a
*e^2-b*d*e+c*d^2)^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c*d+10*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*(x+d/e)+2*((a*e^2-b*d*e
+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*c^2*d^2+80/(a*e
^2-b*d*e+c*d^2)^2*c^3/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*d^2+
10/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^2*c^2
*d^2+20/3*e^2/(a*e^2-b*d*e+c*d^2)^2*c/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2
)^(1/2)*b^4+2/e/(a*e^2-b*d*e+c*d^2)/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*
c*d-2*c*e^2/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2
)*b^2-20/3*c^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/
2)*x*b-160/3*c^3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)
^(1/2)*x*b-5/6*e^2/(a*e^2-b*d*e+c*d^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^2
-5/2*e^4/(a*e^2-b*d*e+c*d^2)^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2-10/3/(a*e
^2-b*d*e+c*d^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c^2*d^2-1/(a*e^2-b*d*e+c*d
^2)/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b+2*c*e^2/(a*e^2-b*d*e+c*d^2)^2/
((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+2/3*c/(a*e^2-b*d*e+c*d^2)/((x+d/e)^2*c+(b*e-
2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-2*c*e^2/(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*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)
/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))+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*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*
c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b^2+5/6*e^2/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)
^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^4-10/3*c/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^
2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^2-80/3*c^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)^2/((x+d/
e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2-10*e^2/(a*e^2-b*d*e+c*d^2)^3/((x+d/e)^2*c+(b*e
-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c^2*d^2+5/2*e^4/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*
c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^4-20/3/e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2
*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b*c^3*d^3+40/3*e^2/(a*e^2-b*d*e+c*d^2)^2*c^2/(4*a*c-b^
2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^3-320/3/e/(a*e^2-b*d*e+c*d^2)^2*c^5
/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d^3-10*e/(a*e^2-b*d*e+c*d^2
)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*x*c^2*b^2*d-30*e^3/(a*e^2-b*
d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^2*c^2*d+60*e^2/
(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*c^3*d^
2-80*e/(a*e^2-b*d*e+c*d^2)^2*c^3/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/
2)*x*b^2*d+40/3*c^3/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e
^2)^(3/2)*x*d+20/3*c^2/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2
)/e^2)^(3/2)*b*d+320/3*c^4/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e
+c*d^2)/e^2)^(1/2)*x*d+160/3*c^3/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2
-b*d*e+c*d^2)/e^2)^(1/2)*b*d-4*c^2*e^2/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a
*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b+8*c^3*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+
(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d+4*c^2*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e
)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*d+30*e^2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/
e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*c^2*d^2-160/3/e/(a*e^2-b*d*e+c*d^2)^2*c^4/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*
e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*d^3+5*e^4/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(
b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^3*c-20*e/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*
c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c^3*d^3-40*e/(a*e^2-b*d*e+c*d^2)^2*c^2/(4*a*c-b^2)^2/
((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^3*d-15*e^3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^
2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^3*c*d-5*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-
b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^3*c*d-40*e/(a*e^2-b*d*e+c*d^2)^3/(4*a
*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^4*d^3+20/(a*e^2-b*d*e+c*d^2)^2/(
4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*x*c^3*b*d^2+160/(a*e^2-b*d*e+c*d^
2)^2*c^4/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*d^2-10*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*(x+d/e)+2*((a*e^2-
b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*c*d*b+5/3*
e^2/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*x*c*b^
3-40/3/e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*x
*c^4*d^3

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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((2*c*x+b)/(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 = 84.3529, size = 15165, normalized size = 35.85 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/12*(3*(16*(a^2*b^2*c^2 - 4*a^3*c^3)*d^3*e^3 - 16*(a^2*b^3*c - 4*a^3*b*c^2)*d^2*e^4 + (5*a^2*b^4 - 24*a^3*b
^2*c + 16*a^4*c^2)*d*e^5 + (16*(b^2*c^4 - 4*a*c^5)*d^2*e^4 - 16*(b^3*c^3 - 4*a*b*c^4)*d*e^5 + (5*b^4*c^2 - 24*
a*b^2*c^3 + 16*a^2*c^4)*e^6)*x^5 + (16*(b^2*c^4 - 4*a*c^5)*d^3*e^3 + 16*(b^3*c^3 - 4*a*b*c^4)*d^2*e^4 - (27*b^
4*c^2 - 104*a*b^2*c^3 - 16*a^2*c^4)*d*e^5 + 2*(5*b^5*c - 24*a*b^3*c^2 + 16*a^2*b*c^3)*e^6)*x^4 + (32*(b^3*c^3
- 4*a*b*c^4)*d^3*e^3 - 16*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 2*(3*b^5*c + 8*a*b^3*c^2 - 80*a^2*b*c^
3)*d*e^5 + (5*b^6 - 14*a*b^4*c - 32*a^2*b^2*c^2 + 32*a^3*c^3)*e^6)*x^3 + (16*(b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^
4)*d^3*e^3 - 16*(b^5*c - 4*a*b^3*c^2)*d^2*e^4 + (5*b^6 - 46*a*b^4*c + 96*a^2*b^2*c^2 + 32*a^3*c^3)*d*e^5 + 2*(
5*a*b^5 - 24*a^2*b^3*c + 16*a^3*b*c^2)*e^6)*x^2 + (32*(a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^3 - 16*(2*a*b^4*c - 9*a^
2*b^2*c^2 + 4*a^3*c^3)*d^2*e^4 + 2*(5*a*b^5 - 32*a^2*b^3*c + 48*a^3*b*c^2)*d*e^5 + (5*a^2*b^4 - 24*a^3*b^2*c +
16*a^4*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)*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*(2*(b^2*c^4 -
4*a*c^5)*d^7 - 8*(b^3*c^3 - 3*a*b*c^4)*d^6*e + 4*(3*b^4*c^2 - 2*a*b^2*c^3 - 20*a^2*c^4)*d^5*e^2 - 8*(b^5*c + 5
*a*b^3*c^2 - 26*a^2*b*c^3)*d^4*e^3 + 2*(b^6 + 24*a*b^4*c - 90*a^2*b^2*c^2 - 8*a^3*c^3)*d^3*e^4 - (16*a*b^5 - 4
1*a^2*b^3*c - 52*a^3*b*c^2)*d^2*e^5 + (11*a^2*b^4 - 54*a^3*b^2*c + 56*a^4*c^2)*d*e^6 + 3*(a^3*b^3 - 4*a^4*b*c)
*e^7 - (16*c^6*d^5*e^2 - 40*b*c^5*d^4*e^3 + 2*(31*b^2*c^4 - 44*a*c^5)*d^3*e^4 - (53*b^3*c^3 - 132*a*b*c^4)*d^2
*e^5 + (15*b^4*c^2 - 14*a*b^2*c^3 - 104*a^2*c^4)*d*e^6 - (15*a*b^3*c^2 - 52*a^2*b*c^3)*e^7)*x^4 - 2*(8*c^6*d^6
*e - 8*b*c^5*d^5*e^2 - (11*b^2*c^4 - 4*a*c^5)*d^4*e^3 + 4*(11*b^3*c^3 - 24*a*b*c^4)*d^3*e^4 - 2*(24*b^4*c^2 -
73*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^5 + (15*b^5*c - 24*a*b^3*c^2 - 88*a^2*b*c^3)*d*e^6 - 3*(5*a*b^4*c - 19*a^2*b^2
*c^2 + 4*a^3*c^3)*e^7)*x^3 - 3*(8*b*c^5*d^6*e - 2*(11*b^2*c^4 - 12*a*c^5)*d^5*e^2 + 8*(3*b^3*c^3 - 7*a*b*c^4)*
d^4*e^3 - 4*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^3*e^4 - (11*b^5*c - 26*a*b^3*c^2 - 32*a^2*b*c^3)*d^2*e^5 + (
5*b^6 - 8*a*b^4*c - 18*a^2*b^2*c^2 - 56*a^3*c^3)*d*e^6 - (5*a*b^5 - 14*a^2*b^3*c - 16*a^3*b*c^2)*e^7)*x^2 - 2*
((5*b^2*c^4 + 4*a*c^5)*d^6*e - 4*(5*b^3*c^3 - 6*a*b*c^4)*d^5*e^2 + 2*(15*b^4*c^2 - 34*a*b^2*c^3 - 4*a^2*c^4)*d
^4*e^3 - 4*(5*b^5*c - 17*a*b^3*c^2 + 8*a^2*b*c^3)*d^3*e^4 + (5*b^6 - 33*a*b^4*c + 69*a^2*b^2*c^2 - 28*a^3*c^3)
*d^2*e^5 + (5*a*b^5 - 4*a^2*b^3*c - 56*a^3*b*c^2)*d*e^6 - 2*(5*a^2*b^4 - 21*a^3*b^2*c + 8*a^4*c^2)*e^7)*x)*sqr
t(c*x^2 + b*x + a))/((a^2*b^2*c^4 - 4*a^3*c^5)*d^9 - 4*(a^2*b^3*c^3 - 4*a^3*b*c^4)*d^8*e + 2*(3*a^2*b^4*c^2 -
10*a^3*b^2*c^3 - 8*a^4*c^4)*d^7*e^2 - 4*(a^2*b^5*c - a^3*b^3*c^2 - 12*a^4*b*c^3)*d^6*e^3 + (a^2*b^6 + 8*a^3*b^
4*c - 42*a^4*b^2*c^2 - 24*a^5*c^3)*d^5*e^4 - 4*(a^3*b^5 - a^4*b^3*c - 12*a^5*b*c^2)*d^4*e^5 + 2*(3*a^4*b^4 - 1
0*a^5*b^2*c - 8*a^6*c^2)*d^3*e^6 - 4*(a^5*b^3 - 4*a^6*b*c)*d^2*e^7 + (a^6*b^2 - 4*a^7*c)*d*e^8 + ((b^2*c^6 - 4
*a*c^7)*d^8*e - 4*(b^3*c^5 - 4*a*b*c^6)*d^7*e^2 + 2*(3*b^4*c^4 - 10*a*b^2*c^5 - 8*a^2*c^6)*d^6*e^3 - 4*(b^5*c^
3 - a*b^3*c^4 - 12*a^2*b*c^5)*d^5*e^4 + (b^6*c^2 + 8*a*b^4*c^3 - 42*a^2*b^2*c^4 - 24*a^3*c^5)*d^4*e^5 - 4*(a*b
^5*c^2 - a^2*b^3*c^3 - 12*a^3*b*c^4)*d^3*e^6 + 2*(3*a^2*b^4*c^2 - 10*a^3*b^2*c^3 - 8*a^4*c^4)*d^2*e^7 - 4*(a^3
*b^3*c^2 - 4*a^4*b*c^3)*d*e^8 + (a^4*b^2*c^2 - 4*a^5*c^3)*e^9)*x^5 + ((b^2*c^6 - 4*a*c^7)*d^9 - 2*(b^3*c^5 - 4
*a*b*c^6)*d^8*e - 2*(b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6)*d^7*e^2 + 4*(2*b^5*c^3 - 9*a*b^3*c^4 + 4*a^2*b*c^5)*d^
6*e^3 - (7*b^6*c^2 - 16*a*b^4*c^3 - 54*a^2*b^2*c^4 + 24*a^3*c^5)*d^5*e^4 + 2*(b^7*c + 6*a*b^5*c^2 - 40*a^2*b^3
*c^3)*d^4*e^5 - 2*(4*a*b^6*c - 7*a^2*b^4*c^2 - 38*a^3*b^2*c^3 + 8*a^4*c^4)*d^3*e^6 + 4*(3*a^2*b^5*c - 11*a^3*b
^3*c^2 - 4*a^4*b*c^3)*d^2*e^7 - (8*a^3*b^4*c - 33*a^4*b^2*c^2 + 4*a^5*c^3)*d*e^8 + 2*(a^4*b^3*c - 4*a^5*b*c^2)
*e^9)*x^4 + (2*(b^3*c^5 - 4*a*b*c^6)*d^9 - (7*b^4*c^4 - 30*a*b^2*c^5 + 8*a^2*c^6)*d^8*e + 8*(b^5*c^3 - 4*a*b^3
*c^4)*d^7*e^2 - 2*(b^6*c^2 - 20*a^2*b^2*c^4 + 16*a^3*c^5)*d^6*e^3 - 2*(b^7*c - 6*a*b^5*c^2 + 14*a^2*b^3*c^3 -
24*a^3*b*c^4)*d^5*e^4 + (b^8 + 2*a*b^6*c - 18*a^2*b^4*c^2 - 12*a^3*b^2*c^3 - 48*a^4*c^4)*d^4*e^5 - 4*(a*b^7 -
2*a^2*b^5*c - 4*a^3*b^3*c^2 - 16*a^4*b*c^3)*d^3*e^6 + 2*(3*a^2*b^6 - 8*a^3*b^4*c - 12*a^4*b^2*c^2 - 16*a^5*c^3
)*d^2*e^7 - 2*(2*a^3*b^5 - 5*a^4*b^3*c - 12*a^5*b*c^2)*d*e^8 + (a^4*b^4 - 2*a^5*b^2*c - 8*a^6*c^2)*e^9)*x^3 +
((b^4*c^4 - 2*a*b^2*c^5 - 8*a^2*c^6)*d^9 - 2*(2*b^5*c^3 - 5*a*b^3*c^4 - 12*a^2*b*c^5)*d^8*e + 2*(3*b^6*c^2 - 8
*a*b^4*c^3 - 12*a^2*b^2*c^4 - 16*a^3*c^5)*d^7*e^2 - 4*(b^7*c - 2*a*b^5*c^2 - 4*a^2*b^3*c^3 - 16*a^3*b*c^4)*d^6
*e^3 + (b^8 + 2*a*b^6*c - 18*a^2*b^4*c^2 - 12*a^3*b^2*c^3 - 48*a^4*c^4)*d^5*e^4 - 2*(a*b^7 - 6*a^2*b^5*c + 14*
a^3*b^3*c^2 - 24*a^4*b*c^3)*d^4*e^5 - 2*(a^2*b^6 - 20*a^4*b^2*c^2 + 16*a^5*c^3)*d^3*e^6 + 8*(a^3*b^5 - 4*a^4*b
^3*c)*d^2*e^7 - (7*a^4*b^4 - 30*a^5*b^2*c + 8*a^6*c^2)*d*e^8 + 2*(a^5*b^3 - 4*a^6*b*c)*e^9)*x^2 + (2*(a*b^3*c^
4 - 4*a^2*b*c^5)*d^9 - (8*a*b^4*c^3 - 33*a^2*b^2*c^4 + 4*a^3*c^5)*d^8*e + 4*(3*a*b^5*c^2 - 11*a^2*b^3*c^3 - 4*
a^3*b*c^4)*d^7*e^2 - 2*(4*a*b^6*c - 7*a^2*b^4*c^2 - 38*a^3*b^2*c^3 + 8*a^4*c^4)*d^6*e^3 + 2*(a*b^7 + 6*a^2*b^5
*c - 40*a^3*b^3*c^2)*d^5*e^4 - (7*a^2*b^6 - 16*a^3*b^4*c - 54*a^4*b^2*c^2 + 24*a^5*c^3)*d^4*e^5 + 4*(2*a^3*b^5
- 9*a^4*b^3*c + 4*a^5*b*c^2)*d^3*e^6 - 2*(a^4*b^4 - 6*a^5*b^2*c + 8*a^6*c^2)*d^2*e^7 - 2*(a^5*b^3 - 4*a^6*b*c
)*d*e^8 + (a^6*b^2 - 4*a^7*c)*e^9)*x), -1/6*(3*(16*(a^2*b^2*c^2 - 4*a^3*c^3)*d^3*e^3 - 16*(a^2*b^3*c - 4*a^3*b
*c^2)*d^2*e^4 + (5*a^2*b^4 - 24*a^3*b^2*c + 16*a^4*c^2)*d*e^5 + (16*(b^2*c^4 - 4*a*c^5)*d^2*e^4 - 16*(b^3*c^3
- 4*a*b*c^4)*d*e^5 + (5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*e^6)*x^5 + (16*(b^2*c^4 - 4*a*c^5)*d^3*e^3 + 16*(
b^3*c^3 - 4*a*b*c^4)*d^2*e^4 - (27*b^4*c^2 - 104*a*b^2*c^3 - 16*a^2*c^4)*d*e^5 + 2*(5*b^5*c - 24*a*b^3*c^2 + 1
6*a^2*b*c^3)*e^6)*x^4 + (32*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 - 16*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 2
*(3*b^5*c + 8*a*b^3*c^2 - 80*a^2*b*c^3)*d*e^5 + (5*b^6 - 14*a*b^4*c - 32*a^2*b^2*c^2 + 32*a^3*c^3)*e^6)*x^3 +
(16*(b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d^3*e^3 - 16*(b^5*c - 4*a*b^3*c^2)*d^2*e^4 + (5*b^6 - 46*a*b^4*c + 96*
a^2*b^2*c^2 + 32*a^3*c^3)*d*e^5 + 2*(5*a*b^5 - 24*a^2*b^3*c + 16*a^3*b*c^2)*e^6)*x^2 + (32*(a*b^3*c^2 - 4*a^2*
b*c^3)*d^3*e^3 - 16*(2*a*b^4*c - 9*a^2*b^2*c^2 + 4*a^3*c^3)*d^2*e^4 + 2*(5*a*b^5 - 32*a^2*b^3*c + 48*a^3*b*c^2
)*d*e^5 + (5*a^2*b^4 - 24*a^3*b^2*c + 16*a^4*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^2*c^4 - 4*a*c^5)*d^7 - 8*(b^3*c^3 - 3
*a*b*c^4)*d^6*e + 4*(3*b^4*c^2 - 2*a*b^2*c^3 - 20*a^2*c^4)*d^5*e^2 - 8*(b^5*c + 5*a*b^3*c^2 - 26*a^2*b*c^3)*d^
4*e^3 + 2*(b^6 + 24*a*b^4*c - 90*a^2*b^2*c^2 - 8*a^3*c^3)*d^3*e^4 - (16*a*b^5 - 41*a^2*b^3*c - 52*a^3*b*c^2)*d
^2*e^5 + (11*a^2*b^4 - 54*a^3*b^2*c + 56*a^4*c^2)*d*e^6 + 3*(a^3*b^3 - 4*a^4*b*c)*e^7 - (16*c^6*d^5*e^2 - 40*b
*c^5*d^4*e^3 + 2*(31*b^2*c^4 - 44*a*c^5)*d^3*e^4 - (53*b^3*c^3 - 132*a*b*c^4)*d^2*e^5 + (15*b^4*c^2 - 14*a*b^2
*c^3 - 104*a^2*c^4)*d*e^6 - (15*a*b^3*c^2 - 52*a^2*b*c^3)*e^7)*x^4 - 2*(8*c^6*d^6*e - 8*b*c^5*d^5*e^2 - (11*b^
2*c^4 - 4*a*c^5)*d^4*e^3 + 4*(11*b^3*c^3 - 24*a*b*c^4)*d^3*e^4 - 2*(24*b^4*c^2 - 73*a*b^2*c^3 + 8*a^2*c^4)*d^2
*e^5 + (15*b^5*c - 24*a*b^3*c^2 - 88*a^2*b*c^3)*d*e^6 - 3*(5*a*b^4*c - 19*a^2*b^2*c^2 + 4*a^3*c^3)*e^7)*x^3 -
3*(8*b*c^5*d^6*e - 2*(11*b^2*c^4 - 12*a*c^5)*d^5*e^2 + 8*(3*b^3*c^3 - 7*a*b*c^4)*d^4*e^3 - 4*(b^4*c^2 - 6*a*b^
2*c^3 + 8*a^2*c^4)*d^3*e^4 - (11*b^5*c - 26*a*b^3*c^2 - 32*a^2*b*c^3)*d^2*e^5 + (5*b^6 - 8*a*b^4*c - 18*a^2*b^
2*c^2 - 56*a^3*c^3)*d*e^6 - (5*a*b^5 - 14*a^2*b^3*c - 16*a^3*b*c^2)*e^7)*x^2 - 2*((5*b^2*c^4 + 4*a*c^5)*d^6*e
- 4*(5*b^3*c^3 - 6*a*b*c^4)*d^5*e^2 + 2*(15*b^4*c^2 - 34*a*b^2*c^3 - 4*a^2*c^4)*d^4*e^3 - 4*(5*b^5*c - 17*a*b^
3*c^2 + 8*a^2*b*c^3)*d^3*e^4 + (5*b^6 - 33*a*b^4*c + 69*a^2*b^2*c^2 - 28*a^3*c^3)*d^2*e^5 + (5*a*b^5 - 4*a^2*b
^3*c - 56*a^3*b*c^2)*d*e^6 - 2*(5*a^2*b^4 - 21*a^3*b^2*c + 8*a^4*c^2)*e^7)*x)*sqrt(c*x^2 + b*x + a))/((a^2*b^2
*c^4 - 4*a^3*c^5)*d^9 - 4*(a^2*b^3*c^3 - 4*a^3*b*c^4)*d^8*e + 2*(3*a^2*b^4*c^2 - 10*a^3*b^2*c^3 - 8*a^4*c^4)*d
^7*e^2 - 4*(a^2*b^5*c - a^3*b^3*c^2 - 12*a^4*b*c^3)*d^6*e^3 + (a^2*b^6 + 8*a^3*b^4*c - 42*a^4*b^2*c^2 - 24*a^5
*c^3)*d^5*e^4 - 4*(a^3*b^5 - a^4*b^3*c - 12*a^5*b*c^2)*d^4*e^5 + 2*(3*a^4*b^4 - 10*a^5*b^2*c - 8*a^6*c^2)*d^3*
e^6 - 4*(a^5*b^3 - 4*a^6*b*c)*d^2*e^7 + (a^6*b^2 - 4*a^7*c)*d*e^8 + ((b^2*c^6 - 4*a*c^7)*d^8*e - 4*(b^3*c^5 -
4*a*b*c^6)*d^7*e^2 + 2*(3*b^4*c^4 - 10*a*b^2*c^5 - 8*a^2*c^6)*d^6*e^3 - 4*(b^5*c^3 - a*b^3*c^4 - 12*a^2*b*c^5)
*d^5*e^4 + (b^6*c^2 + 8*a*b^4*c^3 - 42*a^2*b^2*c^4 - 24*a^3*c^5)*d^4*e^5 - 4*(a*b^5*c^2 - a^2*b^3*c^3 - 12*a^3
*b*c^4)*d^3*e^6 + 2*(3*a^2*b^4*c^2 - 10*a^3*b^2*c^3 - 8*a^4*c^4)*d^2*e^7 - 4*(a^3*b^3*c^2 - 4*a^4*b*c^3)*d*e^8
+ (a^4*b^2*c^2 - 4*a^5*c^3)*e^9)*x^5 + ((b^2*c^6 - 4*a*c^7)*d^9 - 2*(b^3*c^5 - 4*a*b*c^6)*d^8*e - 2*(b^4*c^4
- 6*a*b^2*c^5 + 8*a^2*c^6)*d^7*e^2 + 4*(2*b^5*c^3 - 9*a*b^3*c^4 + 4*a^2*b*c^5)*d^6*e^3 - (7*b^6*c^2 - 16*a*b^4
*c^3 - 54*a^2*b^2*c^4 + 24*a^3*c^5)*d^5*e^4 + 2*(b^7*c + 6*a*b^5*c^2 - 40*a^2*b^3*c^3)*d^4*e^5 - 2*(4*a*b^6*c
- 7*a^2*b^4*c^2 - 38*a^3*b^2*c^3 + 8*a^4*c^4)*d^3*e^6 + 4*(3*a^2*b^5*c - 11*a^3*b^3*c^2 - 4*a^4*b*c^3)*d^2*e^7
- (8*a^3*b^4*c - 33*a^4*b^2*c^2 + 4*a^5*c^3)*d*e^8 + 2*(a^4*b^3*c - 4*a^5*b*c^2)*e^9)*x^4 + (2*(b^3*c^5 - 4*a
*b*c^6)*d^9 - (7*b^4*c^4 - 30*a*b^2*c^5 + 8*a^2*c^6)*d^8*e + 8*(b^5*c^3 - 4*a*b^3*c^4)*d^7*e^2 - 2*(b^6*c^2 -
20*a^2*b^2*c^4 + 16*a^3*c^5)*d^6*e^3 - 2*(b^7*c - 6*a*b^5*c^2 + 14*a^2*b^3*c^3 - 24*a^3*b*c^4)*d^5*e^4 + (b^8
+ 2*a*b^6*c - 18*a^2*b^4*c^2 - 12*a^3*b^2*c^3 - 48*a^4*c^4)*d^4*e^5 - 4*(a*b^7 - 2*a^2*b^5*c - 4*a^3*b^3*c^2 -
16*a^4*b*c^3)*d^3*e^6 + 2*(3*a^2*b^6 - 8*a^3*b^4*c - 12*a^4*b^2*c^2 - 16*a^5*c^3)*d^2*e^7 - 2*(2*a^3*b^5 - 5*
a^4*b^3*c - 12*a^5*b*c^2)*d*e^8 + (a^4*b^4 - 2*a^5*b^2*c - 8*a^6*c^2)*e^9)*x^3 + ((b^4*c^4 - 2*a*b^2*c^5 - 8*a
^2*c^6)*d^9 - 2*(2*b^5*c^3 - 5*a*b^3*c^4 - 12*a^2*b*c^5)*d^8*e + 2*(3*b^6*c^2 - 8*a*b^4*c^3 - 12*a^2*b^2*c^4 -
16*a^3*c^5)*d^7*e^2 - 4*(b^7*c - 2*a*b^5*c^2 - 4*a^2*b^3*c^3 - 16*a^3*b*c^4)*d^6*e^3 + (b^8 + 2*a*b^6*c - 18*
a^2*b^4*c^2 - 12*a^3*b^2*c^3 - 48*a^4*c^4)*d^5*e^4 - 2*(a*b^7 - 6*a^2*b^5*c + 14*a^3*b^3*c^2 - 24*a^4*b*c^3)*d
^4*e^5 - 2*(a^2*b^6 - 20*a^4*b^2*c^2 + 16*a^5*c^3)*d^3*e^6 + 8*(a^3*b^5 - 4*a^4*b^3*c)*d^2*e^7 - (7*a^4*b^4 -
30*a^5*b^2*c + 8*a^6*c^2)*d*e^8 + 2*(a^5*b^3 - 4*a^6*b*c)*e^9)*x^2 + (2*(a*b^3*c^4 - 4*a^2*b*c^5)*d^9 - (8*a*b
^4*c^3 - 33*a^2*b^2*c^4 + 4*a^3*c^5)*d^8*e + 4*(3*a*b^5*c^2 - 11*a^2*b^3*c^3 - 4*a^3*b*c^4)*d^7*e^2 - 2*(4*a*b
^6*c - 7*a^2*b^4*c^2 - 38*a^3*b^2*c^3 + 8*a^4*c^4)*d^6*e^3 + 2*(a*b^7 + 6*a^2*b^5*c - 40*a^3*b^3*c^2)*d^5*e^4
- (7*a^2*b^6 - 16*a^3*b^4*c - 54*a^4*b^2*c^2 + 24*a^5*c^3)*d^4*e^5 + 4*(2*a^3*b^5 - 9*a^4*b^3*c + 4*a^5*b*c^2)
*d^3*e^6 - 2*(a^4*b^4 - 6*a^5*b^2*c + 8*a^6*c^2)*d^2*e^7 - 2*(a^5*b^3 - 4*a^6*b*c)*d*e^8 + (a^6*b^2 - 4*a^7*c)
*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((2*c*x+b)/(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((2*c*x+b)/(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

Timed out