3.32.44 \(\int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} (c+b x^2)^2} \, dx\) [3144]

Optimal. Leaf size=1293 \[ \frac {\left (-a^2 b c-a b^2 c+b^3 c+a b^3 c-b^4 x+a^3 c x-a^2 b c x-a^2 b^2 c x+a b^3 c x\right ) \sqrt {c+b x+a x^2}}{2 c \left (b^3+a^2 c-2 a b c+b^2 c\right ) \left (c+b x^2\right )}-\frac {\text {RootSum}\left [b^2 c+b c^2-4 \sqrt {a} b c \text {$\#$1}+4 a c \text {$\#$1}^2-2 b c \text {$\#$1}^2+b \text {$\#$1}^4\& ,\frac {-a^2 b^3 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-a b^4 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-2 b^5 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-3 a b^5 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+a^3 b c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-5 a^2 b^2 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-4 a^3 b^2 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+2 a b^3 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+7 a^2 b^3 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-a b^4 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+2 a^{5/2} b^2 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a^{3/2} b^3 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 \sqrt {a} b^4 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+6 a^{3/2} b^4 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a^{7/2} c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 a^{5/2} b c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+8 a^{7/2} b c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-12 a^{5/2} b^2 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 a^{3/2} b^3 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+b^5 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a^3 b c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+3 a^2 b^2 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-2 a b^3 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a^2 b^3 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a b^4 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} b^4 c-a^{5/2} b c^2+2 a^{3/2} b^2 c^2-\sqrt {a} b^3 c^2+2 a b^3 c \text {$\#$1}-b^4 c \text {$\#$1}+2 a^3 c^2 \text {$\#$1}-5 a^2 b c^2 \text {$\#$1}+4 a b^2 c^2 \text {$\#$1}-b^3 c^2 \text {$\#$1}+b^4 \text {$\#$1}^3+a^2 b c \text {$\#$1}^3-2 a b^2 c \text {$\#$1}^3+b^3 c \text {$\#$1}^3}\& \right ]}{8 c} \]

[Out]

Unintegrable

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(3187\) vs. \(2(1293)=2586\).
time = 40.81, antiderivative size = 3187, normalized size of antiderivative = 2.46, number of steps used = 6, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1076, 1050, 1044, 214} \begin {gather*} \text {Too large to display} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*b*c - b^2*x + a^2*x^2)/(Sqrt[c + b*x + a*x^2]*(c + b*x^2)^2),x]

[Out]

-1/2*((b*(a^2 + a*(1 - b)*b - b^2)*c + (b^4 - a^2*(a - b)*c + a*(a - b)*b^2*c)*x)*Sqrt[c + b*x + a*x^2])/((b^3
*c + (a*c - b*c)^2)*(c + b*x^2)) - (Sqrt[2*b^5*c - a^4*c^2 - b^4*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] +
a^2*b*c*(b^2 + 7*b*c - 6*b^2*c - 3*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] + b*Sqrt[c]*Sqrt[b^3 + a^2*c -
2*a*b*c + b^2*c]) + a*b^2*c*(3*b^3 - 2*b*c + b^2*c + 2*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] + b*Sqrt[c
]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) - a^3*(4*b*c^2 - 5*b^2*c^2 + c^(3/2)*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c
])]*Sqrt[b^6 - b^5*(c + 4*a*c) - 2*a^3*c^(3/2)*(a*Sqrt[c] - Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) + 2*a^2*b*c^(
3/2)*(2*a*Sqrt[c] + 2*a^2*Sqrt[c] - Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] - 2*a*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2
*c]) + a^2*b^2*Sqrt[c]*(2*a*(1 - 5*c)*Sqrt[c] - 2*c^(3/2) - Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] + 6*c*Sqrt[b^3
+ a^2*c - 2*a*b*c + b^2*c]) + b^3*(a^2*c*(3 + 8*c) - a*Sqrt[c]*(1 + 2*c)*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])
+ b^4*(2*a^2*c - Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] - a*(2*c + 2*c^2 + 3*Sqrt[c]*Sqrt[b^3 + a^2*c -
2*a*b*c + b^2*c]))]*ArcTanh[(Sqrt[b]*(b*c*(b^4 - 2*a^3*(1 - 2*b)*c + a*b^3*(1 + 3*b + 2*c) + a^2*b*(b + 2*c -
6*b*c)) - (b^4 + a^3*c + 3*a^2*b*c - a*(2 + a)*b^2*c - a*b^3*c)*(a*c - b*c + Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*
c + b^2*c]) + (b^2*(b^4 + a^3*c + 3*a^2*b*c - a*(2 + a)*b^2*c - a*b^3*c) + (b^4 - 2*a^3*(1 - 2*b)*c + a*b^3*(1
+ 3*b + 2*c) + a^2*b*(b + 2*c - 6*b*c))*(a*c - b*c - Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]))*x))/(Sqrt[
2]*Sqrt[2*b^5*c - a^4*c^2 - b^4*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] + a^2*b*c*(b^2 + 7*b*c - 6*b^2*c -
3*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] + b*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) + a*b^2*c*(3*b
^3 - 2*b*c + b^2*c + 2*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] + b*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^
2*c]) - a^3*(4*b*c^2 - 5*b^2*c^2 + c^(3/2)*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*Sqrt[b^6 - b^5*(c + 4*a*c) -
2*a^3*c^(3/2)*(a*Sqrt[c] - Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) + 2*a^2*b*c^(3/2)*(2*a*Sqrt[c] + 2*a^2*Sqrt[c]
- Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] - 2*a*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) + a^2*b^2*Sqrt[c]*(2*a*(1 -
5*c)*Sqrt[c] - 2*c^(3/2) - Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] + 6*c*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) + b^
3*(a^2*c*(3 + 8*c) - a*Sqrt[c]*(1 + 2*c)*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) + b^4*(2*a^2*c - Sqrt[c]*Sqrt[b^
3 + a^2*c - 2*a*b*c + b^2*c] - a*(2*c + 2*c^2 + 3*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]))]*Sqrt[c + b*x
+ a*x^2])])/(4*Sqrt[2]*Sqrt[b]*c^(3/2)*(b^3 + a^2*c - 2*a*b*c + b^2*c)^(3/2)) + (Sqrt[b^6 - b^5*(c + 4*a*c) -
2*a^3*c^(3/2)*(a*Sqrt[c] + Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) + a^2*b^2*Sqrt[c]*(2*a*(1 - 5*c)*Sqrt[c] - 2*c
^(3/2) + Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] - 6*c*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) + b^3*(a^2*c*(3 + 8*c)
+ a*Sqrt[c]*(1 + 2*c)*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) + 2*a^2*b*c^(3/2)*(2*a^2*Sqrt[c] + Sqrt[b^3 + a^2*
c - 2*a*b*c + b^2*c] + 2*a*(Sqrt[c] + Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])) + b^4*(2*a^2*c + Sqrt[c]*Sqrt[b^3
+ a^2*c - 2*a*b*c + b^2*c] - a*(2*c + 2*c^2 - 3*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]))]*Sqrt[2*b^5*c -
a^4*c^2 + b^4*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] + a^2*b*c*(b^2 + 7*b*c - 6*b^2*c + 3*Sqrt[c]*Sqrt[b^
3 + a^2*c - 2*a*b*c + b^2*c] - b*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) - a^3*(4*b*c^2 - 5*b^2*c^2 - c^(
3/2)*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) + a*b^2*c*(3*b^3 + b^2*c - 2*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^
2*c] - b*(2*c + Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]))]*ArcTanh[(Sqrt[b]*(b*c*(b^4 - 2*a^3*(1 - 2*b)*c
+ a*b^3*(1 + 3*b + 2*c) + a^2*b*(b + 2*c - 6*b*c)) - (b^4 + a^3*c + 3*a^2*b*c - a*(2 + a)*b^2*c - a*b^3*c)*(a*
c - b*c - Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) + (b^2*(b^4 + a^3*c + 3*a^2*b*c - a*(2 + a)*b^2*c - a*b
^3*c) + (b^4 - 2*a^3*(1 - 2*b)*c + a*b^3*(1 + 3*b + 2*c) + a^2*b*(b + 2*c - 6*b*c))*(a*c - b*c + Sqrt[c]*Sqrt[
b^3 + a^2*c - 2*a*b*c + b^2*c]))*x))/(Sqrt[2]*Sqrt[b^6 - b^5*(c + 4*a*c) - 2*a^3*c^(3/2)*(a*Sqrt[c] + Sqrt[b^3
+ a^2*c - 2*a*b*c + b^2*c]) + a^2*b^2*Sqrt[c]*(2*a*(1 - 5*c)*Sqrt[c] - 2*c^(3/2) + Sqrt[b^3 + a^2*c - 2*a*b*c
+ b^2*c] - 6*c*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) + b^3*(a^2*c*(3 + 8*c) + a*Sqrt[c]*(1 + 2*c)*Sqrt[b^3 + a
^2*c - 2*a*b*c + b^2*c]) + 2*a^2*b*c^(3/2)*(2*a^2*Sqrt[c] + Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] + 2*a*(Sqrt[c]
+ Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])) + b^4*(2*a^2*c + Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] - a*(2*c
+ 2*c^2 - 3*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]))]*Sqrt[2*b^5*c - a^4*c^2 + b^4*Sqrt[c]*Sqrt[b^3 + a^
2*c - 2*a*b*c + b^2*c] + a^2*b*c*(b^2 + 7*b*c - 6*b^2*c + 3*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] - b*Sq
rt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) - a^3*(4*b*c^2 - 5*b^2*c^2 - c^(3/2)*Sqrt[b^3 + a^2*c - 2*a*b*c + b
^2*c]) + a*b^2*c*(3*b^3 + b^2*c - 2*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] - b*(2*c + Sqrt[c]*Sqrt[b^3 +
a^2*c - 2*a*b*c + b^2*c]))]*Sqrt[c + b*x + a*x^...

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 1044

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*
a*g*h, Subst[Int[1/Simp[2*a^2*g*h*c + a*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; F
reeQ[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]

Rule 1050

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Dist[1/(2*q), Int[Simp[(-a)*h*e - g*(c*d - a*f - q) + (h*(c*d - a*f + q) -
g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[Simp[(-a)*h*e - g*(c*d - a*f + q
) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g,
h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]

Rule 1076

Int[((a_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_
Symbol] :> Simp[(a + c*x^2)^(p + 1)*((d + e*x + f*x^2)^(q + 1)/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)))*(
(A*c - a*C)*(2*a*c*e) + ((-a)*B)*(2*c^2*d - c*(2*a*f)) + c*(A*(2*c^2*d - c*(2*a*f)) - B*(-2*a*c*e) + C*(-2*a*(
c*d - a*f)))*x), x] + Dist[1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)), Int[(a + c*x^2)^(p + 1)*(d + e*x +
f*x^2)^q*Simp[(-2*A*c - 2*a*C)*((c*d - a*f)^2 - ((-a)*e)*(c*e))*(p + 1) + (2*(A*c*(c*d - a*f) - a*(c*C*d - B*c
*e - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e) + ((-a)*B)*(2*c^2*d - c*((Plus[2])*a*f)))
*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e) + ((-a)*B)*(2*c^2*d + (-c)*((Plus[2])*a*f)))*(p + q + 2) - (2*(A*c*
(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*((-c)*e*(2*p + q + 4)))*x - c*f*(2*(A*c*(c*d - a*f) - a*(c*C*d - B*c
*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, A, B, C, q}, x] && NeQ[e^2 - 4*d*f, 0]
&& LtQ[p, -1] && NeQ[a*c*e^2 + (c*d - a*f)^2, 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rubi steps

\begin {align*} \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )^2} \, dx &=-\frac {\left (b \left (a^2+a (1-b) b-b^2\right ) c+\left (b^4-a^2 (a-b) c+a (a-b) b^2 c\right ) x\right ) \sqrt {c+b x+a x^2}}{2 \left (b^3 c+(a c-b c)^2\right ) \left (c+b x^2\right )}+\frac {\int \frac {b c^2 \left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right )-b^2 c \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) x}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx}{4 b c \left (b^3 c+(a c-b c)^2\right )}\\ &=-\frac {\left (b \left (a^2+a (1-b) b-b^2\right ) c+\left (b^4-a^2 (a-b) c+a (a-b) b^2 c\right ) x\right ) \sqrt {c+b x+a x^2}}{2 \left (b^3 c+(a c-b c)^2\right ) \left (c+b x^2\right )}+\frac {\int \frac {b c^2 \left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )-b^2 c \left (b c \left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) x}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx}{8 b c^{5/2} \left (b^3+a^2 c-2 a b c+b^2 c\right )^{3/2}}-\frac {\int \frac {b c^2 \left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )-b^2 c \left (b c \left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) x}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx}{8 b c^{5/2} \left (b^3+a^2 c-2 a b c+b^2 c\right )^{3/2}}\\ &=-\frac {\left (b \left (a^2+a (1-b) b-b^2\right ) c+\left (b^4-a^2 (a-b) c+a (a-b) b^2 c\right ) x\right ) \sqrt {c+b x+a x^2}}{2 \left (b^3 c+(a c-b c)^2\right ) \left (c+b x^2\right )}-\frac {\left (b^2 c^{3/2} \left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) \left (b c \left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-2 b^4 c^5 \left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) \left (b c \left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )+b c x^2} \, dx,x,\frac {-b^2 c^2 \left (b c \left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )-b^2 c^2 \left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) x}{\sqrt {c+b x+a x^2}}\right )}{4 \left (b^3+a^2 c-2 a b c+b^2 c\right )^{3/2}}+\frac {\left (b^2 c^{3/2} \left (b c \left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) \left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-2 b^4 c^5 \left (b c \left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) \left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )+b c x^2} \, dx,x,\frac {-b^2 c^2 \left (b c \left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )-b^2 c^2 \left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) x}{\sqrt {c+b x+a x^2}}\right )}{4 \left (b^3+a^2 c-2 a b c+b^2 c\right )^{3/2}}\\ &=-\frac {\left (b \left (a^2+a (1-b) b-b^2\right ) c+\left (b^4-a^2 (a-b) c+a (a-b) b^2 c\right ) x\right ) \sqrt {c+b x+a x^2}}{2 \left (b^3 c+(a c-b c)^2\right ) \left (c+b x^2\right )}-\frac {\sqrt {2 b^5 c-a^4 c^2-b^4 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+a^2 b c \left (b^2+7 b c-6 b^2 c-3 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+b \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+a b^2 c \left (3 b^3-2 b c+b^2 c+2 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+b \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )-a^3 \left (4 b c^2-5 b^2 c^2+c^{3/2} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )} \sqrt {b^6-b^5 (c+4 a c)-2 a^3 c^{3/2} \left (a \sqrt {c}-\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+2 a^2 b c^{3/2} \left (2 a \sqrt {c}+2 a^2 \sqrt {c}-\sqrt {b^3+a^2 c-2 a b c+b^2 c}-2 a \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+a^2 b^2 \sqrt {c} \left (2 a (1-5 c) \sqrt {c}-2 c^{3/2}-\sqrt {b^3+a^2 c-2 a b c+b^2 c}+6 c \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+b^3 \left (a^2 c (3+8 c)-a \sqrt {c} (1+2 c) \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+b^4 \left (2 a^2 c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}-a \left (2 c+2 c^2+3 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )} \tanh ^{-1}\left (\frac {\sqrt {b} \left (b c \left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+\left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) x\right )}{\sqrt {2} \sqrt {2 b^5 c-a^4 c^2-b^4 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+a^2 b c \left (b^2+7 b c-6 b^2 c-3 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+b \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+a b^2 c \left (3 b^3-2 b c+b^2 c+2 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+b \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )-a^3 \left (4 b c^2-5 b^2 c^2+c^{3/2} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )} \sqrt {b^6-b^5 (c+4 a c)-2 a^3 c^{3/2} \left (a \sqrt {c}-\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+2 a^2 b c^{3/2} \left (2 a \sqrt {c}+2 a^2 \sqrt {c}-\sqrt {b^3+a^2 c-2 a b c+b^2 c}-2 a \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+a^2 b^2 \sqrt {c} \left (2 a (1-5 c) \sqrt {c}-2 c^{3/2}-\sqrt {b^3+a^2 c-2 a b c+b^2 c}+6 c \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+b^3 \left (a^2 c (3+8 c)-a \sqrt {c} (1+2 c) \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+b^4 \left (2 a^2 c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}-a \left (2 c+2 c^2+3 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )} \sqrt {c+b x+a x^2}}\right )}{4 \sqrt {2} \sqrt {b} c^{3/2} \left (b^3+a^2 c-2 a b c+b^2 c\right )^{3/2}}+\frac {\sqrt {b^6-b^5 (c+4 a c)-2 a^3 c^{3/2} \left (a \sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+a^2 b^2 \sqrt {c} \left (2 a (1-5 c) \sqrt {c}-2 c^{3/2}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}-6 c \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+b^3 \left (a^2 c (3+8 c)+a \sqrt {c} (1+2 c) \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+2 a^2 b c^{3/2} \left (2 a^2 \sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}+2 a \left (\sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )+b^4 \left (2 a^2 c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}-a \left (2 c+2 c^2-3 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )} \sqrt {2 b^5 c-a^4 c^2+b^4 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+a^2 b c \left (b^2+7 b c-6 b^2 c+3 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}-b \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )-a^3 \left (4 b c^2-5 b^2 c^2-c^{3/2} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+a b^2 c \left (3 b^3+b^2 c-2 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}-b \left (2 c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )} \tanh ^{-1}\left (\frac {\sqrt {b} \left (b c \left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+\left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (b^4-2 a^3 (1-2 b) c+a b^3 (1+3 b+2 c)+a^2 b (b+2 c-6 b c)\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) x\right )}{\sqrt {2} \sqrt {b^6-b^5 (c+4 a c)-2 a^3 c^{3/2} \left (a \sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+a^2 b^2 \sqrt {c} \left (2 a (1-5 c) \sqrt {c}-2 c^{3/2}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}-6 c \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+b^3 \left (a^2 c (3+8 c)+a \sqrt {c} (1+2 c) \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+2 a^2 b c^{3/2} \left (2 a^2 \sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}+2 a \left (\sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )+b^4 \left (2 a^2 c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}-a \left (2 c+2 c^2-3 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )} \sqrt {2 b^5 c-a^4 c^2+b^4 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+a^2 b c \left (b^2+7 b c-6 b^2 c+3 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}-b \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )-a^3 \left (4 b c^2-5 b^2 c^2-c^{3/2} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+a b^2 c \left (3 b^3+b^2 c-2 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}-b \left (2 c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )} \sqrt {c+b x+a x^2}}\right )}{4 \sqrt {2} \sqrt {b} c^{3/2} \left (b^3+a^2 c-2 a b c+b^2 c\right )^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 1.86, size = 1546, normalized size = 1.20 \begin {gather*} \frac {1}{8} \left (\frac {4 \sqrt {c+x (b+a x)} \left (a^3 c x+b^3 (c-b x)+a b^2 c (-1+b+b x)-a^2 b c (1+x+b x)\right )}{c \left (b^3+a^2 c-2 a b c+b^2 c\right ) \left (c+b x^2\right )}-\frac {4 a \text {RootSum}\left [b^2 c+b c^2-4 \sqrt {a} b c \text {$\#$1}+4 a c \text {$\#$1}^2-2 b c \text {$\#$1}^2+b \text {$\#$1}^4\&,\frac {-a b \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+4 b^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+2 a^{3/2} \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}}{-\sqrt {a} b c+2 a c \text {$\#$1}-b c \text {$\#$1}+b \text {$\#$1}^3}\&\right ]}{b}+\frac {\text {RootSum}\left [b^2 c+b c^2-4 \sqrt {a} b c \text {$\#$1}+4 a c \text {$\#$1}^2-2 b c \text {$\#$1}^2+b \text {$\#$1}^4\&,\frac {-3 a^2 b^4 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+17 a b^5 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+2 b^6 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+3 a b^6 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-4 a^4 b c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+23 a^3 b^2 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-31 a^2 b^3 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+4 a^3 b^3 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+14 a b^4 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-7 a^2 b^4 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+a b^5 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+6 a^{5/2} b^3 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a^{3/2} b^4 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 \sqrt {a} b^5 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-6 a^{3/2} b^5 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+8 a^{9/2} c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-12 a^{7/2} b c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 a^{5/2} b^2 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-8 a^{7/2} b^2 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+12 a^{5/2} b^3 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a^{3/2} b^4 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-b^6 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a^3 b^2 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-3 a^2 b^3 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 a b^4 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a^2 b^4 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a b^5 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} b^4 c-a^{5/2} b c^2+2 a^{3/2} b^2 c^2-\sqrt {a} b^3 c^2+2 a b^3 c \text {$\#$1}-b^4 c \text {$\#$1}+2 a^3 c^2 \text {$\#$1}-5 a^2 b c^2 \text {$\#$1}+4 a b^2 c^2 \text {$\#$1}-b^3 c^2 \text {$\#$1}+b^4 \text {$\#$1}^3+a^2 b c \text {$\#$1}^3-2 a b^2 c \text {$\#$1}^3+b^3 c \text {$\#$1}^3}\&\right ]}{b c}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*b*c - b^2*x + a^2*x^2)/(Sqrt[c + b*x + a*x^2]*(c + b*x^2)^2),x]

[Out]

((4*Sqrt[c + x*(b + a*x)]*(a^3*c*x + b^3*(c - b*x) + a*b^2*c*(-1 + b + b*x) - a^2*b*c*(1 + x + b*x)))/(c*(b^3
+ a^2*c - 2*a*b*c + b^2*c)*(c + b*x^2)) - (4*a*RootSum[b^2*c + b*c^2 - 4*Sqrt[a]*b*c*#1 + 4*a*c*#1^2 - 2*b*c*#
1^2 + b*#1^4 & , (-(a*b*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]) + 4*b^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*
x + a*x^2] - #1] + 2*a^(3/2)*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1)/(-(Sqrt[a]*b*c) + 2*a*c*#1 - b
*c*#1 + b*#1^3) & ])/b + RootSum[b^2*c + b*c^2 - 4*Sqrt[a]*b*c*#1 + 4*a*c*#1^2 - 2*b*c*#1^2 + b*#1^4 & , (-3*a
^2*b^4*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + 17*a*b^5*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2]
- #1] + 2*b^6*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + 3*a*b^6*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a
*x^2] - #1] - 4*a^4*b*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + 23*a^3*b^2*c^2*Log[-(Sqrt[a]*x) + S
qrt[c + b*x + a*x^2] - #1] - 31*a^2*b^3*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + 4*a^3*b^3*c^2*Log
[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + 14*a*b^4*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] - 7*
a^2*b^4*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + a*b^5*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2
] - #1] + 6*a^(5/2)*b^3*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - 2*a^(3/2)*b^4*c*Log[-(Sqrt[a]*x)
+ Sqrt[c + b*x + a*x^2] - #1]*#1 - 2*Sqrt[a]*b^5*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - 6*a^(3
/2)*b^5*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 + 8*a^(9/2)*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x +
a*x^2] - #1]*#1 - 12*a^(7/2)*b*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 + 4*a^(5/2)*b^2*c^2*Log[-
(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - 8*a^(7/2)*b^2*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1
]*#1 + 12*a^(5/2)*b^3*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - 4*a^(3/2)*b^4*c^2*Log[-(Sqrt[a]*
x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - b^6*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2 - a^3*b^2*c*Log[
-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2 - 3*a^2*b^3*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#
1^2 + 2*a*b^4*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2 + a^2*b^4*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*
x + a*x^2] - #1]*#1^2 + a*b^5*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2)/(-(Sqrt[a]*b^4*c) - a^(5/
2)*b*c^2 + 2*a^(3/2)*b^2*c^2 - Sqrt[a]*b^3*c^2 + 2*a*b^3*c*#1 - b^4*c*#1 + 2*a^3*c^2*#1 - 5*a^2*b*c^2*#1 + 4*a
*b^2*c^2*#1 - b^3*c^2*#1 + b^4*#1^3 + a^2*b*c*#1^3 - 2*a*b^2*c*#1^3 + b^3*c*#1^3) & ]/(b*c))/8

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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.19, size = 1250, normalized size = 0.97

method	result	size
default	\(\frac {a \left (b^{2}+a \right ) \ln \left (\frac {-\frac {2 \left (\sqrt {-b c}\, b +a c -b c \right )}{b}-\frac {\sqrt {-b c}\, \left (\sqrt {-b c}\, b +2 a c \right ) \left (x +\frac {\sqrt {-b c}}{b}\right )}{b c}+2 \sqrt {-\frac {\sqrt {-b c}\, b +a c -b c}{b}}\, \sqrt {a \left (x +\frac {\sqrt {-b c}}{b}\right )^{2}-\frac {\sqrt {-b c}\, \left (\sqrt {-b c}\, b +2 a c \right ) \left (x +\frac {\sqrt {-b c}}{b}\right )}{b c}-\frac {\sqrt {-b c}\, b +a c -b c}{b}}}{x +\frac {\sqrt {-b c}}{b}}\right )}{4 \sqrt {-b c}\, b \sqrt {-\frac {\sqrt {-b c}\, b +a c -b c}{b}}}-\frac {a \left (b^{2}+a \right ) \ln \left (\frac {-\frac {2 \left (-\sqrt {-b c}\, b +a c -b c \right )}{b}+\frac {\sqrt {-b c}\, \left (-\sqrt {-b c}\, b +2 a c \right ) \left (x -\frac {\sqrt {-b c}}{b}\right )}{b c}+2 \sqrt {-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}\, \sqrt {a \left (x -\frac {\sqrt {-b c}}{b}\right )^{2}+\frac {\sqrt {-b c}\, \left (-\sqrt {-b c}\, b +2 a c \right ) \left (x -\frac {\sqrt {-b c}}{b}\right )}{b c}-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}}{x -\frac {\sqrt {-b c}}{b}}\right )}{4 \sqrt {-b c}\, b \sqrt {-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}}+\frac {\left (-a \,b^{2} c +\sqrt {-b c}\, b^{2}+a^{2} c \right ) \left (\frac {b \sqrt {a \left (x -\frac {\sqrt {-b c}}{b}\right )^{2}+\frac {\sqrt {-b c}\, \left (-\sqrt {-b c}\, b +2 a c \right ) \left (x -\frac {\sqrt {-b c}}{b}\right )}{b c}-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}}{\left (-\sqrt {-b c}\, b +a c -b c \right ) \left (x -\frac {\sqrt {-b c}}{b}\right )}-\frac {\sqrt {-b c}\, \left (-\sqrt {-b c}\, b +2 a c \right ) \ln \left (\frac {-\frac {2 \left (-\sqrt {-b c}\, b +a c -b c \right )}{b}+\frac {\sqrt {-b c}\, \left (-\sqrt {-b c}\, b +2 a c \right ) \left (x -\frac {\sqrt {-b c}}{b}\right )}{b c}+2 \sqrt {-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}\, \sqrt {a \left (x -\frac {\sqrt {-b c}}{b}\right )^{2}+\frac {\sqrt {-b c}\, \left (-\sqrt {-b c}\, b +2 a c \right ) \left (x -\frac {\sqrt {-b c}}{b}\right )}{b c}-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}}{x -\frac {\sqrt {-b c}}{b}}\right )}{2 c \left (-\sqrt {-b c}\, b +a c -b c \right ) \sqrt {-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}}\right )}{4 c \,b^{2}}+\frac {\left (-a \,b^{2} c -\sqrt {-b c}\, b^{2}+a^{2} c \right ) \left (\frac {b \sqrt {a \left (x +\frac {\sqrt {-b c}}{b}\right )^{2}-\frac {\sqrt {-b c}\, \left (\sqrt {-b c}\, b +2 a c \right ) \left (x +\frac {\sqrt {-b c}}{b}\right )}{b c}-\frac {\sqrt {-b c}\, b +a c -b c}{b}}}{\left (\sqrt {-b c}\, b +a c -b c \right ) \left (x +\frac {\sqrt {-b c}}{b}\right )}+\frac {\sqrt {-b c}\, \left (\sqrt {-b c}\, b +2 a c \right ) \ln \left (\frac {-\frac {2 \left (\sqrt {-b c}\, b +a c -b c \right )}{b}-\frac {\sqrt {-b c}\, \left (\sqrt {-b c}\, b +2 a c \right ) \left (x +\frac {\sqrt {-b c}}{b}\right )}{b c}+2 \sqrt {-\frac {\sqrt {-b c}\, b +a c -b c}{b}}\, \sqrt {a \left (x +\frac {\sqrt {-b c}}{b}\right )^{2}-\frac {\sqrt {-b c}\, \left (\sqrt {-b c}\, b +2 a c \right ) \left (x +\frac {\sqrt {-b c}}{b}\right )}{b c}-\frac {\sqrt {-b c}\, b +a c -b c}{b}}}{x +\frac {\sqrt {-b c}}{b}}\right )}{2 c \left (\sqrt {-b c}\, b +a c -b c \right ) \sqrt {-\frac {\sqrt {-b c}\, b +a c -b c}{b}}}\right )}{4 c \,b^{2}}\)	\(1250\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*x^2+a*b*c-b^2*x)/(a*x^2+b*x+c)^(1/2)/(b*x^2+c)^2,x,method=_RETURNVERBOSE)

[Out]

1/4*a*(b^2+a)/(-b*c)^(1/2)/b/(-((-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2)*ln((-2*((-b*c)^(1/2)*b+a*c-b*c)/b-(-b*c)^(1/2
)/b/c*((-b*c)^(1/2)*b+2*a*c)*(x+(-b*c)^(1/2)/b)+2*(-((-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2)*(a*(x+(-b*c)^(1/2)/b)^2-
(-b*c)^(1/2)/b/c*((-b*c)^(1/2)*b+2*a*c)*(x+(-b*c)^(1/2)/b)-((-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2))/(x+(-b*c)^(1/2)/
b))-1/4*a*(b^2+a)/(-b*c)^(1/2)/b/(-(-(-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2)*ln((-2*(-(-b*c)^(1/2)*b+a*c-b*c)/b+(-b*c
)^(1/2)/b/c*(-(-b*c)^(1/2)*b+2*a*c)*(x-(-b*c)^(1/2)/b)+2*(-(-(-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2)*(a*(x-(-b*c)^(1/
2)/b)^2+(-b*c)^(1/2)/b/c*(-(-b*c)^(1/2)*b+2*a*c)*(x-(-b*c)^(1/2)/b)-(-(-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2))/(x-(-b
*c)^(1/2)/b))+1/4*(-a*b^2*c+(-b*c)^(1/2)*b^2+a^2*c)/c/b^2*(1/(-(-b*c)^(1/2)*b+a*c-b*c)*b/(x-(-b*c)^(1/2)/b)*(a
*(x-(-b*c)^(1/2)/b)^2+(-b*c)^(1/2)/b/c*(-(-b*c)^(1/2)*b+2*a*c)*(x-(-b*c)^(1/2)/b)-(-(-b*c)^(1/2)*b+a*c-b*c)/b)
^(1/2)-1/2*(-b*c)^(1/2)/c*(-(-b*c)^(1/2)*b+2*a*c)/(-(-b*c)^(1/2)*b+a*c-b*c)/(-(-(-b*c)^(1/2)*b+a*c-b*c)/b)^(1/
2)*ln((-2*(-(-b*c)^(1/2)*b+a*c-b*c)/b+(-b*c)^(1/2)/b/c*(-(-b*c)^(1/2)*b+2*a*c)*(x-(-b*c)^(1/2)/b)+2*(-(-(-b*c)
^(1/2)*b+a*c-b*c)/b)^(1/2)*(a*(x-(-b*c)^(1/2)/b)^2+(-b*c)^(1/2)/b/c*(-(-b*c)^(1/2)*b+2*a*c)*(x-(-b*c)^(1/2)/b)
-(-(-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2))/(x-(-b*c)^(1/2)/b)))+1/4*(-a*b^2*c-(-b*c)^(1/2)*b^2+a^2*c)/c/b^2*(1/((-b*
c)^(1/2)*b+a*c-b*c)*b/(x+(-b*c)^(1/2)/b)*(a*(x+(-b*c)^(1/2)/b)^2-(-b*c)^(1/2)/b/c*((-b*c)^(1/2)*b+2*a*c)*(x+(-
b*c)^(1/2)/b)-((-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2)+1/2*(-b*c)^(1/2)/c*((-b*c)^(1/2)*b+2*a*c)/((-b*c)^(1/2)*b+a*c-
b*c)/(-((-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2)*ln((-2*((-b*c)^(1/2)*b+a*c-b*c)/b-(-b*c)^(1/2)/b/c*((-b*c)^(1/2)*b+2*
a*c)*(x+(-b*c)^(1/2)/b)+2*(-((-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2)*(a*(x+(-b*c)^(1/2)/b)^2-(-b*c)^(1/2)/b/c*((-b*c)
^(1/2)*b+2*a*c)*(x+(-b*c)^(1/2)/b)-((-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2))/(x+(-b*c)^(1/2)/b)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a^2*x^2 + a*b*c - b^2*x)/(sqrt(a*x^2 + b*x + c)*(b*x^2 + c)^2), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{2} x^{2} + a b c - b^{2} x}{\left (b x^{2} + c\right )^{2} \sqrt {a x^{2} + b x + c}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*x**2+a*b*c-b**2*x)/(a*x**2+b*x+c)**(1/2)/(b*x**2+c)**2,x)

[Out]

Integral((a**2*x**2 + a*b*c - b**2*x)/((b*x**2 + c)**2*sqrt(a*x**2 + b*x + c)), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a^2\,x^2+c\,a\,b-b^2\,x}{{\left (b\,x^2+c\right )}^2\,\sqrt {a\,x^2+b\,x+c}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*x^2 - b^2*x + a*b*c)/((c + b*x^2)^2*(c + b*x + a*x^2)^(1/2)),x)

[Out]

int((a^2*x^2 - b^2*x + a*b*c)/((c + b*x^2)^2*(c + b*x + a*x^2)^(1/2)), x)

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