3.113 \(\int \frac {1}{\sqrt {a+b x+c x^2} (d+e x+f x^2)^2} \, dx\)

Optimal. Leaf size=789 \[ \frac {\left (f \left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) (2 a f-b e+2 c d)-2 f \left (f \left (-4 a^2 f^2+3 a b e f+b^2 \left (e^2-6 d f\right )\right )-c \left (4 a f \left (e^2-3 d f\right )+b \left (e^3-5 d e f\right )\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right )-b \left (e-\sqrt {e^2-4 d f}\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{2 \sqrt {2} \left (e^2-4 d f\right )^{3/2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac {\left (f \left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) (2 a f-b e+2 c d)-2 f \left (f \left (-4 a^2 f^2+3 a b e f+b^2 \left (e^2-6 d f\right )\right )-c \left (4 a f \left (e^2-3 d f\right )+b \left (e^3-5 d e f\right )\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (\sqrt {e^2-4 d f}+e\right )\right )-b \left (\sqrt {e^2-4 d f}+e\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{2 \sqrt {2} \left (e^2-4 d f\right )^{3/2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac {\sqrt {a+b x+c x^2} \left (f x \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+f \left (-a e f-2 b d f+b e^2\right )-c \left (e^3-3 d e f\right )\right )}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )} \]

[Out]

(f*(-a*e*f-2*b*d*f+b*e^2)-c*(-3*d*e*f+e^3)+f*(f*(-2*a*f+b*e)-c*(-2*d*f+e^2))*x)*(c*x^2+b*x+a)^(1/2)/(-4*d*f+e^
2)/((-a*f+c*d)^2-(-a*e+b*d)*(-b*f+c*e))/(f*x^2+e*x+d)+1/4*arctanh(1/4*(4*a*f+2*x*(b*f-c*(e-(-4*d*f+e^2)^(1/2))
)-b*(e-(-4*d*f+e^2)^(1/2)))*2^(1/2)/(c*x^2+b*x+a)^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(
1/2))^(1/2))*(-2*f*(2*c^2*d*(-4*d*f+e^2)+f*(3*a*b*e*f-4*a^2*f^2+b^2*(-6*d*f+e^2))-c*(4*a*f*(-3*d*f+e^2)+b*(-5*
d*e*f+e^3)))+f*(2*a*f-b*e+2*c*d)*(-b*f+c*e)*(e-(-4*d*f+e^2)^(1/2)))/(-4*d*f+e^2)^(3/2)/((-a*f+c*d)^2-(-a*e+b*d
)*(-b*f+c*e))*2^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)-1/4*arctanh(1/4*(4*a*f
-b*(e+(-4*d*f+e^2)^(1/2))+2*x*(b*f-c*(e+(-4*d*f+e^2)^(1/2))))*2^(1/2)/(c*x^2+b*x+a)^(1/2)/(c*e^2-2*c*d*f-b*e*f
+2*a*f^2+(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2))*(-2*f*(2*c^2*d*(-4*d*f+e^2)+f*(3*a*b*e*f-4*a^2*f^2+b^2*(-6*d*f+
e^2))-c*(4*a*f*(-3*d*f+e^2)+b*(-5*d*e*f+e^3)))+f*(2*a*f-b*e+2*c*d)*(-b*f+c*e)*(e+(-4*d*f+e^2)^(1/2)))/(-4*d*f+
e^2)^(3/2)/((-a*f+c*d)^2-(-a*e+b*d)*(-b*f+c*e))*2^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(-b*f+c*e)*(-4*d*f+e^2)^(
1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 8.21, antiderivative size = 787, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {974, 1032, 724, 206} \[ \frac {\left (f \left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) (2 a f-b e+2 c d)-2 f \left (-4 a^2 f^3+3 a b e f^2-4 a c f \left (e^2-3 d f\right )+b^2 f \left (e^2-6 d f\right )-b c \left (e^3-5 d e f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right )-b \left (e-\sqrt {e^2-4 d f}\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{2 \sqrt {2} \left (e^2-4 d f\right )^{3/2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac {\left (f \left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) (2 a f-b e+2 c d)-2 f \left (-4 a^2 f^3+3 a b e f^2-4 a c f \left (e^2-3 d f\right )+b^2 f \left (e^2-6 d f\right )-b c \left (e^3-5 d e f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (\sqrt {e^2-4 d f}+e\right )\right )-b \left (\sqrt {e^2-4 d f}+e\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{2 \sqrt {2} \left (e^2-4 d f\right )^{3/2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac {\sqrt {a+b x+c x^2} \left (f x \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+f \left (-a e f-2 b d f+b e^2\right )-c \left (e^3-3 d e f\right )\right )}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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 974

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[((2*a
*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p +
1)*(d + e*x + f*x^2)^(q + 1))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), x] - Dist[1/
((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*
x^2)^q*Simp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(
p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^
2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f*(p + 1) - c*e*(2*p +
q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e,
f, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e
- b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 1032

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )^2} \, dx &=\frac {\left (f \left (b e^2-2 b d f-a e f\right )-c \left (e^3-3 d e f\right )+f \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{\left (e^2-4 d f\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (d+e x+f x^2\right )}+\frac {\int \frac {\frac {1}{2} \left (3 a b e f^2-4 a^2 f^3+b^2 f \left (e^2-6 d f\right )+2 c^2 d \left (e^2-4 d f\right )-4 a c f \left (e^2-3 d f\right )-b c \left (e^3-5 d e f\right )\right )+\frac {1}{2} f (2 c d-b e+2 a f) (c e-b f) x}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{\left (e^2-4 d f\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\\ &=\frac {\left (f \left (b e^2-2 b d f-a e f\right )-c \left (e^3-3 d e f\right )+f \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{\left (e^2-4 d f\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (d+e x+f x^2\right )}+\frac {\left (-\frac {1}{2} f (2 c d-b e+2 a f) (c e-b f) \left (e-\sqrt {e^2-4 d f}\right )+f \left (3 a b e f^2-4 a^2 f^3+b^2 f \left (e^2-6 d f\right )+2 c^2 d \left (e^2-4 d f\right )-4 a c f \left (e^2-3 d f\right )-b c \left (e^3-5 d e f\right )\right )\right ) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+b x+c x^2}} \, dx}{\left (e^2-4 d f\right )^{3/2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\left (-\frac {1}{2} f (2 c d-b e+2 a f) (c e-b f) \left (e+\sqrt {e^2-4 d f}\right )+f \left (3 a b e f^2-4 a^2 f^3+b^2 f \left (e^2-6 d f\right )+2 c^2 d \left (e^2-4 d f\right )-4 a c f \left (e^2-3 d f\right )-b c \left (e^3-5 d e f\right )\right )\right ) \int \frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+b x+c x^2}} \, dx}{\left (e^2-4 d f\right )^{3/2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\\ &=\frac {\left (f \left (b e^2-2 b d f-a e f\right )-c \left (e^3-3 d e f\right )+f \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{\left (e^2-4 d f\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (d+e x+f x^2\right )}-\frac {\left (2 \left (-\frac {1}{2} f (2 c d-b e+2 a f) (c e-b f) \left (e-\sqrt {e^2-4 d f}\right )+f \left (3 a b e f^2-4 a^2 f^3+b^2 f \left (e^2-6 d f\right )+2 c^2 d \left (e^2-4 d f\right )-4 a c f \left (e^2-3 d f\right )-b c \left (e^3-5 d e f\right )\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16 a f^2-8 b f \left (e-\sqrt {e^2-4 d f}\right )+4 c \left (e-\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )-\left (-2 b f+2 c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {a+b x+c x^2}}\right )}{\left (e^2-4 d f\right )^{3/2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}+\frac {\left (2 \left (-\frac {1}{2} f (2 c d-b e+2 a f) (c e-b f) \left (e+\sqrt {e^2-4 d f}\right )+f \left (3 a b e f^2-4 a^2 f^3+b^2 f \left (e^2-6 d f\right )+2 c^2 d \left (e^2-4 d f\right )-4 a c f \left (e^2-3 d f\right )-b c \left (e^3-5 d e f\right )\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16 a f^2-8 b f \left (e+\sqrt {e^2-4 d f}\right )+4 c \left (e+\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )-\left (-2 b f+2 c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {a+b x+c x^2}}\right )}{\left (e^2-4 d f\right )^{3/2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\\ &=\frac {\left (f \left (b e^2-2 b d f-a e f\right )-c \left (e^3-3 d e f\right )+f \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{\left (e^2-4 d f\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (d+e x+f x^2\right )}+\frac {\left (f (2 c d-b e+2 a f) (c e-b f) \left (e-\sqrt {e^2-4 d f}\right )-2 f \left (3 a b e f^2-4 a^2 f^3+b^2 f \left (e^2-6 d f\right )+2 c^2 d \left (e^2-4 d f\right )-4 a c f \left (e^2-3 d f\right )-b c \left (e^3-5 d e f\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {2} \left (e^2-4 d f\right )^{3/2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}-\frac {\left (f (2 c d-b e+2 a f) (c e-b f) \left (e+\sqrt {e^2-4 d f}\right )-2 f \left (3 a b e f^2-4 a^2 f^3+b^2 f \left (e^2-6 d f\right )+2 c^2 d \left (e^2-4 d f\right )-4 a c f \left (e^2-3 d f\right )-b c \left (e^3-5 d e f\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {2} \left (e^2-4 d f\right )^{3/2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}}\\ \end {align*}

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Mathematica [A]  time = 6.69, size = 1377, normalized size = 1.75 \[ -\frac {8 \left (c x^2+b x+a\right ) f^3}{\left (e^2-4 d f\right ) \left (4 a f^2-2 b \left (e-\sqrt {e^2-4 d f}\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )^2\right ) \left (e+2 f x-\sqrt {e^2-4 d f}\right ) \sqrt {a+x (b+c x)}}-\frac {8 \left (c x^2+b x+a\right ) f^3}{\left (e^2-4 d f\right ) \left (4 a f^2-2 b \left (e+\sqrt {e^2-4 d f}\right ) f+c \left (e+\sqrt {e^2-4 d f}\right )^2\right ) \left (e+2 f x+\sqrt {e^2-4 d f}\right ) \sqrt {a+x (b+c x)}}+\frac {2 \sqrt {2} \sqrt {c x^2+b x+a} \tanh ^{-1}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right )+f \left (2 a f-b \left (e-\sqrt {e^2-4 d f}\right )\right )} \sqrt {c x^2+b x+a}}\right ) f^2}{\left (e^2-4 d f\right )^{3/2} \sqrt {c \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right )+f \left (2 a f-b \left (e-\sqrt {e^2-4 d f}\right )\right )} \sqrt {a+x (b+c x)}}-\frac {8 \sqrt {2} \sqrt {c e^2-b f e-c \sqrt {e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt {e^2-4 d f}} \left (2 b f+2 c \left (\sqrt {e^2-4 d f}-e\right )\right ) \sqrt {c x^2+b x+a} \tanh ^{-1}\left (\frac {-4 a f-b \left (\sqrt {e^2-4 d f}-e\right )-\left (2 b f+2 c \left (\sqrt {e^2-4 d f}-e\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-b f e-c \sqrt {e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt {e^2-4 d f}} \sqrt {c x^2+b x+a}}\right ) f^2}{\left (e^2-4 d f\right ) \left (4 a f^2+2 b \left (\sqrt {e^2-4 d f}-e\right ) f+c \left (\sqrt {e^2-4 d f}-e\right )^2\right ) \left (16 a f^2+8 b \left (\sqrt {e^2-4 d f}-e\right ) f+4 c \left (\sqrt {e^2-4 d f}-e\right )^2\right ) \sqrt {a+x (b+c x)}}-\frac {2 \sqrt {2} \sqrt {c x^2+b x+a} \tanh ^{-1}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )+f \left (2 a f-b \left (e+\sqrt {e^2-4 d f}\right )\right )} \sqrt {c x^2+b x+a}}\right ) f^2}{\left (e^2-4 d f\right )^{3/2} \sqrt {c \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )+f \left (2 a f-b \left (e+\sqrt {e^2-4 d f}\right )\right )} \sqrt {a+x (b+c x)}}-\frac {8 \sqrt {2} \sqrt {c e^2-b f e+c \sqrt {e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt {e^2-4 d f}} \left (2 c \left (e+\sqrt {e^2-4 d f}\right )-2 b f\right ) \sqrt {c x^2+b x+a} \tanh ^{-1}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )-\left (2 c \left (e+\sqrt {e^2-4 d f}\right )-2 b f\right ) x}{2 \sqrt {2} \sqrt {c e^2-b f e+c \sqrt {e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt {e^2-4 d f}} \sqrt {c x^2+b x+a}}\right ) f^2}{\left (e^2-4 d f\right ) \left (4 a f^2-2 b \left (e+\sqrt {e^2-4 d f}\right ) f+c \left (e+\sqrt {e^2-4 d f}\right )^2\right ) \left (16 a f^2-8 b \left (e+\sqrt {e^2-4 d f}\right ) f+4 c \left (e+\sqrt {e^2-4 d f}\right )^2\right ) \sqrt {a+x (b+c x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-8*f^3*(a + b*x + c*x^2))/((e^2 - 4*d*f)*(4*a*f^2 - 2*b*f*(e - Sqrt[e^2 - 4*d*f]) + c*(e - Sqrt[e^2 - 4*d*f])
^2)*(e - Sqrt[e^2 - 4*d*f] + 2*f*x)*Sqrt[a + x*(b + c*x)]) - (8*f^3*(a + b*x + c*x^2))/((e^2 - 4*d*f)*(4*a*f^2
- 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e + Sqrt[e^2 - 4*d*f])^2)*(e + Sqrt[e^2 - 4*d*f] + 2*f*x)*Sqrt[a + x*(b
+ c*x)]) + (2*Sqrt[2]*f^2*Sqrt[a + b*x + c*x^2]*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - S
qrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e - Sqrt[e^2 - 4*d
*f]))]*Sqrt[a + b*x + c*x^2])])/((e^2 - 4*d*f)^(3/2)*Sqrt[c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b
*(e - Sqrt[e^2 - 4*d*f]))]*Sqrt[a + x*(b + c*x)]) - (8*Sqrt[2]*f^2*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*
e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*(2*b*f + 2*c*(-e + Sqrt[e^2 - 4*d*f]))*Sqrt[a + b*x + c*x^2]*ArcT
anh[(-4*a*f - b*(-e + Sqrt[e^2 - 4*d*f]) - (2*b*f + 2*c*(-e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2
*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/((e^2 - 4*d
*f)*(4*a*f^2 + 2*b*f*(-e + Sqrt[e^2 - 4*d*f]) + c*(-e + Sqrt[e^2 - 4*d*f])^2)*(16*a*f^2 + 8*b*f*(-e + Sqrt[e^2
- 4*d*f]) + 4*c*(-e + Sqrt[e^2 - 4*d*f])^2)*Sqrt[a + x*(b + c*x)]) - (2*Sqrt[2]*f^2*Sqrt[a + b*x + c*x^2]*Arc
Tanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*(e^2 - 2*d*
f + e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e + Sqrt[e^2 - 4*d*f]))]*Sqrt[a + b*x + c*x^2])])/((e^2 - 4*d*f)^(3/2
)*Sqrt[c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e + Sqrt[e^2 - 4*d*f]))]*Sqrt[a + x*(b + c*x)]) -
(8*Sqrt[2]*f^2*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + c*e*Sqrt[e^2 - 4*d*f] - b*f*Sqrt[e^2 - 4*d*f]]*(-2*b*
f + 2*c*(e + Sqrt[e^2 - 4*d*f]))*Sqrt[a + b*x + c*x^2]*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) - (-2*b*f +
2*c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + c*e*Sqrt[e^2 - 4*d*f] - b*
f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/((e^2 - 4*d*f)*(4*a*f^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e +
Sqrt[e^2 - 4*d*f])^2)*(16*a*f^2 - 8*b*f*(e + Sqrt[e^2 - 4*d*f]) + 4*c*(e + Sqrt[e^2 - 4*d*f])^2)*Sqrt[a + x*(
b + c*x)])

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.03, size = 3858, normalized size = 4.89 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{2} + b x + a} {\left (f x^{2} + e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {c\,x^2+b\,x+a}\,{\left (f\,x^2+e\,x+d\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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