3.2.13 \(\int \frac {1}{\sqrt {a+b x+c x^2} (d+e x+f x^2)^2} \, dx\) [113]

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

[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 = 7.68, 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 = {988, 1046, 738, 212} \begin {gather*} \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 )} \end {gather*}

Warning: Unable to verify antiderivative.

[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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

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 988

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 1046

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 ) \text {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 ) \text {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 = 12.95, size = 1369, normalized size = 1.74 \begin {gather*} \frac {-\frac {4 \sqrt {a+x (b+c x)} \left (c \left (e^3-3 d e f+e^2 f x-2 d f^2 x\right )+f \left (a f (e+2 f x)-b \left (e^2-2 d f+e f x\right )\right )\right )}{\left (e^2-4 d f\right ) (d+x (e+f x))}-\frac {\sqrt {2} f \left (8 a^2 f^3+2 a b f^2 \left (-4 e+\sqrt {e^2-4 d f}\right )-b^2 f \left (e^2-12 d f+e \sqrt {e^2-4 d f}\right )-2 c^2 d \left (e^2-8 d f+e \sqrt {e^2-4 d f}\right )-2 a c f \left (-5 e^2+12 d f+e \sqrt {e^2-4 d f}\right )+b c \left (e^3-12 d e f+e^2 \sqrt {e^2-4 d f}+2 d f \sqrt {e^2-4 d f}\right )\right ) \log \left (-e+\sqrt {e^2-4 d f}-2 f x\right )}{\left (e^2-4 d f\right )^{3/2} \sqrt {c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+f \left (2 a f+b \left (-e+\sqrt {e^2-4 d f}\right )\right )}}+\frac {\sqrt {2} f \left (8 a^2 f^3-2 a b f^2 \left (4 e+\sqrt {e^2-4 d f}\right )+2 a c f \left (5 e^2-12 d f+e \sqrt {e^2-4 d f}\right )+2 c^2 d \left (-e^2+8 d f+e \sqrt {e^2-4 d f}\right )+b^2 f \left (-e^2+12 d f+e \sqrt {e^2-4 d f}\right )+b c \left (e^3-12 d e f-e^2 \sqrt {e^2-4 d f}-2 d f \sqrt {e^2-4 d f}\right )\right ) \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\left (e^2-4 d f\right )^{3/2} \sqrt {c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )+f \left (2 a f-b \left (e+\sqrt {e^2-4 d f}\right )\right )}}+\frac {\sqrt {2} f \left (-8 a^2 f^3+2 a b f^2 \left (4 e+\sqrt {e^2-4 d f}\right )+b^2 f \left (e^2-12 d f-e \sqrt {e^2-4 d f}\right )+2 a c f \left (-5 e^2+12 d f-e \sqrt {e^2-4 d f}\right )-2 c^2 d \left (-e^2+8 d f+e \sqrt {e^2-4 d f}\right )+b c \left (-e^3+12 d e f+e^2 \sqrt {e^2-4 d f}+2 d f \sqrt {e^2-4 d f}\right )\right ) \log \left (-4 a f+2 c e x+2 c \sqrt {e^2-4 d f} x+b \left (e+\sqrt {e^2-4 d f}-2 f x\right )-2 \sqrt {2} \sqrt {c \left (e^2-2 d f+e \sqrt {e^2-4 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)}\right )}{\left (e^2-4 d f\right )^{3/2} \sqrt {c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )+f \left (2 a f-b \left (e+\sqrt {e^2-4 d f}\right )\right )}}+\frac {\sqrt {2} f \left (8 a^2 f^3+2 a b f^2 \left (-4 e+\sqrt {e^2-4 d f}\right )-b^2 f \left (e^2-12 d f+e \sqrt {e^2-4 d f}\right )-2 c^2 d \left (e^2-8 d f+e \sqrt {e^2-4 d f}\right )-2 a c f \left (-5 e^2+12 d f+e \sqrt {e^2-4 d f}\right )+b c \left (e^3-12 d e f+e^2 \sqrt {e^2-4 d f}+2 d f \sqrt {e^2-4 d f}\right )\right ) \log \left (b \left (-e+\sqrt {e^2-4 d f}+2 f x\right )+2 \left (2 a f-c e x+c \sqrt {e^2-4 d f} x+\sqrt {2} \sqrt {f \left (-b e+2 a f+b \sqrt {e^2-4 d f}\right )+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+x (b+c x)}\right )\right )}{\left (e^2-4 d f\right )^{3/2} \sqrt {c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+f \left (2 a f+b \left (-e+\sqrt {e^2-4 d f}\right )\right )}}}{4 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*Sqrt[a + x*(b + c*x)]*(c*(e^3 - 3*d*e*f + e^2*f*x - 2*d*f^2*x) + f*(a*f*(e + 2*f*x) - b*(e^2 - 2*d*f + e*
f*x))))/((e^2 - 4*d*f)*(d + x*(e + f*x))) - (Sqrt[2]*f*(8*a^2*f^3 + 2*a*b*f^2*(-4*e + Sqrt[e^2 - 4*d*f]) - b^2
*f*(e^2 - 12*d*f + e*Sqrt[e^2 - 4*d*f]) - 2*c^2*d*(e^2 - 8*d*f + e*Sqrt[e^2 - 4*d*f]) - 2*a*c*f*(-5*e^2 + 12*d
*f + e*Sqrt[e^2 - 4*d*f]) + b*c*(e^3 - 12*d*e*f + e^2*Sqrt[e^2 - 4*d*f] + 2*d*f*Sqrt[e^2 - 4*d*f]))*Log[-e + S
qrt[e^2 - 4*d*f] - 2*f*x])/((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[2]*f*(8*a^2*f^3 - 2*a*b*f^2*(4*e + Sqrt[e^2 - 4*d*f]) + 2*a*c*f*(5*e^2 - 12*d*
f + e*Sqrt[e^2 - 4*d*f]) + 2*c^2*d*(-e^2 + 8*d*f + e*Sqrt[e^2 - 4*d*f]) + b^2*f*(-e^2 + 12*d*f + e*Sqrt[e^2 -
4*d*f]) + b*c*(e^3 - 12*d*e*f - e^2*Sqrt[e^2 - 4*d*f] - 2*d*f*Sqrt[e^2 - 4*d*f]))*Log[e + Sqrt[e^2 - 4*d*f] +
2*f*x])/((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[2]*f*(-8*a^2*f^3 + 2*a*b*f^2*(4*e + Sqrt[e^2 - 4*d*f]) + b^2*f*(e^2 - 12*d*f - e*Sqrt[e^2 - 4*d*f]
) + 2*a*c*f*(-5*e^2 + 12*d*f - e*Sqrt[e^2 - 4*d*f]) - 2*c^2*d*(-e^2 + 8*d*f + e*Sqrt[e^2 - 4*d*f]) + b*c*(-e^3
+ 12*d*e*f + e^2*Sqrt[e^2 - 4*d*f] + 2*d*f*Sqrt[e^2 - 4*d*f]))*Log[-4*a*f + 2*c*e*x + 2*c*Sqrt[e^2 - 4*d*f]*x
+ b*(e + Sqrt[e^2 - 4*d*f] - 2*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 + x*(b + c*x)]])/((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[2]*f*(8*a^2*f^3 + 2*a*b*f^2*(-4*e + Sqrt[e^2 - 4*d*f]) - b^
2*f*(e^2 - 12*d*f + e*Sqrt[e^2 - 4*d*f]) - 2*c^2*d*(e^2 - 8*d*f + e*Sqrt[e^2 - 4*d*f]) - 2*a*c*f*(-5*e^2 + 12*
d*f + e*Sqrt[e^2 - 4*d*f]) + b*c*(e^3 - 12*d*e*f + e^2*Sqrt[e^2 - 4*d*f] + 2*d*f*Sqrt[e^2 - 4*d*f]))*Log[b*(-e
+ Sqrt[e^2 - 4*d*f] + 2*f*x) + 2*(2*a*f - c*e*x + c*Sqrt[e^2 - 4*d*f]*x + Sqrt[2]*Sqrt[f*(-(b*e) + 2*a*f + b*
Sqrt[e^2 - 4*d*f]) + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + x*(b + c*x)])])/((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]))]))/(4*(c^2*d^2 - b*c*d*e + f*(
b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f)))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2107\) vs. \(2(735)=1470\).
time = 0.16, size = 2108, normalized size = 2.67

method	result	size
default	\(\text {Expression too large to display}\)	\(2108\)

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,method=_RETURNVERBOSE)

[Out]

2*f/(4*d*f-e^2)/(-4*d*f+e^2)^(1/2)*2^(1/2)/((-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*
d*f+c*e^2)/f^2)^(1/2)*ln(((-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2+1/f
*(-c*(-4*d*f+e^2)^(1/2)+b*f-c*e)*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*2^(1/2)*((-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*
f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2)*(4*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c+4/f*(-c*(-4*d
*f+e^2)^(1/2)+b*f-c*e)*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+2*(-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*
f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2))/(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f))-2*f/(4*d*f-e^2)/(-4*d*f+e^2)^(1/2)*2^(1
/2)/((b*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2)*ln(((b*f*(-4*d*f+e
^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2+(c*(-4*d*f+e^2)^(1/2)+b*f-c*e)/f*(x-1/2/f*(-
e+(-4*d*f+e^2)^(1/2)))+1/2*2^(1/2)*((b*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2
)/f^2)^(1/2)*(4*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2*c+4*(c*(-4*d*f+e^2)^(1/2)+b*f-c*e)/f*(x-1/2/f*(-e+(-4*d*f+
e^2)^(1/2)))+2*(b*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2))/(x-1/2/
f*(-e+(-4*d*f+e^2)^(1/2))))-1/(4*d*f-e^2)*(-2/(b*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c
*d*f+c*e^2)*f^2/(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))*((x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2*c+(c*(-4*d*f+e^2)^(1/2)
+b*f-c*e)/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*(b*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f
-2*c*d*f+c*e^2)/f^2)^(1/2)+(c*(-4*d*f+e^2)^(1/2)+b*f-c*e)*f/(b*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a
*f^2-b*e*f-2*c*d*f+c*e^2)*2^(1/2)/((b*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)
/f^2)^(1/2)*ln(((b*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2+(c*(-4*d*f+e^2
)^(1/2)+b*f-c*e)/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*2^(1/2)*((b*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c
*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2)*(4*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2*c+4*(c*(-4*d*f+e^2)^(1/2)+b*
f-c*e)/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+2*(b*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*
d*f+c*e^2)/f^2)^(1/2))/(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))))-1/(4*d*f-e^2)*(-2/(-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+
e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)*f^2/(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)*((x+1/2*(e+(-4*d*f+e^2)^(1/2)
)/f)^2*c+1/f*(-c*(-4*d*f+e^2)^(1/2)+b*f-c*e)*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*(-b*f*(-4*d*f+e^2)^(1/2)+(-4
*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2)+f*(-c*(-4*d*f+e^2)^(1/2)+b*f-c*e)/(-b*f*(-4*d*f+e^
2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)*2^(1/2)/((-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1
/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2)*ln(((-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*
e*f-2*c*d*f+c*e^2)/f^2+1/f*(-c*(-4*d*f+e^2)^(1/2)+b*f-c*e)*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*2^(1/2)*((-b*f
*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2)*(4*(x+1/2*(e+(-4*d*f+e^2)^(
1/2))/f)^2*c+4/f*(-c*(-4*d*f+e^2)^(1/2)+b*f-c*e)*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+2*(-b*f*(-4*d*f+e^2)^(1/2)+(
-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

[Out]

Timed out

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

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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