\(\int \frac {x^2 \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx\) [109]

Optimal result

Integrand size = 30, antiderivative size = 761 \[ \int \frac {x^2 \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx=-\frac {(4 c e-b f-2 c f x) \sqrt {a+b x+c x^2}}{4 c f^2}-\frac {\left (b^2 f^2+4 c f (b e-a f)-8 c^2 \left (e^2-d f\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2} f^3}-\frac {\left (c \left (e^4-4 d e^2 f+2 d^2 f^2-e^3 \sqrt {e^2-4 d f}+2 d e f \sqrt {e^2-4 d f}\right )+f \left (a f \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )-b \left (e^3-3 d e f-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 d f}\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+f \left (2 a f-b \left (e-\sqrt {e^2-4 d f}\right )\right )}}+\frac {\left (c \left (e^4-4 d e^2 f+2 d^2 f^2+e^3 \sqrt {e^2-4 d f}-2 d e f \sqrt {e^2-4 d f}\right )+f \left (a f \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )-b \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )+f \left (2 a f-b \left (e+\sqrt {e^2-4 d f}\right )\right )}} \]

[Out]

-1/8*(b^2*f^2+4*c*f*(-a*f+b*e)-8*c^2*(-d*f+e^2))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)/f^
3-1/4*(-2*c*f*x-b*f+4*c*e)*(c*x^2+b*x+a)^(1/2)/c/f^2-1/2*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))*(c*(e^4-4*d*e^2*f+2*d^2*f^2-e^3*(-4*d*f+e^2)^(1/2)+2*d*e*f*(-4*d*f+e^2)^(1/2))+f*(a*f*(e^2-2*d*f-e
*(-4*d*f+e^2)^(1/2))-b*(e^3-3*d*e*f-e^2*(-4*d*f+e^2)^(1/2)+d*f*(-4*d*f+e^2)^(1/2))))/f^3*2^(1/2)/(-4*d*f+e^2)^
(1/2)/(f*(2*a*f-b*(e-(-4*d*f+e^2)^(1/2)))+c*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))^(1/2)+1/2*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))*(c*(e^4-4*d*e^2*f+2*d^2*f^2+e^3*(-4*d*f+e^2)^(1/2)-2*d*e*f*(-4*d*f+
e^2)^(1/2))+f*(a*f*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2))-b*(e^3-3*d*e*f+e^2*(-4*d*f+e^2)^(1/2)-d*f*(-4*d*f+e^2)^(1/
2))))/f^3*2^(1/2)/(-4*d*f+e^2)^(1/2)/(c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2))+f*(2*a*f-b*(e+(-4*d*f+e^2)^(1/2))))^(
1/2)

Rubi [A] (verified)

Time = 2.02 (sec) , antiderivative size = 761, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1081, 1090, 635, 212, 1046, 738} \[ \int \frac {x^2 \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx=-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 c f (b e-a f)+b^2 f^2-8 c^2 \left (e^2-d f\right )\right )}{8 c^{3/2} f^3}-\frac {\left (f \left (a f \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )-b \left (-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 d f}-3 d e f+e^3\right )\right )+c \left (2 d^2 f^2-4 d e^2 f+2 d e f \sqrt {e^2-4 d f}-e^3 \sqrt {e^2-4 d f}+e^4\right )\right ) \text {arctanh}\left (\frac {4 a f+2 x \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right )-b \left (e-\sqrt {e^2-4 d f}\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {f \left (2 a f-b \left (e-\sqrt {e^2-4 d f}\right )\right )+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\left (f \left (a f \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )-b \left (e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}-3 d e f+e^3\right )\right )+c \left (2 d^2 f^2-4 d e^2 f-2 d e f \sqrt {e^2-4 d f}+e^3 \sqrt {e^2-4 d f}+e^4\right )\right ) \text {arctanh}\left (\frac {4 a f+2 x \left (b f-c \left (\sqrt {e^2-4 d f}+e\right )\right )-b \left (\sqrt {e^2-4 d f}+e\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {f \left (2 a f-b \left (\sqrt {e^2-4 d f}+e\right )\right )+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {\sqrt {a+b x+c x^2} (-b f+4 c e-2 c f x)}{4 c f^2} \]

[In]

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

[Out]

-1/4*((4*c*e - b*f - 2*c*f*x)*Sqrt[a + b*x + c*x^2])/(c*f^2) - ((b^2*f^2 + 4*c*f*(b*e - a*f) - 8*c^2*(e^2 - d*
f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(3/2)*f^3) - ((c*(e^4 - 4*d*e^2*f + 2*d^2*f^2
- e^3*Sqrt[e^2 - 4*d*f] + 2*d*e*f*Sqrt[e^2 - 4*d*f]) + f*(a*f*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) - b*(e^3 -
3*d*e*f - e^2*Sqrt[e^2 - 4*d*f] + d*f*Sqrt[e^2 - 4*d*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])])/(Sqrt[2]*f^3*Sqrt[e^2 - 4*d*f]*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]))]) + ((c*(e^4 - 4*d*e^2*f + 2*d^2*f^2 + e^3*Sqrt[e^2 - 4*d*f] - 2*d*e*f*Sqrt
[e^2 - 4*d*f]) + f*(a*f*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) - b*(e^3 - 3*d*e*f + e^2*Sqrt[e^2 - 4*d*f] - d*f*S
qrt[e^2 - 4*d*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*Sqr
t[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])])/(Sqrt[2]
*f^3*Sqrt[e^2 - 4*d*f]*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]))])

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 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 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]

Rule 1081

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

Rule 1090

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x
_)^2]), x_Symbol] :> Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + (B*c - b*C)*x)
/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c
, 0] && NeQ[e^2 - 4*d*f, 0]

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 1.14 (sec) , antiderivative size = 864, normalized size of antiderivative = 1.14 \[ \int \frac {x^2 \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx=\frac {2 \sqrt {c} f (-4 c e+b f+2 c f x) \sqrt {a+x (b+c x)}+\left (-b^2 f^2+4 c f (-b e+a f)+8 c^2 \left (e^2-d f\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-8 c^{3/2} \text {RootSum}\left [b^2 d-a b e+a^2 f-4 b \sqrt {c} d \text {$\#$1}+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2+b e \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {-b c d e^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a c e^3 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+b c d^2 f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+b^2 d e f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 a c d e f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a b e^2 f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a^2 e f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+2 c^{3/2} d e^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 c^{3/2} d^2 f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 b \sqrt {c} d e f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a \sqrt {c} d f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-c e^3 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 c d e f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+b e^2 f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-b d f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a e f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{2 b \sqrt {c} d-a \sqrt {c} e-4 c d \text {$\#$1}-b e \text {$\#$1}+2 a f \text {$\#$1}+3 \sqrt {c} e \text {$\#$1}^2-2 f \text {$\#$1}^3}\&\right ]}{8 c^{3/2} f^3} \]

[In]

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

[Out]

(2*Sqrt[c]*f*(-4*c*e + b*f + 2*c*f*x)*Sqrt[a + x*(b + c*x)] + (-(b^2*f^2) + 4*c*f*(-(b*e) + a*f) + 8*c^2*(e^2
- d*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - 8*c^(3/2)*RootSum[b^2*d - a*b*e + a^2*f - 4*b
*Sqrt[c]*d*#1 + 2*a*Sqrt[c]*e*#1 + 4*c*d*#1^2 + b*e*#1^2 - 2*a*f*#1^2 - 2*Sqrt[c]*e*#1^3 + f*#1^4 & , (-(b*c*d
*e^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]) + a*c*e^3*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]
+ b*c*d^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] + b^2*d*e*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2
] - #1] - 2*a*c*d*e*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - a*b*e^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + b
*x + c*x^2] - #1] + a^2*e*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] + 2*c^(3/2)*d*e^2*Log[-(Sqrt[c]*x
) + Sqrt[a + b*x + c*x^2] - #1]*#1 - 2*c^(3/2)*d^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - 2*b*S
qrt[c]*d*e*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 + 2*a*Sqrt[c]*d*f^2*Log[-(Sqrt[c]*x) + Sqrt[a +
b*x + c*x^2] - #1]*#1 - c*e^3*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2 + 2*c*d*e*f*Log[-(Sqrt[c]*x
) + Sqrt[a + b*x + c*x^2] - #1]*#1^2 + b*e^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2 - b*d*f^2*L
og[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2 - a*e*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1
^2)/(2*b*Sqrt[c]*d - a*Sqrt[c]*e - 4*c*d*#1 - b*e*#1 + 2*a*f*#1 + 3*Sqrt[c]*e*#1^2 - 2*f*#1^3) & ])/(8*c^(3/2)
*f^3)

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 1148, normalized size of antiderivative = 1.51

method	result	size
risch	\(\text {Expression too large to display}\)	\(1148\)
default	\(\text {Expression too large to display}\)	\(1666\)

[In]

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

[Out]

1/4*(2*c*f*x+b*f-4*c*e)*(c*x^2+b*x+a)^(1/2)/c/f^2+1/8/c/f^2*(1/f*(4*a*c*f^2-b^2*f^2-4*b*c*e*f-8*c^2*d*f+8*c^2*
e^2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+4/f^2*c*(a*e*f^2*(-4*d*f+e^2)^(1/2)+b*d*f^2*(-4*d*f+e
^2)^(1/2)-b*e^2*f*(-4*d*f+e^2)^(1/2)-2*c*d*e*f*(-4*d*f+e^2)^(1/2)+c*e^3*(-4*d*f+e^2)^(1/2)+2*a*d*f^3-a*e^2*f^2
-3*b*d*e*f^2+b*e^3*f-2*c*d^2*f^2+4*c*d*e^2*f-c*e^4)/(-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))))+4/f^2*c
*(a*e*f^2*(-4*d*f+e^2)^(1/2)+b*d*f^2*(-4*d*f+e^2)^(1/2)-b*e^2*f*(-4*d*f+e^2)^(1/2)-2*c*d*e*f*(-4*d*f+e^2)^(1/2
)+c*e^3*(-4*d*f+e^2)^(1/2)-2*a*d*f^3+a*e^2*f^2+3*b*d*e*f^2-b*e^3*f+2*c*d^2*f^2-4*c*d*e^2*f+c*e^4)/(-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)))

Fricas [F(-1)]

Timed out. \[ \int \frac {x^2 \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {x^2 \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx=\int \frac {x^{2} \sqrt {a + b x + c x^{2}}}{d + e x + f x^{2}}\, dx \]

[In]

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

[Out]

Integral(x**2*sqrt(a + b*x + c*x**2)/(d + e*x + f*x**2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*d*f-e^2>0)', see `assume?` f
or more deta

Giac [F(-2)]

Exception generated. \[ \int \frac {x^2 \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx=\int \frac {x^2\,\sqrt {c\,x^2+b\,x+a}}{f\,x^2+e\,x+d} \,d x \]

[In]

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

[Out]

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