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

Optimal. Leaf size=761 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (4 c f (b e-a f)+b^2 f^2-8 c^2 \left (e^2-d f\right )\right )}{8 c^{3/2} f^3}-\frac{\left (f \left (a f \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )-b \left (-e^2 \sqrt{e^2-4 d f}+d f \sqrt{e^2-4 d f}-3 d e f+e^3\right )\right )+c \left (2 d^2 f^2-e^3 \sqrt{e^2-4 d f}-4 d e^2 f+2 d e f \sqrt{e^2-4 d f}+e^4\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{f \left (2 a f-b \left (e-\sqrt{e^2-4 d f}\right )\right )+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\left (f \left (a f \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )-b \left (e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right )\right )+c \left (2 d^2 f^2+e^3 \sqrt{e^2-4 d f}-4 d e^2 f-2 d e f \sqrt{e^2-4 d f}+e^4\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{f \left (2 a f-b \left (\sqrt{e^2-4 d f}+e\right )\right )+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\sqrt{a+b x+c x^2} (-b f+4 c e-2 c f x)}{4 c f^2} \]

[Out]

-((4*c*e - b*f - 2*c*f*x)*Sqrt[a + b*x + c*x^2])/(4*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*Sqr
t[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]))])

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Rubi [A]  time = 3.13535, antiderivative size = 761, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1067, 1076, 621, 206, 1032, 724} \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (4 c f (b e-a f)+b^2 f^2-8 c^2 \left (e^2-d f\right )\right )}{8 c^{3/2} f^3}-\frac{\left (f \left (a f \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )-b \left (-e^2 \sqrt{e^2-4 d f}+d f \sqrt{e^2-4 d f}-3 d e f+e^3\right )\right )+c \left (2 d^2 f^2-e^3 \sqrt{e^2-4 d f}-4 d e^2 f+2 d e f \sqrt{e^2-4 d f}+e^4\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{f \left (2 a f-b \left (e-\sqrt{e^2-4 d f}\right )\right )+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\left (f \left (a f \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )-b \left (e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right )\right )+c \left (2 d^2 f^2+e^3 \sqrt{e^2-4 d f}-4 d e^2 f-2 d e f \sqrt{e^2-4 d f}+e^4\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{f \left (2 a f-b \left (\sqrt{e^2-4 d f}+e\right )\right )+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\sqrt{a+b x+c x^2} (-b f+4 c e-2 c f x)}{4 c f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((4*c*e - b*f - 2*c*f*x)*Sqrt[a + b*x + c*x^2])/(4*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*Sqr
t[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]))])

Rule 1067

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 1076

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]

Rule 621

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 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 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]

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]

Rubi steps

\begin{align*} \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{\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 ) \operatorname{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 ) \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 )}{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 ) \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 )}{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 [A]  time = 2.39381, size = 552, normalized size = 0.73 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \left (4 c f (a f-b e)-b^2 f^2+8 c^2 \left (e^2-d f\right )\right )}{8 c^{3/2} f^3}+\frac{f \sqrt{e^2-4 d f} \sqrt{a+x (b+c x)} (b f-4 c e+2 c f x)+\sqrt{2} c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right ) \sqrt{f \left (2 a f-b \left (\sqrt{e^2-4 d f}+e\right )\right )+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac{4 a f-b \left (\sqrt{e^2-4 d f}+e-2 f x\right )-2 c x \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+x (b+c x)} \sqrt{f \left (2 a f-b \left (\sqrt{e^2-4 d f}+e\right )\right )+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )+\sqrt{2} c \left (e \sqrt{e^2-4 d f}+2 d f-e^2\right ) \sqrt{f \left (2 a f+b \left (\sqrt{e^2-4 d f}-e\right )\right )+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac{4 a f+b \left (\sqrt{e^2-4 d f}-e+2 f x\right )+2 c x \left (\sqrt{e^2-4 d f}-e\right )}{2 \sqrt{2} \sqrt{a+x (b+c x)} \sqrt{f \left (2 a f+b \left (\sqrt{e^2-4 d f}-e\right )\right )+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{4 c f^3 \sqrt{e^2-4 d f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.312, size = 14815, normalized size = 19.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \sqrt{a + b x + c x^{2}}}{d + e x + f x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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