3.107 \(\int \frac{(a+b x+c x^2)^{3/2}}{(d+e x+f x^2)^2} \, dx\)

Optimal. Leaf size=704 \[ -\frac{\left (\left (e-\sqrt{e^2-4 d f}\right ) (c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right )-2 f \left (f \left (-b e (3 a f+c d)+4 a f (a f+c d)+2 b^2 d f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{2 \sqrt{2} f^2 \left (e^2-4 d f\right )^{3/2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{\left (\left (\sqrt{e^2-4 d f}+e\right ) (c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right )-2 f \left (f \left (-b e (3 a f+c d)+4 a f (a f+c d)+2 b^2 d f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{2 \sqrt{2} f^2 \left (e^2-4 d f\right )^{3/2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{f^2}-\frac{\sqrt{a+b x+c x^2} (-2 b f+c e-2 c f x)}{f \left (e^2-4 d f\right )}-\frac{(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 11.9496, antiderivative size = 704, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {971, 1066, 1076, 621, 206, 1032, 724} \[ -\frac{\left (\left (e-\sqrt{e^2-4 d f}\right ) (c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right )-2 f \left (f \left (-b e (3 a f+c d)+4 a f (a f+c d)+2 b^2 d f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{2 \sqrt{2} f^2 \left (e^2-4 d f\right )^{3/2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{\left (\left (\sqrt{e^2-4 d f}+e\right ) (c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right )-2 f \left (f \left (-b e (3 a f+c d)+4 a f (a f+c d)+2 b^2 d f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{2 \sqrt{2} f^2 \left (e^2-4 d f\right )^{3/2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{f^2}-\frac{\sqrt{a+b x+c x^2} (-2 b f+c e-2 c f x)}{f \left (e^2-4 d f\right )}-\frac{(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2)^2,x]

[Out]

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

Rule 971

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

Rule 1066

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[((B*c*f*(2*p + 2*q + 3) + 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 - B*f)*(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*
(B*e - 2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(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 - B*e + 2*A*f)*(2*p + 2*q +
3))))*x + (p*(c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(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 - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, 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{\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^2} \, dx &=-\frac{(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}-\frac{\int \frac{\sqrt{a+b x+c x^2} \left (\frac{1}{2} (3 b e-4 a f)+(3 c e+b f) x+4 c f x^2\right )}{d+e x+f x^2} \, dx}{-e^2+4 d f}\\ &=-\frac{(c e-2 b f-2 c f x) \sqrt{a+b x+c x^2}}{f \left (e^2-4 d f\right )}-\frac{(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}-\frac{\int \frac{c f \left (2 b^2 d f+4 a f (c d+a f)-b e (c d+3 a f)\right )-c f \left (2 c^2 d e+2 a c e f+b f (b e-2 a f)+b c \left (e^2-10 d f\right )\right ) x-2 c^3 f \left (e^2-4 d f\right ) x^2}{\sqrt{a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{2 c f^2 \left (e^2-4 d f\right )}\\ &=-\frac{(c e-2 b f-2 c f x) \sqrt{a+b x+c x^2}}{f \left (e^2-4 d f\right )}-\frac{(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}+\frac{c^2 \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{f^2}-\frac{\int \frac{2 c^3 d f \left (e^2-4 d f\right )+c f^2 \left (2 b^2 d f+4 a f (c d+a f)-b e (c d+3 a f)\right )+\left (2 c^3 e f \left (e^2-4 d f\right )-c f^2 \left (2 c^2 d e+2 a c e f+b f (b e-2 a f)+b c \left (e^2-10 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 \left (e^2-4 d f\right )}\\ &=-\frac{(c e-2 b f-2 c f x) \sqrt{a+b x+c x^2}}{f \left (e^2-4 d f\right )}-\frac{(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}+\frac{\left (2 c^2\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 )}{f^2}+\frac{\left ((c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right ) \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 (2 b^2 d f+4 a f (c d+a f)-b e (c d+3 a 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}{2 f^2 \left (e^2-4 d f\right )^{3/2}}-\frac{\left ((c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right ) \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 (2 b^2 d f+4 a f (c d+a f)-b e (c d+3 a 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}{2 f^2 \left (e^2-4 d f\right )^{3/2}}\\ &=-\frac{(c e-2 b f-2 c f x) \sqrt{a+b x+c x^2}}{f \left (e^2-4 d f\right )}-\frac{(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}+\frac{c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{f^2}-\frac{\left ((c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right ) \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 (2 b^2 d f+4 a f (c d+a f)-b e (c d+3 a f)\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^2 \left (e^2-4 d f\right )^{3/2}}+\frac{\left ((c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right ) \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 (2 b^2 d f+4 a f (c d+a f)-b e (c d+3 a f)\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^2 \left (e^2-4 d f\right )^{3/2}}\\ &=-\frac{(c e-2 b f-2 c f x) \sqrt{a+b x+c x^2}}{f \left (e^2-4 d f\right )}-\frac{(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}+\frac{c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{f^2}-\frac{\left ((c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right ) \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 (2 b^2 d f+4 a f (c d+a f)-b e (c d+3 a 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} f^2 \left (e^2-4 d f\right )^{3/2} \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}}}+\frac{\left ((c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right ) \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 (2 b^2 d f+4 a f (c d+a f)-b e (c d+3 a 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} f^2 \left (e^2-4 d f\right )^{3/2} \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}}}\\ \end{align*}

Mathematica [B]  time = 6.81996, size = 2843, normalized size = 4.04 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2)^2,x]

[Out]

(-2*f*(a + x*(b + c*x))^(3/2))/((e^2 - 4*d*f)*(e - Sqrt[e^2 - 4*d*f] + 2*f*x)) - (2*f*(a + x*(b + c*x))^(3/2))
/((e^2 - 4*d*f)*(e + Sqrt[e^2 - 4*d*f] + 2*f*x)) - (3*f*(a + x*(b + c*x))^(3/2)*(((-4*b*c*f - 2*c*(b*f + 2*c*(
-e + Sqrt[e^2 - 4*d*f])) - 4*c^2*f*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^2) - ((2*Sqrt[c]*(b^2*f^2 + 4*c^2*(e^2 - 2
*d*f - e*Sqrt[e^2 - 4*d*f]) + 4*c*f*(a*f - b*(e - Sqrt[e^2 - 4*d*f])))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a +
b*x + c*x^2])])/f + (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]]*(4*c*f*(8*a*b*f^2 - 3*b^2*f*(e - Sqrt[e^2 - 4*d*f]) - 4*a*c*f*(e - Sqrt[e^2 - 4*d*f]) + 4*b*c*(e^2 -
2*d*f - e*Sqrt[e^2 - 4*d*f])) + 4*c*(-e + Sqrt[e^2 - 4*d*f])*(b^2*f^2 + 4*c^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d
*f]) + 4*c*f*(a*f - b*(e - Sqrt[e^2 - 4*d*f]))))*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*Sq
rt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(f*(16*a*f^2 + 8*b*f*(-e + Sqrt[e^2 - 4*d*f]) + 4*c*(-e + Sqrt[e^2 -
4*d*f])^2)))/(16*c*f^2)))/((e^2 - 4*d*f)*(a + b*x + c*x^2)^(3/2)) + (f*(a + x*(b + c*x))^(3/2)*(((-4*c*f*(4*a
*f - b*(e - Sqrt[e^2 - 4*d*f])) - 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*(b*f + 2*c*(-e + Sqrt[e^2 - 4*d*f])) - 4
*c*f*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^2) - ((-2*(b*f - c*(e - Sqrt[e^2 - 4*d
*f]))*(b^2*f^2 - 4*c^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) - 4*c*f*(3*a*f - b*(e - Sqrt[e^2 - 4*d*f])))*ArcTan
h[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*f) + (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]]*(-4*(-e + Sqrt[e^2 - 4*d*f])*(b*f - c*(e - Sqrt[e^2 - 4*d
*f]))*(b^2*f^2 - 4*c^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) - 4*c*f*(3*a*f - b*(e - Sqrt[e^2 - 4*d*f]))) + 4*f*
(2*c*f*(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]))^2 - (e - Sqrt[e^2 - 4*d*f])*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*(b^2*
f + 4*a*c*f - 2*b*c*(e - Sqrt[e^2 - 4*d*f]))))*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])])/(f*(16*a*f^2 + 8*b*f*(-e + Sqrt[e^2 - 4*d*f]) + 4*c*(-e + Sqrt[e^2 - 4
*d*f])^2)))/(16*c*f^2)))/((e^2 - 4*d*f)^(3/2)*(a + b*x + c*x^2)^(3/2)) - (f*(a + x*(b + c*x))^(3/2)*(((4*c*f*(
-4*a*f + b*(e + Sqrt[e^2 - 4*d*f])) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*(-(b*f) + 2*c*(e + Sqrt[e^2 - 4*d*f]
)) - 4*c*f*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^2) - ((-2*(b*f - c*(e + Sqrt[e^2
- 4*d*f]))*(b^2*f^2 - 4*c^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) - 4*c*f*(3*a*f - b*(e + Sqrt[e^2 - 4*d*f])))*
ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*f) - (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]]*(4*(e + Sqrt[e^2 - 4*d*f])*(b*f - c*(e + Sqrt[e^2 -
4*d*f]))*(b^2*f^2 - 4*c^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) - 4*c*f*(3*a*f - b*(e + Sqrt[e^2 - 4*d*f]))) +
4*f*(2*c*f*(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]))^2 - (e + Sqrt[e^2 - 4*d*f])*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*(
b^2*f + 4*a*c*f - 2*b*c*(e + Sqrt[e^2 - 4*d*f]))))*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*Sq
rt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(f*(16*a*f^2 - 8*b*f*(e + Sqrt[e^2 - 4*d*f]) + 4*c*(e + Sqrt[e^2 - 4
*d*f])^2)))/(16*c*f^2)))/((e^2 - 4*d*f)^(3/2)*(a + b*x + c*x^2)^(3/2)) + (3*f*(a + x*(b + c*x))^(3/2)*(((4*b*c
*f - 2*c*(-(b*f) + 2*c*(e + Sqrt[e^2 - 4*d*f])) + 4*c^2*f*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^2) - ((-2*Sqrt[c]*(
b^2*f^2 + 4*c^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + 4*c*f*(a*f - b*(e + Sqrt[e^2 - 4*d*f])))*ArcTanh[(b + 2*
c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/f - (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]]*(4*c*(e + Sqrt[e^2 - 4*d*f])*(b^2*f^2 + 4*c^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*
d*f]) + 4*c*f*(a*f - b*(e + Sqrt[e^2 - 4*d*f]))) + 4*c*f*(3*b^2*f*(e + Sqrt[e^2 - 4*d*f]) + 4*a*c*f*(e + Sqrt[
e^2 - 4*d*f]) - 4*b*(2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))))*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])])/(f*(16*a*f^2 - 8*b*f*(e + Sqrt[e^2 - 4*d*f]) +
4*c*(e + Sqrt[e^2 - 4*d*f])^2)))/(16*c*f^2)))/((e^2 - 4*d*f)*(a + b*x + c*x^2)^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.369, size = 72576, normalized size = 103.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d)^2,x)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (f x^{2} + e x + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d)^2,x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)/(f*x^2 + e*x + d)^2, x)

________________________________________________________________________________________

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((c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d)^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [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((c*x**2+b*x+a)**(3/2)/(f*x**2+e*x+d)**2,x)

[Out]

Timed out

________________________________________________________________________________________

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((c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError