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

Optimal. Leaf size=795 \[ -\frac{(4 e-3 f x) \left (c x^2+a\right )^{3/2}}{12 f^2}-\frac{\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt{c x^2+a}}{8 f^4}+\frac{\left (3 a^2 f^4+12 a c \left (e^2-d f\right ) f^2+8 c^2 \left (e^4-3 d f e^2+d^2 f^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{c x^2+a}}\right )}{8 \sqrt{c} f^5}-\frac{\left (a^2 \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right ) f^4+2 a c \left (e^4-\sqrt{e^2-4 d f} e^3-4 d f e^2+2 d f \sqrt{e^2-4 d f} e+2 d^2 f^2\right ) f^2+c^2 \left (e^6-\sqrt{e^2-4 d f} e^5-6 d f e^4+4 d f \sqrt{e^2-4 d f} e^3+9 d^2 f^2 e^2-3 d^2 f^2 \sqrt{e^2-4 d f} e-2 d^3 f^3\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c \left (e-\sqrt{e^2-4 d f}\right ) x}{\sqrt{2} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}}\right )}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )}}+\frac{\left (a^2 \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right ) f^4+2 a c \left (e^4+\sqrt{e^2-4 d f} e^3-4 d f e^2-2 d f \sqrt{e^2-4 d f} e+2 d^2 f^2\right ) f^2+c^2 \left (e^6+\sqrt{e^2-4 d f} e^5-6 d f e^4-4 d f \sqrt{e^2-4 d f} e^3+9 d^2 f^2 e^2+3 d^2 f^2 \sqrt{e^2-4 d f} e-2 d^3 f^3\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c \left (e+\sqrt{e^2-4 d f}\right ) x}{\sqrt{2} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}}\right )}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )}} \]

[Out]

-((8*e*(a*f^2 + c*(e^2 - 2*d*f)) - f*(3*a*f^2 + 4*c*(e^2 - d*f))*x)*Sqrt[a + c*x^2])/(8*f^4) - ((4*e - 3*f*x)*
(a + c*x^2)^(3/2))/(12*f^2) + ((3*a^2*f^4 + 12*a*c*f^2*(e^2 - d*f) + 8*c^2*(e^4 - 3*d*e^2*f + d^2*f^2))*ArcTan
h[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(8*Sqrt[c]*f^5) - ((a^2*f^4*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + 2*a*c*f^2*(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]) + c^2*(e^6 - 6*d*e^4*f + 9*d^2
*e^2*f^2 - 2*d^3*f^3 - e^5*Sqrt[e^2 - 4*d*f] + 4*d*e^3*f*Sqrt[e^2 - 4*d*f] - 3*d^2*e*f^2*Sqrt[e^2 - 4*d*f]))*A
rcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sq
rt[a + c*x^2])])/(Sqrt[2]*f^5*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((a^2
*f^4*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + 2*a*c*f^2*(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]) + c^2*(e^6 - 6*d*e^4*f + 9*d^2*e^2*f^2 - 2*d^3*f^3 + e^5*Sqrt[e^2 - 4*d*f] - 4*d*e^3*
f*Sqrt[e^2 - 4*d*f] + 3*d^2*e*f^2*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*S
qrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^5*Sqrt[e^2 - 4*d*f]*Sqrt[2*
a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])

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Rubi [A]  time = 4.26402, antiderivative size = 795, 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 = {1069, 1068, 1080, 217, 206, 1034, 725} \[ -\frac{(4 e-3 f x) \left (c x^2+a\right )^{3/2}}{12 f^2}-\frac{\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt{c x^2+a}}{8 f^4}+\frac{\left (3 a^2 f^4+12 a c \left (e^2-d f\right ) f^2+8 c^2 \left (e^4-3 d f e^2+d^2 f^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{c x^2+a}}\right )}{8 \sqrt{c} f^5}-\frac{\left (a^2 \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right ) f^4+2 a c \left (e^4-\sqrt{e^2-4 d f} e^3-4 d f e^2+2 d f \sqrt{e^2-4 d f} e+2 d^2 f^2\right ) f^2+c^2 \left (e^6-\sqrt{e^2-4 d f} e^5-6 d f e^4+4 d f \sqrt{e^2-4 d f} e^3+9 d^2 f^2 e^2-3 d^2 f^2 \sqrt{e^2-4 d f} e-2 d^3 f^3\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c \left (e-\sqrt{e^2-4 d f}\right ) x}{\sqrt{2} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}}\right )}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )}}+\frac{\left (a^2 \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right ) f^4+2 a c \left (e^4+\sqrt{e^2-4 d f} e^3-4 d f e^2-2 d f \sqrt{e^2-4 d f} e+2 d^2 f^2\right ) f^2+c^2 \left (e^6+\sqrt{e^2-4 d f} e^5-6 d f e^4-4 d f \sqrt{e^2-4 d f} e^3+9 d^2 f^2 e^2+3 d^2 f^2 \sqrt{e^2-4 d f} e-2 d^3 f^3\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c \left (e+\sqrt{e^2-4 d f}\right ) x}{\sqrt{2} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}}\right )}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((8*e*(a*f^2 + c*(e^2 - 2*d*f)) - f*(3*a*f^2 + 4*c*(e^2 - d*f))*x)*Sqrt[a + c*x^2])/(8*f^4) - ((4*e - 3*f*x)*
(a + c*x^2)^(3/2))/(12*f^2) + ((3*a^2*f^4 + 12*a*c*f^2*(e^2 - d*f) + 8*c^2*(e^4 - 3*d*e^2*f + d^2*f^2))*ArcTan
h[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(8*Sqrt[c]*f^5) - ((a^2*f^4*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + 2*a*c*f^2*(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]) + c^2*(e^6 - 6*d*e^4*f + 9*d^2
*e^2*f^2 - 2*d^3*f^3 - e^5*Sqrt[e^2 - 4*d*f] + 4*d*e^3*f*Sqrt[e^2 - 4*d*f] - 3*d^2*e*f^2*Sqrt[e^2 - 4*d*f]))*A
rcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sq
rt[a + c*x^2])])/(Sqrt[2]*f^5*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((a^2
*f^4*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + 2*a*c*f^2*(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]) + c^2*(e^6 - 6*d*e^4*f + 9*d^2*e^2*f^2 - 2*d^3*f^3 + e^5*Sqrt[e^2 - 4*d*f] - 4*d*e^3*
f*Sqrt[e^2 - 4*d*f] + 3*d^2*e*f^2*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*S
qrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^5*Sqrt[e^2 - 4*d*f]*Sqrt[2*
a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])

Rule 1069

Int[((a_) + (c_.)*(x_)^2)^(p_)*((A_.) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Si
mp[((C*(-(c*e*(2*p + q + 2))) + 2*c*C*f*(p + q + 1)*x)*(a + 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 + c*x^2)^(p - 1)*(d + e*x +
f*x^2)^q*Simp[p*(-(a*e))*(C*(c*e)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p + q + 1)*(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)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p +
q + 1)*(C*e*f*p*(-4*a*c)))*x + (p*(c*e)*(C*(c*e)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(-
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
, c, d, e, f, A, C, q}, x] && 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 1068

Int[((a_) + (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*(-(c*e*(2*p + q + 2))) + 2*c*C*f*(p + q + 1)*x)*(a + 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 + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[p*(-(a*e))*(C*(c*e)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3
)) + (p + q + 1)*(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)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(-4*a*c)))*x + (p*(c*e)*(C*(c*e)*(q + 1
) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(-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] /; FreeQ[{a, c, d, e, f, A, B, C, q}, x] && 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 1080

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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 1034

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

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rubi steps

\begin{align*} \int \frac{x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx &=-\frac{(4 e-3 f x) \left (a+c x^2\right )^{3/2}}{12 f^2}-\frac{\int \frac{\sqrt{a+c x^2} \left (3 a c d f-3 c e (4 c d-a f) x-3 c \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x^2\right )}{d+e x+f x^2} \, dx}{12 c f^2}\\ &=-\frac{\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt{a+c x^2}}{8 f^4}-\frac{(4 e-3 f x) \left (a+c x^2\right )^{3/2}}{12 f^2}+\frac{\int \frac{-3 a c^2 d f \left (5 a f^2+4 c \left (e^2-d f\right )\right )-3 c^2 e \left (5 a^2 f^3+4 a c f \left (e^2-5 d f\right )-8 c^2 d \left (e^2-2 d f\right )\right ) x+3 c^2 \left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right ) x^2}{\sqrt{a+c x^2} \left (d+e x+f x^2\right )} \, dx}{24 c^2 f^4}\\ &=-\frac{\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt{a+c x^2}}{8 f^4}-\frac{(4 e-3 f x) \left (a+c x^2\right )^{3/2}}{12 f^2}+\frac{\int \frac{-3 a c^2 d f^2 \left (5 a f^2+4 c \left (e^2-d f\right )\right )-3 c^2 d \left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right )+\left (-3 c^2 e f \left (5 a^2 f^3+4 a c f \left (e^2-5 d f\right )-8 c^2 d \left (e^2-2 d f\right )\right )-3 c^2 e \left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right )\right ) x}{\sqrt{a+c x^2} \left (d+e x+f x^2\right )} \, dx}{24 c^2 f^5}+\frac{\left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{8 f^5}\\ &=-\frac{\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt{a+c x^2}}{8 f^4}-\frac{(4 e-3 f x) \left (a+c x^2\right )^{3/2}}{12 f^2}+\frac{\left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{8 f^5}+\frac{\left (a^2 f^4 \left (e^2-2 d f-e \sqrt{e^2-4 d f}\right )+2 a c f^2 \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 )+c^2 \left (e^6-6 d e^4 f+9 d^2 e^2 f^2-2 d^3 f^3-e^5 \sqrt{e^2-4 d f}+4 d e^3 f \sqrt{e^2-4 d f}-3 d^2 e f^2 \sqrt{e^2-4 d f}\right )\right ) \int \frac{1}{\left (e-\sqrt{e^2-4 d f}+2 f x\right ) \sqrt{a+c x^2}} \, dx}{f^5 \sqrt{e^2-4 d f}}-\frac{\left (a^2 f^4 \left (e^2-2 d f+e \sqrt{e^2-4 d f}\right )+2 a c f^2 \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 )+c^2 \left (e^6-6 d e^4 f+9 d^2 e^2 f^2-2 d^3 f^3+e^5 \sqrt{e^2-4 d f}-4 d e^3 f \sqrt{e^2-4 d f}+3 d^2 e f^2 \sqrt{e^2-4 d f}\right )\right ) \int \frac{1}{\left (e+\sqrt{e^2-4 d f}+2 f x\right ) \sqrt{a+c x^2}} \, dx}{f^5 \sqrt{e^2-4 d f}}\\ &=-\frac{\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt{a+c x^2}}{8 f^4}-\frac{(4 e-3 f x) \left (a+c x^2\right )^{3/2}}{12 f^2}+\frac{\left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c} f^5}-\frac{\left (a^2 f^4 \left (e^2-2 d f-e \sqrt{e^2-4 d f}\right )+2 a c f^2 \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 )+c^2 \left (e^6-6 d e^4 f+9 d^2 e^2 f^2-2 d^3 f^3-e^5 \sqrt{e^2-4 d f}+4 d e^3 f \sqrt{e^2-4 d f}-3 d^2 e f^2 \sqrt{e^2-4 d f}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a f^2+c \left (e-\sqrt{e^2-4 d f}\right )^2-x^2} \, dx,x,\frac{2 a f-c \left (e-\sqrt{e^2-4 d f}\right ) x}{\sqrt{a+c x^2}}\right )}{f^5 \sqrt{e^2-4 d f}}+\frac{\left (a^2 f^4 \left (e^2-2 d f+e \sqrt{e^2-4 d f}\right )+2 a c f^2 \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 )+c^2 \left (e^6-6 d e^4 f+9 d^2 e^2 f^2-2 d^3 f^3+e^5 \sqrt{e^2-4 d f}-4 d e^3 f \sqrt{e^2-4 d f}+3 d^2 e f^2 \sqrt{e^2-4 d f}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a f^2+c \left (e+\sqrt{e^2-4 d f}\right )^2-x^2} \, dx,x,\frac{2 a f-c \left (e+\sqrt{e^2-4 d f}\right ) x}{\sqrt{a+c x^2}}\right )}{f^5 \sqrt{e^2-4 d f}}\\ &=-\frac{\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt{a+c x^2}}{8 f^4}-\frac{(4 e-3 f x) \left (a+c x^2\right )^{3/2}}{12 f^2}+\frac{\left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c} f^5}-\frac{\left (a^2 f^4 \left (e^2-2 d f-e \sqrt{e^2-4 d f}\right )+2 a c f^2 \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 )+c^2 \left (e^6-6 d e^4 f+9 d^2 e^2 f^2-2 d^3 f^3-e^5 \sqrt{e^2-4 d f}+4 d e^3 f \sqrt{e^2-4 d f}-3 d^2 e f^2 \sqrt{e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c \left (e-\sqrt{e^2-4 d f}\right ) x}{\sqrt{2} \sqrt{2 a f^2+c \left (e^2-2 d f-e \sqrt{e^2-4 d f}\right )} \sqrt{a+c x^2}}\right )}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-2 d f-e \sqrt{e^2-4 d f}\right )}}+\frac{\left (a^2 f^4 \left (e^2-2 d f+e \sqrt{e^2-4 d f}\right )+2 a c f^2 \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 )+c^2 \left (e^6-6 d e^4 f+9 d^2 e^2 f^2-2 d^3 f^3+e^5 \sqrt{e^2-4 d f}-4 d e^3 f \sqrt{e^2-4 d f}+3 d^2 e f^2 \sqrt{e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c \left (e+\sqrt{e^2-4 d f}\right ) x}{\sqrt{2} \sqrt{2 a f^2+c \left (e^2-2 d f+e \sqrt{e^2-4 d f}\right )} \sqrt{a+c x^2}}\right )}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-2 d f+e \sqrt{e^2-4 d f}\right )}}\\ \end{align*}

Mathematica [A]  time = 3.7611, size = 793, normalized size = 1. \[ \frac{3 f \sqrt{a+c x^2} \left (\frac{3 a^{3/2} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{c} \sqrt{\frac{c x^2}{a}+1}}+5 a x+2 c x^3\right )-\frac{3 \left (\frac{2 d f-e^2}{\sqrt{e^2-4 d f}}+e\right ) \left (\frac{2 \left (2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )\right ) \left (-\sqrt{4 a f^2-2 c e \sqrt{e^2-4 d f}-4 c d f+2 c e^2} \tanh ^{-1}\left (\frac{2 a f+c x \left (\sqrt{e^2-4 d f}-e\right )}{\sqrt{a+c x^2} \sqrt{4 a f^2-2 c \left (e \sqrt{e^2-4 d f}+2 d f-e^2\right )}}\right )+\sqrt{c} \left (\sqrt{e^2-4 d f}-e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )+2 f \sqrt{a+c x^2}\right )}{f^2}+\frac{2 \sqrt{c} \sqrt{a+c x^2} \left (\sqrt{e^2-4 d f}-e\right ) \left (\sqrt{c} x \sqrt{\frac{c x^2}{a}+1}+\sqrt{a} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )\right )}{\sqrt{\frac{c x^2}{a}+1}}\right )}{2 f}+\frac{3 \left (\frac{e^2-2 d f}{\sqrt{e^2-4 d f}}+e\right ) \left (\frac{2 \left (2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )\right ) \left (\sqrt{4 a f^2+2 c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{a+c x^2} \sqrt{4 a f^2+2 c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )+\sqrt{c} \left (\sqrt{e^2-4 d f}+e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )-2 f \sqrt{a+c x^2}\right )}{f^2}+\frac{2 \sqrt{c} \sqrt{a+c x^2} \left (\sqrt{e^2-4 d f}+e\right ) \left (\sqrt{c} x \sqrt{\frac{c x^2}{a}+1}+\sqrt{a} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )\right )}{\sqrt{\frac{c x^2}{a}+1}}\right )}{2 f}-4 \left (a+c x^2\right )^{3/2} \left (\frac{e^2-2 d f}{\sqrt{e^2-4 d f}}+e\right )-4 \left (a+c x^2\right )^{3/2} \left (\frac{2 d f-e^2}{\sqrt{e^2-4 d f}}+e\right )}{24 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(e + (e^2 - 2*d*f)/Sqrt[e^2 - 4*d*f])*(a + c*x^2)^(3/2) - 4*(e + (-e^2 + 2*d*f)/Sqrt[e^2 - 4*d*f])*(a + c*
x^2)^(3/2) + 3*f*Sqrt[a + c*x^2]*(5*a*x + 2*c*x^3 + (3*a^(3/2)*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[c]*Sqrt[1 +
(c*x^2)/a])) - (3*(e + (-e^2 + 2*d*f)/Sqrt[e^2 - 4*d*f])*((2*Sqrt[c]*(-e + Sqrt[e^2 - 4*d*f])*Sqrt[a + c*x^2]
*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/a] + Sqrt[a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/Sqrt[1 + (c*x^2)/a] + (2*(2*a*f^2 + c
*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]))*(2*f*Sqrt[a + c*x^2] + Sqrt[c]*(-e + Sqrt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]
*x)/Sqrt[a + c*x^2]] - Sqrt[2*c*e^2 - 4*c*d*f + 4*a*f^2 - 2*c*e*Sqrt[e^2 - 4*d*f]]*ArcTanh[(2*a*f + c*(-e + Sq
rt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 - 2*c*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])]))/f^2))/(2*f)
+ (3*(e + (e^2 - 2*d*f)/Sqrt[e^2 - 4*d*f])*((2*Sqrt[c]*(e + Sqrt[e^2 - 4*d*f])*Sqrt[a + c*x^2]*(Sqrt[c]*x*Sqrt
[1 + (c*x^2)/a] + Sqrt[a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/Sqrt[1 + (c*x^2)/a] + (2*(2*a*f^2 + c*(e^2 - 2*d*f +
e*Sqrt[e^2 - 4*d*f]))*(-2*f*Sqrt[a + c*x^2] + Sqrt[c]*(e + Sqrt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x
^2]] + Sqrt[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(
Sqrt[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])]))/f^2))/(2*f))/(24*f^2)

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Maple [B]  time = 0.275, size = 19148, normalized size = 24.1 \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+a)^(3/2)/(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+a)^(3/2)/(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+a)^(3/2)/(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} \left (a + c x^{2}\right )^{\frac{3}{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+a)**(3/2)/(f*x**2+e*x+d),x)

[Out]

Integral(x**2*(a + c*x**2)**(3/2)/(d + e*x + f*x**2), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage2