3.9 \(\int \frac{A+B x}{(a+b x+c x^2)^{5/2} (d-f x^2)} \, dx\)

Optimal. Leaf size=797 \[ -\frac{\left (B \sqrt{d}-A \sqrt{f}\right ) \tanh ^{-1}\left (\frac{-2 \sqrt{f} a+\left (2 c \sqrt{d}-b \sqrt{f}\right ) x+b \sqrt{d}}{2 \sqrt{-\sqrt{d} \sqrt{f} b+c d+a f} \sqrt{c x^2+b x+a}}\right ) f^{3/2}}{2 \sqrt{d} \left (-\sqrt{d} \sqrt{f} b+c d+a f\right )^{5/2}}+\frac{\left (\sqrt{f} A+B \sqrt{d}\right ) \tanh ^{-1}\left (\frac{2 \sqrt{f} a+\left (\sqrt{f} b+2 c \sqrt{d}\right ) x+b \sqrt{d}}{2 \sqrt{\sqrt{d} \sqrt{f} b+c d+a f} \sqrt{c x^2+b x+a}}\right ) f^{3/2}}{2 \sqrt{d} \left (\sqrt{d} \sqrt{f} b+c d+a f\right )^{5/2}}-\frac{2 \left (3 B d f^2 b^6-A f^2 (7 c d+6 a f) b^5-B f \left (7 c^2 d^2+14 a c f d-3 a^2 f^2\right ) b^4+A c f \left (15 c^2 d^2+46 a c f d+43 a^2 f^2\right ) b^3+2 B c \left (2 c^3 d^3+5 a c^2 f d^2+4 a^2 c f^2 d-11 a^3 f^3\right ) b^2-4 A c^2 \left (2 c^3 d^3+9 a c^2 f d^2+24 a^2 c f^2 d+17 a^3 f^3\right ) b+24 a^2 B c^2 f (c d+a f)^2+c \left (3 B d f^2 b^5-2 A f^2 (4 c d+3 a f) b^4-B f \left (17 c^2 d^2+10 a c f d-3 a^2 f^2\right ) b^3+2 A c f \left (15 c^2 d^2+22 a c f d+19 a^2 f^2\right ) b^2+4 B c \left (2 c^3 d^3+11 a c^2 f d^2+4 a^2 c f^2 d-5 a^3 f^3\right ) b-8 A c^2 (c d+a f)^2 (2 c d+5 a f)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c^2 d^2+2 a c f d-f \left (b^2 d-a^2 f\right )\right )^2 \sqrt{c x^2+b x+a}}-\frac{2 \left (a B \left (-f b^2+2 c^2 d+2 a c f\right )+A \left (b^3 f-b c (c d+3 a f)\right )+c \left (A f b^2+B (c d-a f) b-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (c x^2+b x+a\right )^{3/2}} \]

[Out]

(-2*(a*B*(2*c^2*d - b^2*f + 2*a*c*f) + A*(b^3*f - b*c*(c*d + 3*a*f)) + c*(A*b^2*f + b*B*(c*d - a*f) - 2*A*c*(c
*d + a*f))*x))/(3*(b^2 - 4*a*c)*(b^2*d*f - (c*d + a*f)^2)*(a + b*x + c*x^2)^(3/2)) - (2*(3*b^6*B*d*f^2 + 24*a^
2*B*c^2*f*(c*d + a*f)^2 - A*b^5*f^2*(7*c*d + 6*a*f) - b^4*B*f*(7*c^2*d^2 + 14*a*c*d*f - 3*a^2*f^2) + A*b^3*c*f
*(15*c^2*d^2 + 46*a*c*d*f + 43*a^2*f^2) + 2*b^2*B*c*(2*c^3*d^3 + 5*a*c^2*d^2*f + 4*a^2*c*d*f^2 - 11*a^3*f^3) -
4*A*b*c^2*(2*c^3*d^3 + 9*a*c^2*d^2*f + 24*a^2*c*d*f^2 + 17*a^3*f^3) + c*(3*b^5*B*d*f^2 - 2*A*b^4*f^2*(4*c*d +
3*a*f) - 8*A*c^2*(c*d + a*f)^2*(2*c*d + 5*a*f) - b^3*B*f*(17*c^2*d^2 + 10*a*c*d*f - 3*a^2*f^2) + 2*A*b^2*c*f*
(15*c^2*d^2 + 22*a*c*d*f + 19*a^2*f^2) + 4*b*B*c*(2*c^3*d^3 + 11*a*c^2*d^2*f + 4*a^2*c*d*f^2 - 5*a^3*f^3))*x))
/(3*(b^2 - 4*a*c)^2*(c^2*d^2 + 2*a*c*d*f - f*(b^2*d - a^2*f))^2*Sqrt[a + b*x + c*x^2]) - ((B*Sqrt[d] - A*Sqrt[
f])*f^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] +
a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(5/2)) + ((B*Sqrt[d] + A*Sqrt[f])*f^(
3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sq
rt[a + b*x + c*x^2])])/(2*Sqrt[d]*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 1.87008, antiderivative size = 796, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1018, 1064, 1033, 724, 206} \[ -\frac{\left (B \sqrt{d}-A \sqrt{f}\right ) \tanh ^{-1}\left (\frac{-2 \sqrt{f} a+\left (2 c \sqrt{d}-b \sqrt{f}\right ) x+b \sqrt{d}}{2 \sqrt{-\sqrt{d} \sqrt{f} b+c d+a f} \sqrt{c x^2+b x+a}}\right ) f^{3/2}}{2 \sqrt{d} \left (-\sqrt{d} \sqrt{f} b+c d+a f\right )^{5/2}}+\frac{\left (\sqrt{f} A+B \sqrt{d}\right ) \tanh ^{-1}\left (\frac{2 \sqrt{f} a+\left (\sqrt{f} b+2 c \sqrt{d}\right ) x+b \sqrt{d}}{2 \sqrt{\sqrt{d} \sqrt{f} b+c d+a f} \sqrt{c x^2+b x+a}}\right ) f^{3/2}}{2 \sqrt{d} \left (\sqrt{d} \sqrt{f} b+c d+a f\right )^{5/2}}-\frac{2 \left (3 B d f^2 b^6-A f^2 (7 c d+6 a f) b^5-B f \left (7 c^2 d^2+14 a c f d-3 a^2 f^2\right ) b^4+A c f \left (15 c^2 d^2+46 a c f d+43 a^2 f^2\right ) b^3+2 B c \left (2 c^3 d^3+5 a c^2 f d^2+4 a^2 c f^2 d-11 a^3 f^3\right ) b^2-4 A c^2 \left (2 c^3 d^3+9 a c^2 f d^2+24 a^2 c f^2 d+17 a^3 f^3\right ) b+24 a^2 B c^2 f (c d+a f)^2+c \left (3 B d f^2 b^5-2 A f^2 (4 c d+3 a f) b^4-B f \left (17 c^2 d^2+10 a c f d-3 a^2 f^2\right ) b^3+2 A c f \left (15 c^2 d^2+22 a c f d+19 a^2 f^2\right ) b^2+4 B c \left (2 c^3 d^3+11 a c^2 f d^2+4 a^2 c f^2 d-5 a^3 f^3\right ) b-8 A c^2 (c d+a f)^2 (2 c d+5 a f)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c^2 d^2+2 a c f d-f \left (b^2 d-a^2 f\right )\right )^2 \sqrt{c x^2+b x+a}}-\frac{2 \left (A f b^3-A c (c d+3 a f) b+a B \left (-f b^2+2 c^2 d+2 a c f\right )+c \left (A f b^2+B (c d-a f) b-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (c x^2+b x+a\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((a + b*x + c*x^2)^(5/2)*(d - f*x^2)),x]

[Out]

(-2*(A*b^3*f - A*b*c*(c*d + 3*a*f) + a*B*(2*c^2*d - b^2*f + 2*a*c*f) + c*(A*b^2*f + b*B*(c*d - a*f) - 2*A*c*(c
*d + a*f))*x))/(3*(b^2 - 4*a*c)*(b^2*d*f - (c*d + a*f)^2)*(a + b*x + c*x^2)^(3/2)) - (2*(3*b^6*B*d*f^2 + 24*a^
2*B*c^2*f*(c*d + a*f)^2 - A*b^5*f^2*(7*c*d + 6*a*f) - b^4*B*f*(7*c^2*d^2 + 14*a*c*d*f - 3*a^2*f^2) + A*b^3*c*f
*(15*c^2*d^2 + 46*a*c*d*f + 43*a^2*f^2) + 2*b^2*B*c*(2*c^3*d^3 + 5*a*c^2*d^2*f + 4*a^2*c*d*f^2 - 11*a^3*f^3) -
4*A*b*c^2*(2*c^3*d^3 + 9*a*c^2*d^2*f + 24*a^2*c*d*f^2 + 17*a^3*f^3) + c*(3*b^5*B*d*f^2 - 2*A*b^4*f^2*(4*c*d +
3*a*f) - 8*A*c^2*(c*d + a*f)^2*(2*c*d + 5*a*f) - b^3*B*f*(17*c^2*d^2 + 10*a*c*d*f - 3*a^2*f^2) + 2*A*b^2*c*f*
(15*c^2*d^2 + 22*a*c*d*f + 19*a^2*f^2) + 4*b*B*c*(2*c^3*d^3 + 11*a*c^2*d^2*f + 4*a^2*c*d*f^2 - 5*a^3*f^3))*x))
/(3*(b^2 - 4*a*c)^2*(c^2*d^2 + 2*a*c*d*f - f*(b^2*d - a^2*f))^2*Sqrt[a + b*x + c*x^2]) - ((B*Sqrt[d] - A*Sqrt[
f])*f^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] +
a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(5/2)) + ((B*Sqrt[d] + A*Sqrt[f])*f^(
3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sq
rt[a + b*x + c*x^2])])/(2*Sqrt[d]*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(5/2))

Rule 1018

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp
[((a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q + 1)*((g*c)*(-(b*(c*d + a*f))) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(
2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(2*a*f)) - h*(b*c*d + a*b*f))*x))/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)
*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + f
*x^2)^q*Simp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d)*(-(b*f)))*(p + 1) + (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*
(c*d - a*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f*((g*c)*(-(b*(c*d + a*f))) + (g*b - a*h)*(2*c^2*d + b^2*f - c*
(2*a*f)))*(p + q + 2) - (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(b*f*(p + 1)))*x - c*f*(b^2*(g*f
) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}
, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] &&  !( !IntegerQ[p] && ILtQ[q, -1
])

Rule 1064

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (f_.)*(x_)^2)^(q_), x_
Symbol] :> Simp[((a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q + 1)*((A*c - a*C)*(-(b*(c*d + a*f))) + (A*b - a*B)*(
2*c^2*d + b^2*f - c*(2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(2*a*f)) - B*(b*c*d + a*b*f) + C*(b^2*d - 2*a*(c*d -
a*f)))*x))/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f
)^2)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d)*(
-(b*f)))*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(a*f*(p + 1
) - c*d*(p + 2)) - (2*f*((A*c - a*C)*(-(b*(c*d + a*f))) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(2*a*f)))*(p + q +
2) - (b^2*(C*d + A*f) - b*(B*c*d + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(b*f*(p + 1)))*x - c*f*(b
^2*(C*d + A*f) - b*(B*c*d + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]
/; FreeQ[{a, b, c, d, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2,
0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 1033

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

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])

Rubi steps

\begin{align*} \int \frac{A+B x}{\left (a+b x+c x^2\right )^{5/2} \left (d-f x^2\right )} \, dx &=-\frac{2 \left (A b^3 f-A b c (c d+3 a f)+a B \left (2 c^2 d-b^2 f+2 a c f\right )+c \left (A b^2 f+b B (c d-a f)-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{2 \int \frac{\frac{1}{2} \left (3 b^3 B d f-4 b B c d (c d+2 a f)-A b^2 f (7 c d+3 a f)+4 A c \left (2 c^2 d^2+5 a c d f+3 a^2 f^2\right )\right )+\frac{3}{2} \left (b^2-4 a c\right ) f (A b f-B (c d+a f)) x+2 c f \left (A b^2 f+b B (c d-a f)-2 A c (c d+a f)\right ) x^2}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right )}\\ &=-\frac{2 \left (A b^3 f-A b c (c d+3 a f)+a B \left (2 c^2 d-b^2 f+2 a c f\right )+c \left (A b^2 f+b B (c d-a f)-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (3 b^6 B d f^2+24 a^2 B c^2 f (c d+a f)^2-A b^5 f^2 (7 c d+6 a f)-b^4 B f \left (7 c^2 d^2+14 a c d f-3 a^2 f^2\right )+A b^3 c f \left (15 c^2 d^2+46 a c d f+43 a^2 f^2\right )+2 b^2 B c \left (2 c^3 d^3+5 a c^2 d^2 f+4 a^2 c d f^2-11 a^3 f^3\right )-4 A b c^2 \left (2 c^3 d^3+9 a c^2 d^2 f+24 a^2 c d f^2+17 a^3 f^3\right )+c \left (3 b^5 B d f^2-2 A b^4 f^2 (4 c d+3 a f)-8 A c^2 (c d+a f)^2 (2 c d+5 a f)-b^3 B f \left (17 c^2 d^2+10 a c d f-3 a^2 f^2\right )+2 A b^2 c f \left (15 c^2 d^2+22 a c d f+19 a^2 f^2\right )+4 b B c \left (2 c^3 d^3+11 a c^2 d^2 f+4 a^2 c d f^2-5 a^3 f^3\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (b^2 d f-(c d+a f)^2\right )^2 \sqrt{a+b x+c x^2}}+\frac{4 \int \frac{\frac{3}{4} \left (b^2-4 a c\right )^2 f^2 \left (A b^2 d f-2 b B d (c d+a f)+A (c d+a f)^2\right )-\frac{3}{4} \left (b^2-4 a c\right )^2 f^2 \left (2 A b f (c d+a f)-B \left (c^2 d^2+2 a c d f+f \left (b^2 d+a^2 f\right )\right )\right ) x}{\sqrt{a+b x+c x^2} \left (d-f x^2\right )} \, dx}{3 \left (b^2-4 a c\right )^2 \left (b^2 d f-(c d+a f)^2\right )^2}\\ &=-\frac{2 \left (A b^3 f-A b c (c d+3 a f)+a B \left (2 c^2 d-b^2 f+2 a c f\right )+c \left (A b^2 f+b B (c d-a f)-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (3 b^6 B d f^2+24 a^2 B c^2 f (c d+a f)^2-A b^5 f^2 (7 c d+6 a f)-b^4 B f \left (7 c^2 d^2+14 a c d f-3 a^2 f^2\right )+A b^3 c f \left (15 c^2 d^2+46 a c d f+43 a^2 f^2\right )+2 b^2 B c \left (2 c^3 d^3+5 a c^2 d^2 f+4 a^2 c d f^2-11 a^3 f^3\right )-4 A b c^2 \left (2 c^3 d^3+9 a c^2 d^2 f+24 a^2 c d f^2+17 a^3 f^3\right )+c \left (3 b^5 B d f^2-2 A b^4 f^2 (4 c d+3 a f)-8 A c^2 (c d+a f)^2 (2 c d+5 a f)-b^3 B f \left (17 c^2 d^2+10 a c d f-3 a^2 f^2\right )+2 A b^2 c f \left (15 c^2 d^2+22 a c d f+19 a^2 f^2\right )+4 b B c \left (2 c^3 d^3+11 a c^2 d^2 f+4 a^2 c d f^2-5 a^3 f^3\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (b^2 d f-(c d+a f)^2\right )^2 \sqrt{a+b x+c x^2}}+\frac{\left (\left (B \sqrt{d}-A \sqrt{f}\right ) f^2\right ) \int \frac{1}{\left (-\sqrt{d} \sqrt{f}-f x\right ) \sqrt{a+b x+c x^2}} \, dx}{2 \sqrt{d} \left (c d-b \sqrt{d} \sqrt{f}+a f\right )^2}+\frac{\left (\left (B \sqrt{d}+A \sqrt{f}\right ) f^2\right ) \int \frac{1}{\left (\sqrt{d} \sqrt{f}-f x\right ) \sqrt{a+b x+c x^2}} \, dx}{2 \sqrt{d} \left (c d+b \sqrt{d} \sqrt{f}+a f\right )^2}\\ &=-\frac{2 \left (A b^3 f-A b c (c d+3 a f)+a B \left (2 c^2 d-b^2 f+2 a c f\right )+c \left (A b^2 f+b B (c d-a f)-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (3 b^6 B d f^2+24 a^2 B c^2 f (c d+a f)^2-A b^5 f^2 (7 c d+6 a f)-b^4 B f \left (7 c^2 d^2+14 a c d f-3 a^2 f^2\right )+A b^3 c f \left (15 c^2 d^2+46 a c d f+43 a^2 f^2\right )+2 b^2 B c \left (2 c^3 d^3+5 a c^2 d^2 f+4 a^2 c d f^2-11 a^3 f^3\right )-4 A b c^2 \left (2 c^3 d^3+9 a c^2 d^2 f+24 a^2 c d f^2+17 a^3 f^3\right )+c \left (3 b^5 B d f^2-2 A b^4 f^2 (4 c d+3 a f)-8 A c^2 (c d+a f)^2 (2 c d+5 a f)-b^3 B f \left (17 c^2 d^2+10 a c d f-3 a^2 f^2\right )+2 A b^2 c f \left (15 c^2 d^2+22 a c d f+19 a^2 f^2\right )+4 b B c \left (2 c^3 d^3+11 a c^2 d^2 f+4 a^2 c d f^2-5 a^3 f^3\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (b^2 d f-(c d+a f)^2\right )^2 \sqrt{a+b x+c x^2}}-\frac{\left (\left (B \sqrt{d}-A \sqrt{f}\right ) f^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d f-4 b \sqrt{d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac{b \sqrt{d} \sqrt{f}-2 a f-\left (-2 c \sqrt{d} \sqrt{f}+b f\right ) x}{\sqrt{a+b x+c x^2}}\right )}{\sqrt{d} \left (c d-b \sqrt{d} \sqrt{f}+a f\right )^2}-\frac{\left (\left (B \sqrt{d}+A \sqrt{f}\right ) f^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d f+4 b \sqrt{d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac{-b \sqrt{d} \sqrt{f}-2 a f-\left (2 c \sqrt{d} \sqrt{f}+b f\right ) x}{\sqrt{a+b x+c x^2}}\right )}{\sqrt{d} \left (c d+b \sqrt{d} \sqrt{f}+a f\right )^2}\\ &=-\frac{2 \left (A b^3 f-A b c (c d+3 a f)+a B \left (2 c^2 d-b^2 f+2 a c f\right )+c \left (A b^2 f+b B (c d-a f)-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (3 b^6 B d f^2+24 a^2 B c^2 f (c d+a f)^2-A b^5 f^2 (7 c d+6 a f)-b^4 B f \left (7 c^2 d^2+14 a c d f-3 a^2 f^2\right )+A b^3 c f \left (15 c^2 d^2+46 a c d f+43 a^2 f^2\right )+2 b^2 B c \left (2 c^3 d^3+5 a c^2 d^2 f+4 a^2 c d f^2-11 a^3 f^3\right )-4 A b c^2 \left (2 c^3 d^3+9 a c^2 d^2 f+24 a^2 c d f^2+17 a^3 f^3\right )+c \left (3 b^5 B d f^2-2 A b^4 f^2 (4 c d+3 a f)-8 A c^2 (c d+a f)^2 (2 c d+5 a f)-b^3 B f \left (17 c^2 d^2+10 a c d f-3 a^2 f^2\right )+2 A b^2 c f \left (15 c^2 d^2+22 a c d f+19 a^2 f^2\right )+4 b B c \left (2 c^3 d^3+11 a c^2 d^2 f+4 a^2 c d f^2-5 a^3 f^3\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (b^2 d f-(c d+a f)^2\right )^2 \sqrt{a+b x+c x^2}}-\frac{\left (B \sqrt{d}-A \sqrt{f}\right ) f^{3/2} \tanh ^{-1}\left (\frac{b \sqrt{d}-2 a \sqrt{f}+\left (2 c \sqrt{d}-b \sqrt{f}\right ) x}{2 \sqrt{c d-b \sqrt{d} \sqrt{f}+a f} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{d} \left (c d-b \sqrt{d} \sqrt{f}+a f\right )^{5/2}}+\frac{\left (B \sqrt{d}+A \sqrt{f}\right ) f^{3/2} \tanh ^{-1}\left (\frac{b \sqrt{d}+2 a \sqrt{f}+\left (2 c \sqrt{d}+b \sqrt{f}\right ) x}{2 \sqrt{c d+b \sqrt{d} \sqrt{f}+a f} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{d} \left (c d+b \sqrt{d} \sqrt{f}+a f\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 5.16498, size = 674, normalized size = 0.85 \[ \frac{2 \left (-\frac{3 f \left (-b^2 \left (B \left (-a^2 f^2+2 a c d f+c^2 d^2\right )+2 a A c f^2 x\right )+b^3 f (B c d x-A (2 a f+c d))+b c (a f+c d) (5 a A f+a B f x+A c d-3 B c d x)+2 c (a f+c d)^2 (A c x-a B)+b^4 B d f\right )}{\sqrt{a+x (b+c x)} \left (f \left (a^2 f-b^2 d\right )+2 a c d f+c^2 d^2\right )}+\frac{B \left (2 a^2 c f+a \left (b^2 (-f)-b c f x+2 c^2 d\right )+b c^2 d x\right )+A \left (-b c (3 a f+c d)-2 c^2 x (a f+c d)+b^2 c f x+b^3 f\right )}{(a+x (b+c x))^{3/2}}+\frac{3 f^{3/2} \left (b^2-4 a c\right ) \left (\frac{\left (A \sqrt{f}-B \sqrt{d}\right ) \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^2 \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+b \left (\sqrt{d}-\sqrt{f} x\right )+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{\sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}-\frac{\left (A \sqrt{f}+B \sqrt{d}\right ) \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^2 \tanh ^{-1}\left (\frac{-2 \left (a \sqrt{f}+c \sqrt{d} x\right )-b \left (\sqrt{d}+\sqrt{f} x\right )}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{4 \sqrt{d} \left ((a f+c d)^2-b^2 d f\right )}+\frac{4 c (b+2 c x) \left (2 A c (a f+c d)+b B (a f-c d)-A b^2 f\right )}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}}\right )}{3 \left (b^2-4 a c\right ) \left ((a f+c d)^2-b^2 d f\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((a + b*x + c*x^2)^(5/2)*(d - f*x^2)),x]

[Out]

(2*((4*c*(-(A*b^2*f) + b*B*(-(c*d) + a*f) + 2*A*c*(c*d + a*f))*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x
)]) - (3*f*(b^4*B*d*f + 2*c*(c*d + a*f)^2*(-(a*B) + A*c*x) + b^3*f*(-(A*(c*d + 2*a*f)) + B*c*d*x) + b*c*(c*d +
a*f)*(A*c*d + 5*a*A*f - 3*B*c*d*x + a*B*f*x) - b^2*(B*(c^2*d^2 + 2*a*c*d*f - a^2*f^2) + 2*a*A*c*f^2*x)))/((c^
2*d^2 + 2*a*c*d*f + f*(-(b^2*d) + a^2*f))*Sqrt[a + x*(b + c*x)]) + (A*(b^3*f - b*c*(c*d + 3*a*f) + b^2*c*f*x -
2*c^2*(c*d + a*f)*x) + B*(2*a^2*c*f + b*c^2*d*x + a*(2*c^2*d - b^2*f - b*c*f*x)))/(a + x*(b + c*x))^(3/2) + (
3*(b^2 - 4*a*c)*f^(3/2)*(((-(B*Sqrt[d]) + A*Sqrt[f])*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^2*ArcTanh[(-2*a*Sqrt[f] +
2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt
[c*d - b*Sqrt[d]*Sqrt[f] + a*f] - ((B*Sqrt[d] + A*Sqrt[f])*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^2*ArcTanh[(-2*(a*Sq
rt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])]
)/Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]))/(4*Sqrt[d]*(-(b^2*d*f) + (c*d + a*f)^2))))/(3*(b^2 - 4*a*c)*(-(b^2*d*f
) + (c*d + a*f)^2))

________________________________________________________________________________________

Maple [B]  time = 0.256, size = 6422, normalized size = 8.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(c*x^2+b*x+a)^(5/2)/(-f*x^2+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((B*x+A)/(c*x^2+b*x+a)^(5/2)/(-f*x^2+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((B*x+A)/(c*x^2+b*x+a)^(5/2)/(-f*x^2+d),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((B*x+A)/(c*x**2+b*x+a)**(5/2)/(-f*x**2+d),x)

[Out]

Timed out

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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((B*x+A)/(c*x^2+b*x+a)^(5/2)/(-f*x^2+d),x, algorithm="giac")

[Out]

Exception raised: TypeError