Integrand size = 36, antiderivative size = 249 \[ \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=-\frac {((A b-2 a B) e-b (B e-2 A f) x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}+\frac {(B e-2 A f) \left (8 a e f-b \left (e^2+4 d f\right )\right ) \text {arctanh}\left (\frac {\sqrt {b d-a e} (e+2 f x)}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{2 e^{3/2} (b d-a e)^{3/2} f (b e-4 a f)^{3/2}}+\frac {B \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x+f x^2}}{\sqrt {b d-a e}}\right )}{2 \sqrt {b} (b d-a e)^{3/2} f} \]
1/2*(-2*A*f+B*e)*(8*a*e*f-b*(4*d*f+e^2))*arctanh((2*f*x+e)*(-a*e+b*d)^(1/2 )/e^(1/2)/(-4*a*f+b*e)^(1/2)/(f*x^2+e*x+d)^(1/2))/e^(3/2)/(-a*e+b*d)^(3/2) /f/(-4*a*f+b*e)^(3/2)+1/2*B*arctanh(b^(1/2)*(f*x^2+e*x+d)^(1/2)/(-a*e+b*d) ^(1/2))/(-a*e+b*d)^(3/2)/f/b^(1/2)-((A*b-2*B*a)*e-b*(-2*A*f+B*e)*x)*(f*x^2 +e*x+d)^(1/2)/e/(-a*e+b*d)/(-4*a*f+b*e)/(b*f*x^2+b*e*x+a*e)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 3.53 (sec) , antiderivative size = 2374, normalized size of antiderivative = 9.53 \[ \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\text {Result too large to show} \]
((2*e*Sqrt[d + x*(e + f*x)]*(B*e*(2*a + b*x) - A*b*(e + 2*f*x)))/((b*d - a *e)*(b*e - 4*a*f)*(a*e + b*x*(e + f*x))) - (2*RootSum[a*e*f^2 - 2*b*Sqrt[d ]*e*f*#1 + b*e^2*#1^2 + 4*b*d*f*#1^2 - 2*a*e*f*#1^2 - 2*b*Sqrt[d]*e*#1^3 + a*e*#1^4 & , (-4*A*b^2*d*e*Log[x] + 4*a*b*B*d*e*Log[x] + a*A*b*e^2*Log[x] - a^2*B*e^2*Log[x] + 4*a*A*b*d*f*Log[x] + a^2*A*e*f*Log[x] + 4*A*b^2*d*e* Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] - 4*a*b*B*d*e*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] - a*A*b*e^2*Log[-Sqrt[d] + Sqrt[d + e*x + f* x^2] - x*#1] + a^2*B*e^2*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] - 4* a*A*b*d*f*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] - a^2*A*e*f*Log[-Sq rt[d] + Sqrt[d + e*x + f*x^2] - x*#1] - 2*a*A*b*Sqrt[d]*e*Log[x]*#1 + 2*a^ 2*B*Sqrt[d]*e*Log[x]*#1 + 2*a*A*b*Sqrt[d]*e*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1]*#1 - 2*a^2*B*Sqrt[d]*e*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1]*#1 - a^2*A*e*Log[x]*#1^2 + a^2*A*e*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1]*#1^2)/(-(b*Sqrt[d]*e*f) + b*e^2*#1 + 4*b*d*f*#1 - 2*a*e*f*# 1 - 3*b*Sqrt[d]*e*#1^2 + 2*a*e*#1^3) & ])/a^3 + RootSum[a*e*f^2 - 2*b*Sqrt [d]*e*f*#1 + b*e^2*#1^2 + 4*b*d*f*#1^2 - 2*a*e*f*#1^2 - 2*b*Sqrt[d]*e*#1^3 + a*e*#1^4 & , (-8*A*b^4*d^2*e^2*Log[x] + 8*a*b^3*B*d^2*e^2*Log[x] + 10*a *A*b^3*d*e^3*Log[x] - 10*a^2*b^2*B*d*e^3*Log[x] - 2*a^2*A*b^2*e^4*Log[x] + a^3*b*B*e^4*Log[x] + 40*a*A*b^3*d^2*e*f*Log[x] - 32*a^2*b^2*B*d^2*e*f*Log [x] - 46*a^2*A*b^2*d*e^2*f*Log[x] + 38*a^3*b*B*d*e^2*f*Log[x] + 7*a^3*A...
Time = 0.79 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {1349, 27, 1358, 1313, 221, 1357, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1349 |
| \(\displaystyle -\frac {\int -\frac {b (b d-a e) f^2 \left (2 b B d e-2 a (B e-4 A f) e-B (b e-4 a f) x e-A b \left (e^2+4 d f\right )\right )}{2 \sqrt {f x^2+e x+d} \left (b f x^2+b e x+a e\right )}dx}{b e f^2 (b d-a e)^2 (b e-4 a f)}-\frac {\sqrt {d+e x+f x^2} (e (A b-2 a B)-b x (B e-2 A f))}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 b B d e-2 a (B e-4 A f) e-B (b e-4 a f) x e-A b \left (e^2+4 d f\right )}{\sqrt {f x^2+e x+d} \left (b f x^2+b e x+a e\right )}dx}{2 e (b d-a e) (b e-4 a f)}-\frac {\sqrt {d+e x+f x^2} (e (A b-2 a B)-b x (B e-2 A f))}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}\) |
| \(\Big \downarrow \) 1358 |
| \(\displaystyle \frac {-\frac {(B e-2 A f) \left (8 a e f-b \left (4 d f+e^2\right )\right ) \int \frac {1}{\sqrt {f x^2+e x+d} \left (b f x^2+b e x+a e\right )}dx}{2 f}-\frac {B e (b e-4 a f) \int \frac {e+2 f x}{\sqrt {f x^2+e x+d} \left (b f x^2+b e x+a e\right )}dx}{2 f}}{2 e (b d-a e) (b e-4 a f)}-\frac {\sqrt {d+e x+f x^2} (e (A b-2 a B)-b x (B e-2 A f))}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}\) |
| \(\Big \downarrow \) 1313 |
| \(\displaystyle \frac {\frac {e (B e-2 A f) \left (8 a e f-b \left (4 d f+e^2\right )\right ) \int \frac {1}{e^2 (b e-4 a f)-\frac {e (b d-a e) (e+2 f x)^2}{f x^2+e x+d}}d\frac {e+2 f x}{\sqrt {f x^2+e x+d}}}{f}-\frac {B e (b e-4 a f) \int \frac {e+2 f x}{\sqrt {f x^2+e x+d} \left (b f x^2+b e x+a e\right )}dx}{2 f}}{2 e (b d-a e) (b e-4 a f)}-\frac {\sqrt {d+e x+f x^2} (e (A b-2 a B)-b x (B e-2 A f))}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {(B e-2 A f) \left (8 a e f-b \left (4 d f+e^2\right )\right ) \text {arctanh}\left (\frac {(e+2 f x) \sqrt {b d-a e}}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{\sqrt {e} f \sqrt {b d-a e} \sqrt {b e-4 a f}}-\frac {B e (b e-4 a f) \int \frac {e+2 f x}{\sqrt {f x^2+e x+d} \left (b f x^2+b e x+a e\right )}dx}{2 f}}{2 e (b d-a e) (b e-4 a f)}-\frac {\sqrt {d+e x+f x^2} (e (A b-2 a B)-b x (B e-2 A f))}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}\) |
| \(\Big \downarrow \) 1357 |
| \(\displaystyle \frac {\frac {B e^2 (b e-4 a f) \int \frac {1}{e (b d-a e)-b e \left (f x^2+e x+d\right )}d\sqrt {f x^2+e x+d}}{f}+\frac {(B e-2 A f) \left (8 a e f-b \left (4 d f+e^2\right )\right ) \text {arctanh}\left (\frac {(e+2 f x) \sqrt {b d-a e}}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{\sqrt {e} f \sqrt {b d-a e} \sqrt {b e-4 a f}}}{2 e (b d-a e) (b e-4 a f)}-\frac {\sqrt {d+e x+f x^2} (e (A b-2 a B)-b x (B e-2 A f))}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {(B e-2 A f) \left (8 a e f-b \left (4 d f+e^2\right )\right ) \text {arctanh}\left (\frac {(e+2 f x) \sqrt {b d-a e}}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{\sqrt {e} f \sqrt {b d-a e} \sqrt {b e-4 a f}}+\frac {B e (b e-4 a f) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x+f x^2}}{\sqrt {b d-a e}}\right )}{\sqrt {b} f \sqrt {b d-a e}}}{2 e (b d-a e) (b e-4 a f)}-\frac {\sqrt {d+e x+f x^2} (e (A b-2 a B)-b x (B e-2 A f))}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}\) |
-((((A*b - 2*a*B)*e - b*(B*e - 2*A*f)*x)*Sqrt[d + e*x + f*x^2])/(e*(b*d - a*e)*(b*e - 4*a*f)*(a*e + b*e*x + b*f*x^2))) + (((B*e - 2*A*f)*(8*a*e*f - b*(e^2 + 4*d*f))*ArcTanh[(Sqrt[b*d - a*e]*(e + 2*f*x))/(Sqrt[e]*Sqrt[b*e - 4*a*f]*Sqrt[d + e*x + f*x^2])])/(Sqrt[e]*Sqrt[b*d - a*e]*f*Sqrt[b*e - 4*a *f]) + (B*e*(b*e - 4*a*f)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x + f*x^2])/Sqrt[b*d - a*e]])/(Sqrt[b]*Sqrt[b*d - a*e]*f))/(2*e*(b*d - a*e)*(b*e - 4*a*f))
3.1.35.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*( x_)^2]), x_Symbol] :> Simp[-2*e Subst[Int[1/(e*(b*e - 4*a*f) - (b*d - a*e )*x^2), x], x, (e + 2*f*x)/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0]
Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e _.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(a + b*x + c*x^2)^(p + 1)* ((d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)))*(g*c*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - h*(b* c*d - 2*a*c*e + a*b*f))*x), x] + Simp[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b* d - a*e)*(c*e - b*f))*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f *x^2)^q*Simp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1 ) + (b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a*((-h)*c *e)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g *b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((g*c)*(2 *a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))* (p + q + 2) - (b^2*g*f - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a *((-h)*c*e)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(g*f) - b*(h* c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) + a*h*c*e))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c* e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1])
Int[((g_) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e _.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[-2*g Subst[Int[1/(b*d - a*e - b*x^2), x], x, Sqrt[d + e*x + f*x^2]], 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] && EqQ[c*e - b*f, 0] & & EqQ[h*e - 2*g*f, 0]
Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[-(h*e - 2*g*f)/(2*f) Int[1/ ((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] + Simp[h/(2*f) Int[(e + 2*f*x)/((a + b*x + c*x^2)*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] && EqQ[c *e - b*f, 0] && NeQ[h*e - 2*g*f, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1429\) vs. \(2(219)=438\).
Time = 1.38 (sec) , antiderivative size = 1430, normalized size of antiderivative = 5.74
(2*A*f-B*e)/e/(4*a*f-b*e)/(-b*e*(4*a*f-b*e))^(1/2)/(-(a*e-b*d)/b)^(1/2)*ln ((-2*(a*e-b*d)/b-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2*(b*e+(-b*e*(4*a*f-b*e)) ^(1/2))/b/f)+2*(-(a*e-b*d)/b)^(1/2)*((x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2)) /b/f)^2*f-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2)) /b/f)-(a*e-b*d)/b)^(1/2))/(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f))-(2*A *f-B*e)/e/(4*a*f-b*e)/(-b*e*(4*a*f-b*e))^(1/2)/(-(a*e-b*d)/b)^(1/2)*ln((-2 *(a*e-b*d)/b+(-b*e*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1 /2))/b/f)+2*(-(a*e-b*d)/b)^(1/2)*((x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b /f)^2*f+(-b*e*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/ b/f)-(a*e-b*d)/b)^(1/2))/(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f))-1/2/ f*(2*A*b*f-B*b*e-B*(-b*e*(4*a*f-b*e))^(1/2))/e/(4*a*f-b*e)/b^2*(1/(a*e-b*d )*b/(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)*((x+1/2*(b*e+(-b*e*(4*a*f-b *e))^(1/2))/b/f)^2*f-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2*(b*e+(-b*e*(4*a*f-b *e))^(1/2))/b/f)-(a*e-b*d)/b)^(1/2)+1/2*(-b*e*(4*a*f-b*e))^(1/2)/(a*e-b*d) /(-(a*e-b*d)/b)^(1/2)*ln((-2*(a*e-b*d)/b-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2 *(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)+2*(-(a*e-b*d)/b)^(1/2)*((x+1/2*(b*e+( -b*e*(4*a*f-b*e))^(1/2))/b/f)^2*f-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2*(b*e+( -b*e*(4*a*f-b*e))^(1/2))/b/f)-(a*e-b*d)/b)^(1/2))/(x+1/2*(b*e+(-b*e*(4*a*f -b*e))^(1/2))/b/f)))-1/2/f*(2*A*b*f-B*b*e+B*(-b*e*(4*a*f-b*e))^(1/2))/e/(4 *a*f-b*e)/b^2*(1/(a*e-b*d)*b/(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f...
Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\int { \frac {B x + A}{{\left (b f x^{2} + b e x + a e\right )}^{2} \sqrt {f x^{2} + e x + d}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 5022 vs. \(2 (218) = 436\).
Time = 0.53 (sec) , antiderivative size = 5022, normalized size of antiderivative = 20.17 \[ \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\text {Too large to display} \]
-1/2*((B*b*e^2*(e/sqrt(f) - sqrt((b*e^2*f + 4*b*d*f^2 - 8*a*e*f^2 + 4*sqrt (b^2*d*e^2*f - a*b*e^3*f - 4*a*b*d*e*f^2 + 4*a^2*e^2*f^2)*f)/(b*f^2)))^2 - 4*B*a*e*f*(e/sqrt(f) - sqrt((b*e^2*f + 4*b*d*f^2 - 8*a*e*f^2 + 4*sqrt(b^2 *d*e^2*f - a*b*e^3*f - 4*a*b*d*e*f^2 + 4*a^2*e^2*f^2)*f)/(b*f^2)))^2 + 8*B *b*d*e*sqrt(f)*(e/sqrt(f) - sqrt((b*e^2*f + 4*b*d*f^2 - 8*a*e*f^2 + 4*sqrt (b^2*d*e^2*f - a*b*e^3*f - 4*a*b*d*e*f^2 + 4*a^2*e^2*f^2)*f)/(b*f^2))) - 8 *B*a*e^2*sqrt(f)*(e/sqrt(f) - sqrt((b*e^2*f + 4*b*d*f^2 - 8*a*e*f^2 + 4*sq rt(b^2*d*e^2*f - a*b*e^3*f - 4*a*b*d*e*f^2 + 4*a^2*e^2*f^2)*f)/(b*f^2))) - 4*A*b*e^2*sqrt(f)*(e/sqrt(f) - sqrt((b*e^2*f + 4*b*d*f^2 - 8*a*e*f^2 + 4* sqrt(b^2*d*e^2*f - a*b*e^3*f - 4*a*b*d*e*f^2 + 4*a^2*e^2*f^2)*f)/(b*f^2))) - 16*A*b*d*f^(3/2)*(e/sqrt(f) - sqrt((b*e^2*f + 4*b*d*f^2 - 8*a*e*f^2 + 4 *sqrt(b^2*d*e^2*f - a*b*e^3*f - 4*a*b*d*e*f^2 + 4*a^2*e^2*f^2)*f)/(b*f^2)) ) + 32*A*a*e*f^(3/2)*(e/sqrt(f) - sqrt((b*e^2*f + 4*b*d*f^2 - 8*a*e*f^2 + 4*sqrt(b^2*d*e^2*f - a*b*e^3*f - 4*a*b*d*e*f^2 + 4*a^2*e^2*f^2)*f)/(b*f^2) )) - 12*B*b*d*e^2 + 8*B*a*e^3 + 4*A*b*e^3 + 16*B*a*d*e*f + 16*A*b*d*e*f - 32*A*a*e^2*f)*log(-sqrt(f)*x + sqrt(f*x^2 + e*x + d) - 1/2*e/sqrt(f) + 1/2 *sqrt((b*e^2*f + 4*b*d*f^2 - 8*a*e*f^2 + 4*sqrt(b^2*d*e^2*f - a*b*e^3*f - 4*a*b*d*e*f^2 + 4*a^2*e^2*f^2)*f)/(b*f^2)))/(b*f*(e/sqrt(f) - sqrt((b*e^2* f + 4*b*d*f^2 - 8*a*e*f^2 + 4*sqrt(b^2*d*e^2*f - a*b*e^3*f - 4*a*b*d*e*f^2 + 4*a^2*e^2*f^2)*f)/(b*f^2)))^3 - 3*b*e*sqrt(f)*(e/sqrt(f) - sqrt((b*e...
Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\int \frac {A+B\,x}{{\left (b\,f\,x^2+b\,e\,x+a\,e\right )}^2\,\sqrt {f\,x^2+e\,x+d}} \,d x \]