\(\int \frac {A+B x}{\sqrt {d+e x+f x^2} (a e+b e x+b f x^2)^2} \, dx\) [35]

Optimal result

Integrand size = 36, antiderivative size = 249 \[ \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=-\frac {((A b-2 a B) e-b (B e-2 A f) x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}+\frac {(B e-2 A f) \left (8 a e f-b \left (e^2+4 d f\right )\right ) \text {arctanh}\left (\frac {\sqrt {b d-a e} (e+2 f x)}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{2 e^{3/2} (b d-a e)^{3/2} f (b e-4 a f)^{3/2}}+\frac {B \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x+f x^2}}{\sqrt {b d-a e}}\right )}{2 \sqrt {b} (b d-a e)^{3/2} f} \]

[Out]

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)

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {1030, 1039, 996, 214, 1038} \[ \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\frac {(B e-2 A f) \left (8 a e f-b \left (4 d f+e^2\right )\right ) \text {arctanh}\left (\frac {(e+2 f x) \sqrt {b d-a e}}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{2 e^{3/2} f (b d-a e)^{3/2} (b e-4 a f)^{3/2}}-\frac {\sqrt {d+e x+f x^2} (e (A b-2 a B)-b x (B e-2 A f))}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}+\frac {B \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x+f x^2}}{\sqrt {b d-a e}}\right )}{2 \sqrt {b} f (b d-a e)^{3/2}} \]

[In]

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

[Out]

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

Rule 214

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]

Rule 996

Int[1/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*e, Su
bst[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]

Rule 1030

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Sy
mbol] :> 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] + Dist[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
])

Rule 1038

Int[((g_) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol
] :> Dist[-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]

Rule 1039

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbo
l] :> Dist[-(h*e - 2*g*f)/(2*f), Int[1/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\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 {\int \frac {-\frac {1}{2} b (b d-a e) f^2 \left (2 b B d e-2 a e (B e-4 A f)-A b \left (e^2+4 d f\right )\right )+\frac {1}{2} b B e (b d-a e) f^2 (b e-4 a f) x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )} \, dx}{b e (b d-a e)^2 f^2 (b e-4 a f)} \\ & = -\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 \int \frac {e+2 f x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )} \, dx}{4 (b d-a e) f}-\frac {\left ((B e-2 A f) \left (8 a e f-b \left (e^2+4 d f\right )\right )\right ) \int \frac {1}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )} \, dx}{4 e (b d-a e) f (b e-4 a f)} \\ & = -\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) \text {Subst}\left (\int \frac {1}{b d e-a e^2-b e x^2} \, dx,x,\sqrt {d+e x+f x^2}\right )}{2 (b d-a e) f}+\frac {\left ((B e-2 A f) \left (8 a e f-b \left (e^2+4 d f\right )\right )\right ) \text {Subst}\left (\int \frac {1}{e \left (b e^2-4 a e f\right )-\left (b d e-a e^2\right ) x^2} \, dx,x,\frac {e+2 f x}{\sqrt {d+e x+f x^2}}\right )}{2 (b d-a e) f (b e-4 a f)} \\ & = -\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 ) \tanh ^{-1}\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 \tanh ^{-1}\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} \\ \end{align*}

Mathematica [C] (verified)

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 \[ \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

((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[-Sq
rt[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*L
og[-Sqrt[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*Sq
rt[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*b*e^3*f*Log[x] - 2*a^4*B*e^3*f*Log[x] - 32*a^2*A*b^2*d^2*f^2*L
og[x] + 28*a^3*A*b*d*e*f^2*Log[x] + 8*A*b^4*d^2*e^2*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] - 8*a*b^3*B*d
^2*e^2*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] - 10*a*A*b^3*d*e^3*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] -
x*#1] + 10*a^2*b^2*B*d*e^3*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] + 2*a^2*A*b^2*e^4*Log[-Sqrt[d] + Sqrt[
d + e*x + f*x^2] - x*#1] - a^3*b*B*e^4*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] - 40*a*A*b^3*d^2*e*f*Log[-
Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] + 32*a^2*b^2*B*d^2*e*f*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] +
46*a^2*A*b^2*d*e^2*f*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] - 38*a^3*b*B*d*e^2*f*Log[-Sqrt[d] + Sqrt[d +
e*x + f*x^2] - x*#1] - 7*a^3*A*b*e^3*f*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] + 2*a^4*B*e^3*f*Log[-Sqrt
[d] + Sqrt[d + e*x + f*x^2] - x*#1] + 32*a^2*A*b^2*d^2*f^2*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] - 28*a
^3*A*b*d*e*f^2*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] - 4*a*A*b^3*d^(3/2)*e^2*Log[x]*#1 + 4*a^2*b^2*B*d^
(3/2)*e^2*Log[x]*#1 + 4*a^2*A*b^2*Sqrt[d]*e^3*Log[x]*#1 - 2*a^3*b*B*Sqrt[d]*e^3*Log[x]*#1 + 16*a^2*A*b^2*d^(3/
2)*e*f*Log[x]*#1 - 16*a^3*b*B*d^(3/2)*e*f*Log[x]*#1 - 16*a^3*A*b*Sqrt[d]*e^2*f*Log[x]*#1 + 8*a^4*B*Sqrt[d]*e^2
*f*Log[x]*#1 + 4*a*A*b^3*d^(3/2)*e^2*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1]*#1 - 4*a^2*b^2*B*d^(3/2)*e^2
*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1]*#1 - 4*a^2*A*b^2*Sqrt[d]*e^3*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2
] - x*#1]*#1 + 2*a^3*b*B*Sqrt[d]*e^3*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1]*#1 - 16*a^2*A*b^2*d^(3/2)*e*
f*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1]*#1 + 16*a^3*b*B*d^(3/2)*e*f*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2
] - x*#1]*#1 + 16*a^3*A*b*Sqrt[d]*e^2*f*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1]*#1 - 8*a^4*B*Sqrt[d]*e^2*
f*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1]*#1 - 2*a^2*A*b^2*d*e^2*Log[x]*#1^2 + 2*a^3*b*B*d*e^2*Log[x]*#1^
2 + a^3*A*b*e^3*Log[x]*#1^2 - 2*a^4*B*e^3*Log[x]*#1^2 + 4*a^3*A*b*d*e*f*Log[x]*#1^2 + 2*a^2*A*b^2*d*e^2*Log[-S
qrt[d] + Sqrt[d + e*x + f*x^2] - x*#1]*#1^2 - 2*a^3*b*B*d*e^2*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1]*#1^
2 - a^3*A*b*e^3*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1]*#1^2 + 2*a^4*B*e^3*Log[-Sqrt[d] + Sqrt[d + e*x +
f*x^2] - x*#1]*#1^2 - 4*a^3*A*b*d*e*f*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*(b*d - a*e)*(b*e - 4*a*f)))/(2*e
^2)

Maple [B] (verified)

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

method	result	size
default	\(\text {Expression too large to display}\)	\(1430\)

[In]

int((B*x+A)/(b*f*x^2+b*e*x+a*e)^2/(f*x^2+e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(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)*((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)))

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((B*x+A)/(b*f*x**2+b*e*x+a*e)**2/(f*x**2+e*x+d)**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\int { \frac {B x + A}{{\left (b f x^{2} + b e x + a e\right )}^{2} \sqrt {f x^{2} + e x + d}} \,d x } \]

[In]

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

[Out]

integrate((B*x + A)/((b*f*x^2 + b*e*x + a*e)^2*sqrt(f*x^2 + e*x + d)), x)

Giac [B] (verification not implemented)

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 \[ \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)/(b*f*x^2+b*e*x+a*e)^2/(f*x^2+e*x+d)^(1/2),x, algorithm="giac")

[Out]

-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*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))) - 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^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*d*e*sqrt(f) - 8*a*e^2*sqrt(f) + 2*(b*e^2 - 2*b*d*f + 4*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)))) + (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*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))) - 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*sqr
t((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^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*d*e*sqrt(f) - 8*a*e^2*sqrt(f) + 2*(
b*e^2 - 2*b*d*f + 4*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)))) + (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*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))) - 4*A*b*e^2*sqrt(f)*(e/sq
rt(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^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*d*e*sqrt(f) -
8*a*e^2*sqrt(f) + 2*(b*e^2 - 2*b*d*f + 4*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)))) + (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*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))) - 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) + sq
rt((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 + sqr
t(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*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)))^3 - 3*b*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)))
^2 + 4*b*d*e*sqrt(f) - 8*a*e^2*sqrt(f) + 2*(b*e^2 - 2*b*d*f + 4*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)))))/(b^2*d*e^2 - a*b*
e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f) + ((sqrt(f)*x - sqrt(f*x^2 + e*x + d))^3*B*b*e^2*f - 4*(sqrt(f)*x - sqrt(f*x^
2 + e*x + d))^3*B*a*e*f^2 + (sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*B*b*e^3*sqrt(f) - 2*(sqrt(f)*x - sqrt(f*x^2
+ e*x + d))^2*B*b*d*e*f^(3/2) - 2*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*B*a*e^2*f^(3/2) + (sqrt(f)*x - sqrt(f*
x^2 + e*x + d))^2*A*b*e^2*f^(3/2) + 4*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*A*b*d*f^(5/2) - 8*(sqrt(f)*x - sqr
t(f*x^2 + e*x + d))^2*A*a*e*f^(5/2) - (sqrt(f)*x - sqrt(f*x^2 + e*x + d))*B*b*d*e^2*f + 2*(sqrt(f)*x - sqrt(f*
x^2 + e*x + d))*B*a*e^3*f + (sqrt(f)*x - sqrt(f*x^2 + e*x + d))*A*b*e^3*f - 4*(sqrt(f)*x - sqrt(f*x^2 + e*x +
d))*B*a*d*e*f^2 + 4*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))*A*b*d*e*f^2 - 8*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))*A*
a*e^2*f^2 - B*b*d*e^3*sqrt(f) + B*a*e^4*sqrt(f) + 2*B*b*d^2*e*f^(3/2) - 2*B*a*d*e^2*f^(3/2) + 3*A*b*d*e^2*f^(3
/2) - 2*A*a*e^3*f^(3/2) - 4*A*b*d^2*f^(5/2))/(((sqrt(f)*x - sqrt(f*x^2 + e*x + d))^4*b*f + 2*(sqrt(f)*x - sqrt
(f*x^2 + e*x + d))^3*b*e*sqrt(f) + (sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*b*e^2 - 2*(sqrt(f)*x - sqrt(f*x^2 + e
*x + d))^2*b*d*f + 4*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*a*e*f - 2*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))*b*d*e
*sqrt(f) + 4*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))*a*e^2*sqrt(f) - b*d*e^2 + a*e^3 + b*d^2*f)*(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))

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\int \frac {A+B\,x}{{\left (b\,f\,x^2+b\,e\,x+a\,e\right )}^2\,\sqrt {f\,x^2+e\,x+d}} \,d x \]

[In]

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

[Out]

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