3.46 \(\int \frac{\sqrt{c+d x} \sqrt{e+f x} (A+B x+C x^2)}{(a+b x)^3} \, dx\)

Optimal. Leaf size=658 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right ) \left (3 a^2 b^2 \left (4 B d f (c f+d e)+C \left (5 c^2 f^2+22 c d e f+5 d^2 e^2\right )\right )-8 a^3 b d f (B d f+5 C (c f+d e))+24 a^4 C d^2 f^2-3 a b^3 \left (c^2 f (B f+8 C e)+2 c d e (3 B f+4 C e)+B d^2 e^2\right )-b^4 \left (c^2 \left (-\left (-A f^2+4 B e f+8 C e^2\right )\right )-2 c d e (A f+2 B e)+A d^2 e^2\right )\right )}{4 b^4 (b c-a d)^{3/2} (b e-a f)^{3/2}}-\frac{\sqrt{c+d x} \sqrt{e+f x} \left (-a^2 b f (4 B d f+11 c C f+17 C d e)+12 a^3 C d f^2+a b^2 (B f (3 c f+5 d e)+4 C e (4 c f+d e))-b^3 \left (c \left (-A f^2+4 B e f+4 C e^2\right )+A d e f\right )\right )}{4 b^3 (b c-a d) (b e-a f)^2}+\frac{\sqrt{c+d x} (e+f x)^{3/2} \left (-a^2 b (2 B d f+7 C (c f+d e))+6 a^3 C d f+a b^2 (-2 A d f+3 B c f+3 B d e+8 c C e)-b^3 (-A c f-A d e+4 B c e)\right )}{4 b^2 (a+b x) (b c-a d) (b e-a f)^2}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) (6 a C d f-b (2 B d f+c C f+C d e))}{b^4 \sqrt{d} \sqrt{f}} \]

[Out]

-((12*a^3*C*d*f^2 - a^2*b*f*(17*C*d*e + 11*c*C*f + 4*B*d*f) + a*b^2*(B*f*(5*d*e + 3*c*f) + 4*C*e*(d*e + 4*c*f)
) - b^3*(A*d*e*f + c*(4*C*e^2 + 4*B*e*f - A*f^2)))*Sqrt[c + d*x]*Sqrt[e + f*x])/(4*b^3*(b*c - a*d)*(b*e - a*f)
^2) + ((6*a^3*C*d*f - b^3*(4*B*c*e - A*d*e - A*c*f) + a*b^2*(8*c*C*e + 3*B*d*e + 3*B*c*f - 2*A*d*f) - a^2*b*(2
*B*d*f + 7*C*(d*e + c*f)))*Sqrt[c + d*x]*(e + f*x)^(3/2))/(4*b^2*(b*c - a*d)*(b*e - a*f)^2*(a + b*x)) - ((A*b^
2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(2*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^2) - ((6*a*C*d*f -
b*(C*d*e + c*C*f + 2*B*d*f))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(b^4*Sqrt[d]*Sqrt[f]) -
((24*a^4*C*d^2*f^2 - 3*a*b^3*(B*d^2*e^2 + c^2*f*(8*C*e + B*f) + 2*c*d*e*(4*C*e + 3*B*f)) - 8*a^3*b*d*f*(B*d*f
+ 5*C*(d*e + c*f)) - b^4*(A*d^2*e^2 - 2*c*d*e*(2*B*e + A*f) - c^2*(8*C*e^2 + 4*B*e*f - A*f^2)) + 3*a^2*b^2*(4
*B*d*f*(d*e + c*f) + C*(5*d^2*e^2 + 22*c*d*e*f + 5*c^2*f^2)))*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*
c - a*d]*Sqrt[e + f*x])])/(4*b^4*(b*c - a*d)^(3/2)*(b*e - a*f)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 2.67951, antiderivative size = 657, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1613, 149, 154, 157, 63, 217, 206, 93, 208} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right ) \left (3 a^2 b^2 \left (4 B d f (c f+d e)+C \left (5 c^2 f^2+22 c d e f+5 d^2 e^2\right )\right )-8 a^3 b d f (B d f+5 C (c f+d e))+24 a^4 C d^2 f^2-3 a b^3 \left (c^2 f (B f+8 C e)+2 c d e (3 B f+4 C e)+B d^2 e^2\right )-b^4 \left (c^2 \left (-\left (-A f^2+4 B e f+8 C e^2\right )\right )-2 c d e (A f+2 B e)+A d^2 e^2\right )\right )}{4 b^4 (b c-a d)^{3/2} (b e-a f)^{3/2}}-\frac{\sqrt{c+d x} \sqrt{e+f x} \left (-a^2 b f (4 B d f+11 c C f+17 C d e)+12 a^3 C d f^2+a b^2 (B f (3 c f+5 d e)+4 C e (4 c f+d e))-b^3 \left (c f (4 B e-A f)+A d e f+4 c C e^2\right )\right )}{4 b^3 (b c-a d) (b e-a f)^2}+\frac{\sqrt{c+d x} (e+f x)^{3/2} \left (-a^2 b (2 B d f+7 C (c f+d e))+6 a^3 C d f+a b^2 (-2 A d f+3 B c f+3 B d e+8 c C e)-b^3 (-A c f-A d e+4 B c e)\right )}{4 b^2 (a+b x) (b c-a d) (b e-a f)^2}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) (6 a C d f-b (2 B d f+c C f+C d e))}{b^4 \sqrt{d} \sqrt{f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((12*a^3*C*d*f^2 - a^2*b*f*(17*C*d*e + 11*c*C*f + 4*B*d*f) - b^3*(4*c*C*e^2 + A*d*e*f + c*f*(4*B*e - A*f)) +
a*b^2*(B*f*(5*d*e + 3*c*f) + 4*C*e*(d*e + 4*c*f)))*Sqrt[c + d*x]*Sqrt[e + f*x])/(4*b^3*(b*c - a*d)*(b*e - a*f)
^2) + ((6*a^3*C*d*f - b^3*(4*B*c*e - A*d*e - A*c*f) + a*b^2*(8*c*C*e + 3*B*d*e + 3*B*c*f - 2*A*d*f) - a^2*b*(2
*B*d*f + 7*C*(d*e + c*f)))*Sqrt[c + d*x]*(e + f*x)^(3/2))/(4*b^2*(b*c - a*d)*(b*e - a*f)^2*(a + b*x)) - ((A*b^
2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(2*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^2) - ((6*a*C*d*f -
b*(C*d*e + c*C*f + 2*B*d*f))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(b^4*Sqrt[d]*Sqrt[f]) -
((24*a^4*C*d^2*f^2 - 3*a*b^3*(B*d^2*e^2 + c^2*f*(8*C*e + B*f) + 2*c*d*e*(4*C*e + 3*B*f)) - 8*a^3*b*d*f*(B*d*f
+ 5*C*(d*e + c*f)) - b^4*(A*d^2*e^2 - 2*c*d*e*(2*B*e + A*f) - c^2*(8*C*e^2 + 4*B*e*f - A*f^2)) + 3*a^2*b^2*(4
*B*d*f*(d*e + c*f) + C*(5*d^2*e^2 + 22*c*d*e*f + 5*c^2*f^2)))*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*
c - a*d]*Sqrt[e + f*x])])/(4*b^4*(b*c - a*d)^(3/2)*(b*e - a*f)^(3/2))

Rule 1613

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[{
Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(b*R*(a + b*x)^(m + 1)
*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e
- a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f
*R*(m + 1) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x], x], x]] /; FreeQ[{a, b,
c, d, e, f, n, p}, x] && PolyQ[Px, x] && ILtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sqrt{c+d x} \sqrt{e+f x} \left (A+B x+C x^2\right )}{(a+b x)^3} \, dx &=-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{2 b (b c-a d) (b e-a f) (a+b x)^2}-\frac{\int \frac{\sqrt{c+d x} \sqrt{e+f x} \left (-\frac{3 a^2 C (d e+c f)+b^2 (4 B c e-A d e-A c f)-a b (4 c C e+3 B d e+3 B c f-4 A d f)}{2 b}+\left (a B d f-\frac{3 a^2 C d f}{b}+2 a C (d e+c f)-b (2 c C e+A d f)\right ) x\right )}{(a+b x)^2} \, dx}{2 (b c-a d) (b e-a f)}\\ &=\frac{\left (6 a^3 C d f-b^3 (4 B c e-A d e-A c f)+a b^2 (8 c C e+3 B d e+3 B c f-2 A d f)-a^2 b (2 B d f+7 C (d e+c f))\right ) \sqrt{c+d x} (e+f x)^{3/2}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{2 b (b c-a d) (b e-a f) (a+b x)^2}-\frac{\int \frac{\sqrt{e+f x} \left (\frac{6 a^3 C d f (d e+3 c f)+b^3 \left (A d^2 e^2-2 c d e (2 B e+A f)-c^2 \left (8 C e^2+4 B e f-A f^2\right )\right )+a b^2 \left (d^2 e (3 B e-2 A f)+3 c^2 f (8 C e+B f)+2 c d \left (8 C e^2+5 B e f+A f^2\right )\right )-a^2 b \left (2 B d f (d e+3 c f)+C \left (7 d^2 e^2+34 c d e f+15 c^2 f^2\right )\right )}{4 b}+\frac{d \left (12 a^3 C d f^2-a^2 b f (17 C d e+11 c C f+4 B d f)-b^3 \left (4 c C e^2+A d e f+c f (4 B e-A f)\right )+a b^2 (B f (5 d e+3 c f)+4 C e (d e+4 c f))\right ) x}{2 b}\right )}{(a+b x) \sqrt{c+d x}} \, dx}{2 b (b c-a d) (b e-a f)^2}\\ &=-\frac{\left (12 a^3 C d f^2-a^2 b f (17 C d e+11 c C f+4 B d f)-b^3 \left (4 c C e^2+A d e f+c f (4 B e-A f)\right )+a b^2 (B f (5 d e+3 c f)+4 C e (d e+4 c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{4 b^3 (b c-a d) (b e-a f)^2}+\frac{\left (6 a^3 C d f-b^3 (4 B c e-A d e-A c f)+a b^2 (8 c C e+3 B d e+3 B c f-2 A d f)-a^2 b (2 B d f+7 C (d e+c f))\right ) \sqrt{c+d x} (e+f x)^{3/2}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{2 b (b c-a d) (b e-a f) (a+b x)^2}-\frac{\int \frac{\frac{d (b e-a f) \left (12 a^3 C d f (d e+c f)+a b^2 \left (3 B d^2 e^2+10 c d e (2 C e+B f)+c^2 f (20 C e+3 B f)\right )+b^3 \left (A d^2 e^2-2 c d e (2 B e+A f)-c^2 \left (8 C e^2+4 B e f-A f^2\right )\right )-a^2 b \left (4 B d f (d e+c f)+C \left (11 d^2 e^2+34 c d e f+11 c^2 f^2\right )\right )\right )}{4 b}+\frac{d (b c-a d) (b e-a f)^2 (6 a C d f-b (C d e+c C f+2 B d f)) x}{b}}{(a+b x) \sqrt{c+d x} \sqrt{e+f x}} \, dx}{2 b^2 d (b c-a d) (b e-a f)^2}\\ &=-\frac{\left (12 a^3 C d f^2-a^2 b f (17 C d e+11 c C f+4 B d f)-b^3 \left (4 c C e^2+A d e f+c f (4 B e-A f)\right )+a b^2 (B f (5 d e+3 c f)+4 C e (d e+4 c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{4 b^3 (b c-a d) (b e-a f)^2}+\frac{\left (6 a^3 C d f-b^3 (4 B c e-A d e-A c f)+a b^2 (8 c C e+3 B d e+3 B c f-2 A d f)-a^2 b (2 B d f+7 C (d e+c f))\right ) \sqrt{c+d x} (e+f x)^{3/2}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{2 b (b c-a d) (b e-a f) (a+b x)^2}-\frac{(6 a C d f-b (C d e+c C f+2 B d f)) \int \frac{1}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{2 b^4}+\frac{\left (24 a^4 C d^2 f^2-3 a b^3 \left (B d^2 e^2+c^2 f (8 C e+B f)+2 c d e (4 C e+3 B f)\right )-8 a^3 b d f (B d f+5 C (d e+c f))-b^4 \left (A d^2 e^2-2 c d e (2 B e+A f)-c^2 \left (8 C e^2+4 B e f-A f^2\right )\right )+3 a^2 b^2 \left (4 B d f (d e+c f)+C \left (5 d^2 e^2+22 c d e f+5 c^2 f^2\right )\right )\right ) \int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt{e+f x}} \, dx}{8 b^4 (b c-a d) (b e-a f)}\\ &=-\frac{\left (12 a^3 C d f^2-a^2 b f (17 C d e+11 c C f+4 B d f)-b^3 \left (4 c C e^2+A d e f+c f (4 B e-A f)\right )+a b^2 (B f (5 d e+3 c f)+4 C e (d e+4 c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{4 b^3 (b c-a d) (b e-a f)^2}+\frac{\left (6 a^3 C d f-b^3 (4 B c e-A d e-A c f)+a b^2 (8 c C e+3 B d e+3 B c f-2 A d f)-a^2 b (2 B d f+7 C (d e+c f))\right ) \sqrt{c+d x} (e+f x)^{3/2}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{2 b (b c-a d) (b e-a f) (a+b x)^2}-\frac{(6 a C d f-b (C d e+c C f+2 B d f)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{e-\frac{c f}{d}+\frac{f x^2}{d}}} \, dx,x,\sqrt{c+d x}\right )}{b^4 d}+\frac{\left (24 a^4 C d^2 f^2-3 a b^3 \left (B d^2 e^2+c^2 f (8 C e+B f)+2 c d e (4 C e+3 B f)\right )-8 a^3 b d f (B d f+5 C (d e+c f))-b^4 \left (A d^2 e^2-2 c d e (2 B e+A f)-c^2 \left (8 C e^2+4 B e f-A f^2\right )\right )+3 a^2 b^2 \left (4 B d f (d e+c f)+C \left (5 d^2 e^2+22 c d e f+5 c^2 f^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-b c+a d-(-b e+a f) x^2} \, dx,x,\frac{\sqrt{c+d x}}{\sqrt{e+f x}}\right )}{4 b^4 (b c-a d) (b e-a f)}\\ &=-\frac{\left (12 a^3 C d f^2-a^2 b f (17 C d e+11 c C f+4 B d f)-b^3 \left (4 c C e^2+A d e f+c f (4 B e-A f)\right )+a b^2 (B f (5 d e+3 c f)+4 C e (d e+4 c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{4 b^3 (b c-a d) (b e-a f)^2}+\frac{\left (6 a^3 C d f-b^3 (4 B c e-A d e-A c f)+a b^2 (8 c C e+3 B d e+3 B c f-2 A d f)-a^2 b (2 B d f+7 C (d e+c f))\right ) \sqrt{c+d x} (e+f x)^{3/2}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{2 b (b c-a d) (b e-a f) (a+b x)^2}-\frac{\left (24 a^4 C d^2 f^2-3 a b^3 \left (B d^2 e^2+c^2 f (8 C e+B f)+2 c d e (4 C e+3 B f)\right )-8 a^3 b d f (B d f+5 C (d e+c f))-b^4 \left (A d^2 e^2-2 c d e (2 B e+A f)-c^2 \left (8 C e^2+4 B e f-A f^2\right )\right )+3 a^2 b^2 \left (4 B d f (d e+c f)+C \left (5 d^2 e^2+22 c d e f+5 c^2 f^2\right )\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{c+d x}}{\sqrt{b c-a d} \sqrt{e+f x}}\right )}{4 b^4 (b c-a d)^{3/2} (b e-a f)^{3/2}}-\frac{(6 a C d f-b (C d e+c C f+2 B d f)) \operatorname{Subst}\left (\int \frac{1}{1-\frac{f x^2}{d}} \, dx,x,\frac{\sqrt{c+d x}}{\sqrt{e+f x}}\right )}{b^4 d}\\ &=-\frac{\left (12 a^3 C d f^2-a^2 b f (17 C d e+11 c C f+4 B d f)-b^3 \left (4 c C e^2+A d e f+c f (4 B e-A f)\right )+a b^2 (B f (5 d e+3 c f)+4 C e (d e+4 c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{4 b^3 (b c-a d) (b e-a f)^2}+\frac{\left (6 a^3 C d f-b^3 (4 B c e-A d e-A c f)+a b^2 (8 c C e+3 B d e+3 B c f-2 A d f)-a^2 b (2 B d f+7 C (d e+c f))\right ) \sqrt{c+d x} (e+f x)^{3/2}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{2 b (b c-a d) (b e-a f) (a+b x)^2}-\frac{(6 a C d f-b (C d e+c C f+2 B d f)) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right )}{b^4 \sqrt{d} \sqrt{f}}-\frac{\left (24 a^4 C d^2 f^2-3 a b^3 \left (B d^2 e^2+c^2 f (8 C e+B f)+2 c d e (4 C e+3 B f)\right )-8 a^3 b d f (B d f+5 C (d e+c f))-b^4 \left (A d^2 e^2-2 c d e (2 B e+A f)-c^2 \left (8 C e^2+4 B e f-A f^2\right )\right )+3 a^2 b^2 \left (4 B d f (d e+c f)+C \left (5 d^2 e^2+22 c d e f+5 c^2 f^2\right )\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{c+d x}}{\sqrt{b c-a d} \sqrt{e+f x}}\right )}{4 b^4 (b c-a d)^{3/2} (b e-a f)^{3/2}}\\ \end{align*}

Mathematica [B]  time = 6.44909, size = 2157, normalized size = 3.28 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*(e + f*x)^(3/2))/(2*b^2*(b*e - a*f)*(a + b*x)^2) - ((b*B - 2*a*C)*(c +
d*x)^(3/2)*(e + f*x)^(3/2))/(b*(b*c - a*d)*(b*e - a*f)*(a + b*x)) + (2*C*Sqrt[c + d*x]*Sqrt[e + f*x]*(1 + (d*
f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(1/(2*(1 + (d*f*(c + d*x))/((d*e
- c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(
d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e -
c*f)])])/(2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d
*e - c*f))))^(3/2))))/(b^3*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)]
) + (2*C*(b*c - a*d)*((Sqrt[f]*Sqrt[d*e - c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])
/Sqrt[d*e - c*f]])/(b*d*Sqrt[e + f*x]) + (Sqrt[b*e - a*f]*ArcTan[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c)
+ a*d]*Sqrt[e + f*x])])/(b*Sqrt[-(b*c) + a*d])))/b^3 - ((A*b^2 - a*(b*B - a*C))*(d*e - c*f)*((Sqrt[c + d*x]*Sq
rt[e + f*x])/((b*c - a*d)*(a + b*x)) - ((d*e - c*f)*ArcTan[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]
*Sqrt[e + f*x])])/((-(b*c) + a*d)^(3/2)*Sqrt[b*e - a*f])))/(4*b^2*(b*e - a*f)) - ((b*B - 2*a*C)*((-4*f*(c + d*
x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(
3/(4*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (3*(d*e - c*f)^2*((d^2
*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e -
c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*
e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt
[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(16*d^2*f^2*(c + d*x)^2*(1
+ (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))))))/(3*Sqrt[d/((d^2*e)/(d*e - c*f)
- (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + ((2*a*b*d*f + (b*(-2*a*d*f - b*(d*e + c*f)))/2)*((
2*Sqrt[c + d*x]*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^
(3/2)*(1/(2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (Sqrt[d*e - c*f
]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqr
t[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*(1 + (d*f*(c + d*x))/((d*e -
c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2))))/(b*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*
f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)]) - ((-(b*c) + a*d)*((2*Sqrt[f]*Sqrt[d*e - c*f]*Sqrt[d/((d^2*e)/(d*e - c*f
) - (c*d*f)/(d*e - c*f))]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcS
inh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(b*d^(
3/2)*Sqrt[e + f*x]) - (2*(-(b*e) + a*f)*ArcTan[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*
x])])/(b*Sqrt[-(b*c) + a*d]*Sqrt[b*e - a*f])))/b))/b))/(b^2*(b*c - a*d)*(b*e - a*f))

________________________________________________________________________________________

Maple [B]  time = 0.063, size = 12065, normalized size = 18.3 \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)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^3,x)

[Out]

result too large to display

________________________________________________________________________________________

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((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^3,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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)*(d*x+c)**(1/2)*(f*x+e)**(1/2)/(b*x+a)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 38.8443, size = 11268, normalized size = 17.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^3,x, algorithm="giac")

[Out]

1/4*(15*sqrt(d*f)*C*a^2*b^2*c^2*f^2*abs(d) - 3*sqrt(d*f)*B*a*b^3*c^2*f^2*abs(d) - sqrt(d*f)*A*b^4*c^2*f^2*abs(
d) - 40*sqrt(d*f)*C*a^3*b*c*d*f^2*abs(d) + 12*sqrt(d*f)*B*a^2*b^2*c*d*f^2*abs(d) + 24*sqrt(d*f)*C*a^4*d^2*f^2*
abs(d) - 8*sqrt(d*f)*B*a^3*b*d^2*f^2*abs(d) - 24*sqrt(d*f)*C*a*b^3*c^2*f*abs(d)*e + 4*sqrt(d*f)*B*b^4*c^2*f*ab
s(d)*e + 66*sqrt(d*f)*C*a^2*b^2*c*d*f*abs(d)*e - 18*sqrt(d*f)*B*a*b^3*c*d*f*abs(d)*e + 2*sqrt(d*f)*A*b^4*c*d*f
*abs(d)*e - 40*sqrt(d*f)*C*a^3*b*d^2*f*abs(d)*e + 12*sqrt(d*f)*B*a^2*b^2*d^2*f*abs(d)*e + 8*sqrt(d*f)*C*b^4*c^
2*abs(d)*e^2 - 24*sqrt(d*f)*C*a*b^3*c*d*abs(d)*e^2 + 4*sqrt(d*f)*B*b^4*c*d*abs(d)*e^2 + 15*sqrt(d*f)*C*a^2*b^2
*d^2*abs(d)*e^2 - 3*sqrt(d*f)*B*a*b^3*d^2*abs(d)*e^2 - sqrt(d*f)*A*b^4*d^2*abs(d)*e^2)*arctan(-1/2*(b*c*d*f -
2*a*d^2*f + b*d^2*e - (sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*b)/(sqrt(a*b*c*d*f^2 -
a^2*d^2*f^2 - b^2*c*d*f*e + a*b*d^2*f*e)*d))/((a*b^5*c*f - a^2*b^4*d*f - b^6*c*e + a*b^5*d*e)*sqrt(a*b*c*d*f^
2 - a^2*d^2*f^2 - b^2*c*d*f*e + a*b*d^2*f*e)*d) + 1/2*(9*sqrt(d*f)*C*a^2*b^3*c^5*d^3*f^5*abs(d) - 5*sqrt(d*f)*
B*a*b^4*c^5*d^3*f^5*abs(d) + sqrt(d*f)*A*b^5*c^5*d^3*f^5*abs(d) - 10*sqrt(d*f)*C*a^3*b^2*c^4*d^4*f^5*abs(d) +
6*sqrt(d*f)*B*a^2*b^3*c^4*d^4*f^5*abs(d) - 2*sqrt(d*f)*A*a*b^4*c^4*d^4*f^5*abs(d) - 8*sqrt(d*f)*C*a*b^4*c^5*d^
3*f^4*abs(d)*e + 4*sqrt(d*f)*B*b^5*c^5*d^3*f^4*abs(d)*e - 27*sqrt(d*f)*C*a^2*b^3*c^4*d^4*f^4*abs(d)*e + 15*sqr
t(d*f)*B*a*b^4*c^4*d^4*f^4*abs(d)*e - 3*sqrt(d*f)*A*b^5*c^4*d^4*f^4*abs(d)*e + 40*sqrt(d*f)*C*a^3*b^2*c^3*d^5*
f^4*abs(d)*e - 24*sqrt(d*f)*B*a^2*b^3*c^3*d^5*f^4*abs(d)*e + 8*sqrt(d*f)*A*a*b^4*c^3*d^5*f^4*abs(d)*e - 27*sqr
t(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^2*b^3*c^4*d^2*f^4*abs(d) + 15*sqr
t(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*a*b^4*c^4*d^2*f^4*abs(d) - 3*sqrt(d
*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*A*b^5*c^4*d^2*f^4*abs(d) + 80*sqrt(d*f)*
(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^3*b^2*c^3*d^3*f^4*abs(d) - 44*sqrt(d*f)*
(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*a^2*b^3*c^3*d^3*f^4*abs(d) + 8*sqrt(d*f)*(
sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*A*a*b^4*c^3*d^3*f^4*abs(d) - 56*sqrt(d*f)*(sq
rt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^4*b*c^2*d^4*f^4*abs(d) + 32*sqrt(d*f)*(sqrt
(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*a^3*b^2*c^2*d^4*f^4*abs(d) - 8*sqrt(d*f)*(sqrt(
d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*A*a^2*b^3*c^2*d^4*f^4*abs(d) + 32*sqrt(d*f)*C*a*b^
4*c^4*d^4*f^3*abs(d)*e^2 - 16*sqrt(d*f)*B*b^5*c^4*d^4*f^3*abs(d)*e^2 + 18*sqrt(d*f)*C*a^2*b^3*c^3*d^5*f^3*abs(
d)*e^2 - 10*sqrt(d*f)*B*a*b^4*c^3*d^5*f^3*abs(d)*e^2 + 2*sqrt(d*f)*A*b^5*c^3*d^5*f^3*abs(d)*e^2 - 60*sqrt(d*f)
*C*a^3*b^2*c^2*d^6*f^3*abs(d)*e^2 + 36*sqrt(d*f)*B*a^2*b^3*c^2*d^6*f^3*abs(d)*e^2 - 12*sqrt(d*f)*A*a*b^4*c^2*d
^6*f^3*abs(d)*e^2 + 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a*b^4*c^4
*d^2*f^3*abs(d)*e - 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*b^5*c^4*d
^2*f^3*abs(d)*e - 44*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^2*b^3*c^3
*d^3*f^3*abs(d)*e + 20*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*a*b^4*c^3
*d^3*f^3*abs(d)*e + 4*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*A*b^5*c^3*d^
3*f^3*abs(d)*e - 80*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^3*b^2*c^2*
d^4*f^3*abs(d)*e + 44*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*a^2*b^3*c^
2*d^4*f^3*abs(d)*e - 8*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*A*a*b^4*c^2
*d^4*f^3*abs(d)*e + 112*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^4*b*c*
d^5*f^3*abs(d)*e - 64*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*a^3*b^2*c*
d^5*f^3*abs(d)*e + 16*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*A*a^2*b^3*c*
d^5*f^3*abs(d)*e + 27*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a^2*b^3*c^
3*d*f^3*abs(d) - 15*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*a*b^4*c^3*d*
f^3*abs(d) + 3*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*A*b^5*c^3*d*f^3*abs
(d) - 102*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a^3*b^2*c^2*d^2*f^3*ab
s(d) + 58*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*a^2*b^3*c^2*d^2*f^3*ab
s(d) - 14*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*A*a*b^4*c^2*d^2*f^3*abs(
d) + 152*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a^4*b*c*d^3*f^3*abs(d)
- 88*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*a^3*b^2*c*d^3*f^3*abs(d) +
24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*A*a^2*b^3*c*d^3*f^3*abs(d) - 80
*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a^5*d^4*f^3*abs(d) + 48*sqrt(d*
f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*a^4*b*d^4*f^3*abs(d) - 16*sqrt(d*f)*(sq
rt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*A*a^3*b^2*d^4*f^3*abs(d) - 48*sqrt(d*f)*C*a*b^4
*c^3*d^5*f^2*abs(d)*e^3 + 24*sqrt(d*f)*B*b^5*c^3*d^5*f^2*abs(d)*e^3 + 18*sqrt(d*f)*C*a^2*b^3*c^2*d^6*f^2*abs(d
)*e^3 - 10*sqrt(d*f)*B*a*b^4*c^2*d^6*f^2*abs(d)*e^3 + 2*sqrt(d*f)*A*b^5*c^2*d^6*f^2*abs(d)*e^3 + 40*sqrt(d*f)*
C*a^3*b^2*c*d^7*f^2*abs(d)*e^3 - 24*sqrt(d*f)*B*a^2*b^3*c*d^7*f^2*abs(d)*e^3 + 8*sqrt(d*f)*A*a*b^4*c*d^7*f^2*a
bs(d)*e^3 - 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a*b^4*c^3*d^3*f^2
*abs(d)*e^2 + 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*b^5*c^3*d^3*f^2
*abs(d)*e^2 + 142*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^2*b^3*c^2*d^
4*f^2*abs(d)*e^2 - 70*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*a*b^4*c^2*
d^4*f^2*abs(d)*e^2 - 2*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*A*b^5*c^2*d
^4*f^2*abs(d)*e^2 - 80*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^3*b^2*c
*d^5*f^2*abs(d)*e^2 + 44*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*a^2*b^3
*c*d^5*f^2*abs(d)*e^2 - 8*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*A*a*b^4*
c*d^5*f^2*abs(d)*e^2 - 56*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^4*b*
d^6*f^2*abs(d)*e^2 + 32*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*a^3*b^2*
d^6*f^2*abs(d)*e^2 - 8*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*A*a^2*b^3*d
^6*f^2*abs(d)*e^2 - 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a*b^4*c^3
*d*f^2*abs(d)*e + 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*b^5*c^3*d*f
^2*abs(d)*e + 109*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a^2*b^3*c^2*d^
2*f^2*abs(d)*e - 57*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*a*b^4*c^2*d^
2*f^2*abs(d)*e + 5*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*A*b^5*c^2*d^2*f
^2*abs(d)*e - 228*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a^3*b^2*c*d^3*
f^2*abs(d)*e + 124*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*a^2*b^3*c*d^3
*f^2*abs(d)*e - 20*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*A*a*b^4*c*d^3*f
^2*abs(d)*e + 152*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a^4*b*d^4*f^2*
abs(d)*e - 88*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*a^3*b^2*d^4*f^2*ab
s(d)*e + 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*A*a^2*b^3*d^4*f^2*abs(
d)*e - 9*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*C*a^2*b^3*c^2*f^2*abs(d)
+ 5*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*B*a*b^4*c^2*f^2*abs(d) - sqrt(
d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*A*b^5*c^2*f^2*abs(d) + 32*sqrt(d*f)*(sq
rt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*C*a^3*b^2*c*d*f^2*abs(d) - 20*sqrt(d*f)*(sqrt(d
*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*B*a^2*b^3*c*d*f^2*abs(d) + 8*sqrt(d*f)*(sqrt(d*f)*s
qrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*A*a*b^4*c*d*f^2*abs(d) - 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*
x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*C*a^4*b*d^2*f^2*abs(d) + 16*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c)
- sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*B*a^3*b^2*d^2*f^2*abs(d) - 8*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sq
rt((d*x + c)*d*f - c*d*f + d^2*e))^6*A*a^2*b^3*d^2*f^2*abs(d) + 32*sqrt(d*f)*C*a*b^4*c^2*d^6*f*abs(d)*e^4 - 16
*sqrt(d*f)*B*b^5*c^2*d^6*f*abs(d)*e^4 - 27*sqrt(d*f)*C*a^2*b^3*c*d^7*f*abs(d)*e^4 + 15*sqrt(d*f)*B*a*b^4*c*d^7
*f*abs(d)*e^4 - 3*sqrt(d*f)*A*b^5*c*d^7*f*abs(d)*e^4 - 10*sqrt(d*f)*C*a^3*b^2*d^8*f*abs(d)*e^4 + 6*sqrt(d*f)*B
*a^2*b^3*d^8*f*abs(d)*e^4 - 2*sqrt(d*f)*A*a*b^4*d^8*f*abs(d)*e^4 - 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqr
t((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a*b^4*c^2*d^4*f*abs(d)*e^3 + 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqr
t((d*x + c)*d*f - c*d*f + d^2*e))^2*B*b^5*c^2*d^4*f*abs(d)*e^3 - 44*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(
(d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^2*b^3*c*d^5*f*abs(d)*e^3 + 20*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(
(d*x + c)*d*f - c*d*f + d^2*e))^2*B*a*b^4*c*d^5*f*abs(d)*e^3 + 4*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*
x + c)*d*f - c*d*f + d^2*e))^2*A*b^5*c*d^5*f*abs(d)*e^3 + 80*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x +
c)*d*f - c*d*f + d^2*e))^2*C*a^3*b^2*d^6*f*abs(d)*e^3 - 44*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)
*d*f - c*d*f + d^2*e))^2*B*a^2*b^3*d^6*f*abs(d)*e^3 + 8*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*
f - c*d*f + d^2*e))^2*A*a*b^4*d^6*f*abs(d)*e^3 - 16*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f -
c*d*f + d^2*e))^4*C*a*b^4*c^2*d^2*f*abs(d)*e^2 + 8*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c
*d*f + d^2*e))^4*B*b^5*c^2*d^2*f*abs(d)*e^2 + 109*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*
d*f + d^2*e))^4*C*a^2*b^3*c*d^3*f*abs(d)*e^2 - 57*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*
d*f + d^2*e))^4*B*a*b^4*c*d^3*f*abs(d)*e^2 + 5*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f
+ d^2*e))^4*A*b^5*c*d^3*f*abs(d)*e^2 - 102*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f +
d^2*e))^4*C*a^3*b^2*d^4*f*abs(d)*e^2 + 58*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^
2*e))^4*B*a^2*b^3*d^4*f*abs(d)*e^2 - 14*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*
e))^4*A*a*b^4*d^4*f*abs(d)*e^2 + 8*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6
*C*a*b^4*c^2*f*abs(d)*e - 4*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*B*b^5*
c^2*f*abs(d)*e - 38*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*C*a^2*b^3*c*d*
f*abs(d)*e + 22*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*B*a*b^4*c*d*f*abs(
d)*e - 6*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*A*b^5*c*d*f*abs(d)*e + 32
*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*C*a^3*b^2*d^2*f*abs(d)*e - 20*sqr
t(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*B*a^2*b^3*d^2*f*abs(d)*e + 8*sqrt(d*f
)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*A*a*b^4*d^2*f*abs(d)*e - 8*sqrt(d*f)*C*a*b
^4*c*d^7*abs(d)*e^5 + 4*sqrt(d*f)*B*b^5*c*d^7*abs(d)*e^5 + 9*sqrt(d*f)*C*a^2*b^3*d^8*abs(d)*e^5 - 5*sqrt(d*f)*
B*a*b^4*d^8*abs(d)*e^5 + sqrt(d*f)*A*b^5*d^8*abs(d)*e^5 + 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x +
c)*d*f - c*d*f + d^2*e))^2*C*a*b^4*c*d^5*abs(d)*e^4 - 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d
*f - c*d*f + d^2*e))^2*B*b^5*c*d^5*abs(d)*e^4 - 27*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c
*d*f + d^2*e))^2*C*a^2*b^3*d^6*abs(d)*e^4 + 15*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f
+ d^2*e))^2*B*a*b^4*d^6*abs(d)*e^4 - 3*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*
e))^2*A*b^5*d^6*abs(d)*e^4 - 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*
a*b^4*c*d^3*abs(d)*e^3 + 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*b^5*
c*d^3*abs(d)*e^3 + 27*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a^2*b^3*d^
4*abs(d)*e^3 - 15*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*a*b^4*d^4*abs(
d)*e^3 + 3*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*A*b^5*d^4*abs(d)*e^3 +
8*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*C*a*b^4*c*d*abs(d)*e^2 - 4*sqrt(
d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*B*b^5*c*d*abs(d)*e^2 - 9*sqrt(d*f)*(sqr
t(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*C*a^2*b^3*d^2*abs(d)*e^2 + 5*sqrt(d*f)*(sqrt(d*f
)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*B*a*b^4*d^2*abs(d)*e^2 - sqrt(d*f)*(sqrt(d*f)*sqrt(d*
x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*A*b^5*d^2*abs(d)*e^2)/((a*b^5*c*f - a^2*b^4*d*f - b^6*c*e + a*
b^5*d*e)*(b*c^2*d^2*f^2 - 2*b*c*d^3*f*e - 2*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*
b*c*d*f + 4*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*a*d^2*f + b*d^4*e^2 - 2*(sqrt(d*
f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*b*d^2*e + (sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*
d*f - c*d*f + d^2*e))^4*b)^2) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)*sqrt(d*x + c)*C*abs(d)/(b^3*d^2) - 1/2*(sq
rt(d*f)*C*b*c*f*abs(d) - 6*sqrt(d*f)*C*a*d*f*abs(d) + 2*sqrt(d*f)*B*b*d*f*abs(d) + sqrt(d*f)*C*b*d*abs(d)*e)*l
og((sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2)/(b^4*d^2*f)