\(\int \frac {(e+f x)^3 (A+B x+C x^2)}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx\) [27]

Optimal result

Integrand size = 40, antiderivative size = 501 \[ \int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {\left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (3 B e+4 A f)\right )\right ) (e+f x)^2 \left (a^2-b^2 x^2\right )}{60 b^4 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {(C e-5 B f) (e+f x)^3 \left (a^2-b^2 x^2\right )}{20 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C (e+f x)^4 \left (a^2-b^2 x^2\right )}{5 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left (4 \left (16 a^4 C f^4+4 a^2 b^2 f^2 \left (13 C e^2+5 f (3 B e+A f)\right )-b^4 e^2 \left (3 C e^2-5 f (3 B e+16 A f)\right )\right )+b^2 f \left (a^2 f^2 (71 C e+45 B f)-2 b^2 e \left (3 C e^2-5 f (3 B e+10 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{120 b^6 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (4 A \left (2 b^4 e^3+3 a^2 b^2 e f^2\right )+a^2 \left (3 a^2 f^2 (3 C e+B f)+4 b^2 e^2 (C e+3 B f)\right )\right ) \sqrt {a^2 c-b^2 c x^2} \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{8 b^5 \sqrt {c} \sqrt {a+b x} \sqrt {a c-b c x}} \]

[Out]

-1/60*(16*a^2*C*f^2-b^2*(3*C*e^2-5*f*(4*A*f+3*B*e)))*(f*x+e)^2*(-b^2*x^2+a^2)/b^4/f/(b*x+a)^(1/2)/(-b*c*x+a*c)
^(1/2)+1/20*(-5*B*f+C*e)*(f*x+e)^3*(-b^2*x^2+a^2)/b^2/f/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-1/5*C*(f*x+e)^4*(-b^2
*x^2+a^2)/b^2/f/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-1/120*(64*a^4*C*f^4+16*a^2*b^2*f^2*(13*C*e^2+5*f*(A*f+3*B*e))
-4*b^4*e^2*(3*C*e^2-5*f*(16*A*f+3*B*e))+b^2*f*(a^2*f^2*(45*B*f+71*C*e)-2*b^2*e*(3*C*e^2-5*f*(10*A*f+3*B*e)))*x
)*(-b^2*x^2+a^2)/b^6/f/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)+1/8*(4*A*(3*a^2*b^2*e*f^2+2*b^4*e^3)+a^2*(3*a^2*f^2*(B
*f+3*C*e)+4*b^2*e^2*(3*B*f+C*e)))*arctan(b*x*c^(1/2)/(-b^2*c*x^2+a^2*c)^(1/2))*(-b^2*c*x^2+a^2*c)^(1/2)/b^5/c^
(1/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 496, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1624, 1668, 847, 794, 223, 209} \[ \int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {\left (a^2-b^2 x^2\right ) (e+f x)^2 \left (-\frac {16 a^2 C f^2}{b^2}-5 f (4 A f+3 B e)+3 C e^2\right )}{60 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (a^2-b^2 x^2\right ) (e+f x)^3 (C e-5 B f)}{20 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C \left (a^2-b^2 x^2\right ) (e+f x)^4}{5 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\sqrt {a^2 c-b^2 c x^2} \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right ) \left (3 a^4 f^2 (B f+3 C e)+4 A \left (3 a^2 b^2 e f^2+2 b^4 e^3\right )+4 a^2 b^2 e^2 (3 B f+C e)\right )}{8 b^5 \sqrt {c} \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left (a^2-b^2 x^2\right ) \left (b^2 f x \left (a^2 f^2 (45 B f+71 C e)-b^2 \left (6 C e^3-10 e f (10 A f+3 B e)\right )\right )+4 \left (16 a^4 C f^4+4 a^2 b^2 f^2 \left (5 f (A f+3 B e)+13 C e^2\right )+b^4 \left (-e^2\right ) \left (3 C e^2-5 f (16 A f+3 B e)\right )\right )\right )}{120 b^6 f \sqrt {a+b x} \sqrt {a c-b c x}} \]

[In]

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

[Out]

((3*C*e^2 - (16*a^2*C*f^2)/b^2 - 5*f*(3*B*e + 4*A*f))*(e + f*x)^2*(a^2 - b^2*x^2))/(60*b^2*f*Sqrt[a + b*x]*Sqr
t[a*c - b*c*x]) + ((C*e - 5*B*f)*(e + f*x)^3*(a^2 - b^2*x^2))/(20*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - (C*
(e + f*x)^4*(a^2 - b^2*x^2))/(5*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - ((4*(16*a^4*C*f^4 + 4*a^2*b^2*f^2*(13
*C*e^2 + 5*f*(3*B*e + A*f)) - b^4*e^2*(3*C*e^2 - 5*f*(3*B*e + 16*A*f))) + b^2*f*(a^2*f^2*(71*C*e + 45*B*f) - b
^2*(6*C*e^3 - 10*e*f*(3*B*e + 10*A*f)))*x)*(a^2 - b^2*x^2))/(120*b^6*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + ((3*
a^4*f^2*(3*C*e + B*f) + 4*a^2*b^2*e^2*(C*e + 3*B*f) + 4*A*(2*b^4*e^3 + 3*a^2*b^2*e*f^2))*Sqrt[a^2*c - b^2*c*x^
2]*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(8*b^5*Sqrt[c]*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

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 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 847

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^
m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 1624

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[(a
+ b*x)^FracPart[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]), Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1668

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2 c-b^2 c x^2} \int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {a^2 c-b^2 c x^2}} \, dx}{\sqrt {a+b x} \sqrt {a c-b c x}} \\ & = -\frac {C (e+f x)^4 \left (a^2-b^2 x^2\right )}{5 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\sqrt {a^2 c-b^2 c x^2} \int \frac {(e+f x)^3 \left (-c \left (5 A b^2+4 a^2 C\right ) f^2+b^2 c f (C e-5 B f) x\right )}{\sqrt {a^2 c-b^2 c x^2}} \, dx}{5 b^2 c f^2 \sqrt {a+b x} \sqrt {a c-b c x}} \\ & = \frac {(C e-5 B f) (e+f x)^3 \left (a^2-b^2 x^2\right )}{20 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C (e+f x)^4 \left (a^2-b^2 x^2\right )}{5 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\sqrt {a^2 c-b^2 c x^2} \int \frac {(e+f x)^2 \left (b^2 c^2 f^2 \left (20 A b^2 e+a^2 (13 C e+15 B f)\right )+b^2 c^2 f \left (4 \left (5 A b^2+4 a^2 C\right ) f^2-3 b^2 e (C e-5 B f)\right ) x\right )}{\sqrt {a^2 c-b^2 c x^2}} \, dx}{20 b^4 c^2 f^2 \sqrt {a+b x} \sqrt {a c-b c x}} \\ & = -\frac {\left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (3 B e+4 A f)\right )\right ) (e+f x)^2 \left (a^2-b^2 x^2\right )}{60 b^4 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {(C e-5 B f) (e+f x)^3 \left (a^2-b^2 x^2\right )}{20 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C (e+f x)^4 \left (a^2-b^2 x^2\right )}{5 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\sqrt {a^2 c-b^2 c x^2} \int \frac {(e+f x) \left (-b^2 c^3 f^2 \left (32 a^4 C f^2+3 a^2 b^2 e (11 C e+25 B f)+20 A \left (3 b^4 e^2+2 a^2 b^2 f^2\right )\right )-b^4 c^3 f \left (a^2 f^2 (71 C e+45 B f)-b^2 \left (6 C e^3-10 e f (3 B e+10 A f)\right )\right ) x\right )}{\sqrt {a^2 c-b^2 c x^2}} \, dx}{60 b^6 c^3 f^2 \sqrt {a+b x} \sqrt {a c-b c x}} \\ & = -\frac {\left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (3 B e+4 A f)\right )\right ) (e+f x)^2 \left (a^2-b^2 x^2\right )}{60 b^4 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {(C e-5 B f) (e+f x)^3 \left (a^2-b^2 x^2\right )}{20 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C (e+f x)^4 \left (a^2-b^2 x^2\right )}{5 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left (4 \left (16 a^4 C f^4+4 a^2 b^2 f^2 \left (13 C e^2+5 f (3 B e+A f)\right )-b^4 e^2 \left (3 C e^2-5 f (3 B e+16 A f)\right )\right )+b^2 f \left (a^2 f^2 (71 C e+45 B f)-b^2 \left (6 C e^3-10 e f (3 B e+10 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{120 b^6 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (\left (3 a^4 f^2 (3 C e+B f)+4 a^2 b^2 e^2 (C e+3 B f)+4 A \left (2 b^4 e^3+3 a^2 b^2 e f^2\right )\right ) \sqrt {a^2 c-b^2 c x^2}\right ) \int \frac {1}{\sqrt {a^2 c-b^2 c x^2}} \, dx}{8 b^4 \sqrt {a+b x} \sqrt {a c-b c x}} \\ & = -\frac {\left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (3 B e+4 A f)\right )\right ) (e+f x)^2 \left (a^2-b^2 x^2\right )}{60 b^4 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {(C e-5 B f) (e+f x)^3 \left (a^2-b^2 x^2\right )}{20 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C (e+f x)^4 \left (a^2-b^2 x^2\right )}{5 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left (4 \left (16 a^4 C f^4+4 a^2 b^2 f^2 \left (13 C e^2+5 f (3 B e+A f)\right )-b^4 e^2 \left (3 C e^2-5 f (3 B e+16 A f)\right )\right )+b^2 f \left (a^2 f^2 (71 C e+45 B f)-b^2 \left (6 C e^3-10 e f (3 B e+10 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{120 b^6 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (\left (3 a^4 f^2 (3 C e+B f)+4 a^2 b^2 e^2 (C e+3 B f)+4 A \left (2 b^4 e^3+3 a^2 b^2 e f^2\right )\right ) \sqrt {a^2 c-b^2 c x^2}\right ) \text {Subst}\left (\int \frac {1}{1+b^2 c x^2} \, dx,x,\frac {x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{8 b^4 \sqrt {a+b x} \sqrt {a c-b c x}} \\ & = -\frac {\left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (3 B e+4 A f)\right )\right ) (e+f x)^2 \left (a^2-b^2 x^2\right )}{60 b^4 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {(C e-5 B f) (e+f x)^3 \left (a^2-b^2 x^2\right )}{20 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C (e+f x)^4 \left (a^2-b^2 x^2\right )}{5 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left (4 \left (16 a^4 C f^4+4 a^2 b^2 f^2 \left (13 C e^2+5 f (3 B e+A f)\right )-b^4 e^2 \left (3 C e^2-5 f (3 B e+16 A f)\right )\right )+b^2 f \left (a^2 f^2 (71 C e+45 B f)-b^2 \left (6 C e^3-10 e f (3 B e+10 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{120 b^6 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (3 a^4 f^2 (3 C e+B f)+4 a^2 b^2 e^2 (C e+3 B f)+4 A \left (2 b^4 e^3+3 a^2 b^2 e f^2\right )\right ) \sqrt {a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{8 b^5 \sqrt {c} \sqrt {a+b x} \sqrt {a c-b c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.56 \[ \int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {-\left ((a-b x) \sqrt {a+b x} \left (64 a^4 C f^3+a^2 b^2 f \left (5 f (48 B e+16 A f+9 B f x)+C \left (240 e^2+135 e f x+32 f^2 x^2\right )\right )+2 b^4 \left (10 A f \left (18 e^2+9 e f x+2 f^2 x^2\right )+15 B \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )+3 C x \left (10 e^3+20 e^2 f x+15 e f^2 x^2+4 f^3 x^3\right )\right )\right )\right )+30 b \left (3 a^4 f^2 (3 C e+B f)+4 a^2 b^2 e^2 (C e+3 B f)+4 A \left (2 b^4 e^3+3 a^2 b^2 e f^2\right )\right ) \sqrt {a-b x} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{120 b^6 \sqrt {c (a-b x)}} \]

[In]

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

[Out]

(-((a - b*x)*Sqrt[a + b*x]*(64*a^4*C*f^3 + a^2*b^2*f*(5*f*(48*B*e + 16*A*f + 9*B*f*x) + C*(240*e^2 + 135*e*f*x
+ 32*f^2*x^2)) + 2*b^4*(10*A*f*(18*e^2 + 9*e*f*x + 2*f^2*x^2) + 15*B*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x
^3) + 3*C*x*(10*e^3 + 20*e^2*f*x + 15*e*f^2*x^2 + 4*f^3*x^3)))) + 30*b*(3*a^4*f^2*(3*C*e + B*f) + 4*a^2*b^2*e^
2*(C*e + 3*B*f) + 4*A*(2*b^4*e^3 + 3*a^2*b^2*e*f^2))*Sqrt[a - b*x]*ArcTan[Sqrt[a + b*x]/Sqrt[a - b*x]])/(120*b
^6*Sqrt[c*(a - b*x)])

Maple [A] (verified)

Time = 1.65 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.78

method	result	size
risch	\(-\frac {\left (24 C \,f^{3} x^{4} b^{4}+30 B \,b^{4} f^{3} x^{3}+90 C \,b^{4} e \,f^{2} x^{3}+40 A \,b^{4} f^{3} x^{2}+120 B \,b^{4} e \,f^{2} x^{2}+32 C \,a^{2} b^{2} f^{3} x^{2}+120 C \,b^{4} e^{2} f \,x^{2}+180 A \,b^{4} e \,f^{2} x +45 B \,a^{2} b^{2} f^{3} x +180 B \,b^{4} e^{2} f x +135 C \,a^{2} b^{2} e \,f^{2} x +60 C \,b^{4} e^{3} x +80 A \,a^{2} b^{2} f^{3}+360 A \,b^{4} e^{2} f +240 B \,a^{2} b^{2} e \,f^{2}+120 B \,b^{4} e^{3}+64 C \,a^{4} f^{3}+240 C \,a^{2} b^{2} e^{2} f \right ) \sqrt {b x +a}\, \left (-b x +a \right )}{120 b^{6} \sqrt {-c \left (b x -a \right )}}+\frac {\left (12 A \,a^{2} b^{2} e \,f^{2}+8 A \,b^{4} e^{3}+3 B \,a^{4} f^{3}+12 B \,a^{2} e^{2} f \,b^{2}+9 C \,a^{4} e \,f^{2}+4 C \,a^{2} e^{3} b^{2}\right ) \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}}{8 b^{4} \sqrt {b^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(390\)
default	\(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (-24 C \,b^{4} f^{3} x^{4} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}-30 B \,b^{4} f^{3} x^{3} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}-90 C \,b^{4} e \,f^{2} x^{3} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}+180 A \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) a^{2} b^{4} c e \,f^{2}+120 A \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) b^{6} c \,e^{3}-40 A \,b^{4} f^{3} x^{2} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}+45 B \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) a^{4} b^{2} c \,f^{3}+180 B \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) a^{2} b^{4} c \,e^{2} f -120 B \,b^{4} e \,f^{2} x^{2} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}+135 C \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) a^{4} b^{2} c e \,f^{2}+60 C \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) a^{2} b^{4} c \,e^{3}-32 C \,a^{2} b^{2} f^{3} x^{2} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}-120 C \,b^{4} e^{2} f \,x^{2} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}-180 A \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, b^{4} e \,f^{2} x -45 B \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, a^{2} b^{2} f^{3} x -180 B \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, b^{4} e^{2} f x -135 C \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, a^{2} b^{2} e \,f^{2} x -60 C \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, b^{4} e^{3} x -80 A \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, a^{2} b^{2} f^{3}-360 A \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, b^{4} e^{2} f -240 B \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, a^{2} b^{2} e \,f^{2}-120 B \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, b^{4} e^{3}-64 C \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, a^{4} f^{3}-240 C \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, a^{2} b^{2} e^{2} f \right )}{120 c \,b^{6} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}}\)	\(913\)

[In]

int((f*x+e)^3*(C*x^2+B*x+A)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/120*(24*C*b^4*f^3*x^4+30*B*b^4*f^3*x^3+90*C*b^4*e*f^2*x^3+40*A*b^4*f^3*x^2+120*B*b^4*e*f^2*x^2+32*C*a^2*b^2
*f^3*x^2+120*C*b^4*e^2*f*x^2+180*A*b^4*e*f^2*x+45*B*a^2*b^2*f^3*x+180*B*b^4*e^2*f*x+135*C*a^2*b^2*e*f^2*x+60*C
*b^4*e^3*x+80*A*a^2*b^2*f^3+360*A*b^4*e^2*f+240*B*a^2*b^2*e*f^2+120*B*b^4*e^3+64*C*a^4*f^3+240*C*a^2*b^2*e^2*f
)*(b*x+a)^(1/2)/b^6*(-b*x+a)/(-c*(b*x-a))^(1/2)+1/8*(12*A*a^2*b^2*e*f^2+8*A*b^4*e^3+3*B*a^4*f^3+12*B*a^2*b^2*e
^2*f+9*C*a^4*e*f^2+4*C*a^2*b^2*e^3)/b^4/(b^2*c)^(1/2)*arctan((b^2*c)^(1/2)*x/(-b^2*c*x^2+a^2*c)^(1/2))*(-(b*x+
a)*c*(b*x-a))^(1/2)/(b*x+a)^(1/2)/(-c*(b*x-a))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.40 \[ \int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\left [-\frac {15 \, {\left (12 \, B a^{2} b^{3} e^{2} f + 3 \, B a^{4} b f^{3} + 4 \, {\left (C a^{2} b^{3} + 2 \, A b^{5}\right )} e^{3} + 3 \, {\left (3 \, C a^{4} b + 4 \, A a^{2} b^{3}\right )} e f^{2}\right )} \sqrt {-c} \log \left (2 \, b^{2} c x^{2} - 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {-c} x - a^{2} c\right ) + 2 \, {\left (24 \, C b^{4} f^{3} x^{4} + 120 \, B b^{4} e^{3} + 240 \, B a^{2} b^{2} e f^{2} + 120 \, {\left (2 \, C a^{2} b^{2} + 3 \, A b^{4}\right )} e^{2} f + 16 \, {\left (4 \, C a^{4} + 5 \, A a^{2} b^{2}\right )} f^{3} + 30 \, {\left (3 \, C b^{4} e f^{2} + B b^{4} f^{3}\right )} x^{3} + 8 \, {\left (15 \, C b^{4} e^{2} f + 15 \, B b^{4} e f^{2} + {\left (4 \, C a^{2} b^{2} + 5 \, A b^{4}\right )} f^{3}\right )} x^{2} + 15 \, {\left (4 \, C b^{4} e^{3} + 12 \, B b^{4} e^{2} f + 3 \, B a^{2} b^{2} f^{3} + 3 \, {\left (3 \, C a^{2} b^{2} + 4 \, A b^{4}\right )} e f^{2}\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{240 \, b^{6} c}, -\frac {15 \, {\left (12 \, B a^{2} b^{3} e^{2} f + 3 \, B a^{4} b f^{3} + 4 \, {\left (C a^{2} b^{3} + 2 \, A b^{5}\right )} e^{3} + 3 \, {\left (3 \, C a^{4} b + 4 \, A a^{2} b^{3}\right )} e f^{2}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {c} x}{b^{2} c x^{2} - a^{2} c}\right ) + {\left (24 \, C b^{4} f^{3} x^{4} + 120 \, B b^{4} e^{3} + 240 \, B a^{2} b^{2} e f^{2} + 120 \, {\left (2 \, C a^{2} b^{2} + 3 \, A b^{4}\right )} e^{2} f + 16 \, {\left (4 \, C a^{4} + 5 \, A a^{2} b^{2}\right )} f^{3} + 30 \, {\left (3 \, C b^{4} e f^{2} + B b^{4} f^{3}\right )} x^{3} + 8 \, {\left (15 \, C b^{4} e^{2} f + 15 \, B b^{4} e f^{2} + {\left (4 \, C a^{2} b^{2} + 5 \, A b^{4}\right )} f^{3}\right )} x^{2} + 15 \, {\left (4 \, C b^{4} e^{3} + 12 \, B b^{4} e^{2} f + 3 \, B a^{2} b^{2} f^{3} + 3 \, {\left (3 \, C a^{2} b^{2} + 4 \, A b^{4}\right )} e f^{2}\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{120 \, b^{6} c}\right ] \]

[In]

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

[Out]

[-1/240*(15*(12*B*a^2*b^3*e^2*f + 3*B*a^4*b*f^3 + 4*(C*a^2*b^3 + 2*A*b^5)*e^3 + 3*(3*C*a^4*b + 4*A*a^2*b^3)*e*
f^2)*sqrt(-c)*log(2*b^2*c*x^2 - 2*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*sqrt(-c)*x - a^2*c) + 2*(24*C*b^4*f^3*x^4
+ 120*B*b^4*e^3 + 240*B*a^2*b^2*e*f^2 + 120*(2*C*a^2*b^2 + 3*A*b^4)*e^2*f + 16*(4*C*a^4 + 5*A*a^2*b^2)*f^3 +
30*(3*C*b^4*e*f^2 + B*b^4*f^3)*x^3 + 8*(15*C*b^4*e^2*f + 15*B*b^4*e*f^2 + (4*C*a^2*b^2 + 5*A*b^4)*f^3)*x^2 + 1
5*(4*C*b^4*e^3 + 12*B*b^4*e^2*f + 3*B*a^2*b^2*f^3 + 3*(3*C*a^2*b^2 + 4*A*b^4)*e*f^2)*x)*sqrt(-b*c*x + a*c)*sqr
t(b*x + a))/(b^6*c), -1/120*(15*(12*B*a^2*b^3*e^2*f + 3*B*a^4*b*f^3 + 4*(C*a^2*b^3 + 2*A*b^5)*e^3 + 3*(3*C*a^4
*b + 4*A*a^2*b^3)*e*f^2)*sqrt(c)*arctan(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*sqrt(c)*x/(b^2*c*x^2 - a^2*c)) + (2
4*C*b^4*f^3*x^4 + 120*B*b^4*e^3 + 240*B*a^2*b^2*e*f^2 + 120*(2*C*a^2*b^2 + 3*A*b^4)*e^2*f + 16*(4*C*a^4 + 5*A*
a^2*b^2)*f^3 + 30*(3*C*b^4*e*f^2 + B*b^4*f^3)*x^3 + 8*(15*C*b^4*e^2*f + 15*B*b^4*e*f^2 + (4*C*a^2*b^2 + 5*A*b^
4)*f^3)*x^2 + 15*(4*C*b^4*e^3 + 12*B*b^4*e^2*f + 3*B*a^2*b^2*f^3 + 3*(3*C*a^2*b^2 + 4*A*b^4)*e*f^2)*x)*sqrt(-b
*c*x + a*c)*sqrt(b*x + a))/(b^6*c)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\text {Timed out} \]

[In]

integrate((f*x+e)**3*(C*x**2+B*x+A)/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 471, normalized size of antiderivative = 0.94 \[ \int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {\sqrt {-b^{2} c x^{2} + a^{2} c} C f^{3} x^{4}}{5 \, b^{2} c} - \frac {4 \, \sqrt {-b^{2} c x^{2} + a^{2} c} C a^{2} f^{3} x^{2}}{15 \, b^{4} c} + \frac {A e^{3} \arcsin \left (\frac {b x}{a}\right )}{b \sqrt {c}} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} B e^{3}}{b^{2} c} - \frac {3 \, \sqrt {-b^{2} c x^{2} + a^{2} c} A e^{2} f}{b^{2} c} - \frac {8 \, \sqrt {-b^{2} c x^{2} + a^{2} c} C a^{4} f^{3}}{15 \, b^{6} c} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (3 \, C e f^{2} + B f^{3}\right )} x^{3}}{4 \, b^{2} c} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (3 \, C e^{2} f + 3 \, B e f^{2} + A f^{3}\right )} x^{2}}{3 \, b^{2} c} + \frac {3 \, {\left (3 \, C e f^{2} + B f^{3}\right )} a^{4} \arcsin \left (\frac {b x}{a}\right )}{8 \, b^{5} \sqrt {c}} + \frac {{\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} a^{2} \arcsin \left (\frac {b x}{a}\right )}{2 \, b^{3} \sqrt {c}} - \frac {3 \, \sqrt {-b^{2} c x^{2} + a^{2} c} {\left (3 \, C e f^{2} + B f^{3}\right )} a^{2} x}{8 \, b^{4} c} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} x}{2 \, b^{2} c} - \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} {\left (3 \, C e^{2} f + 3 \, B e f^{2} + A f^{3}\right )} a^{2}}{3 \, b^{4} c} \]

[In]

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

[Out]

-1/5*sqrt(-b^2*c*x^2 + a^2*c)*C*f^3*x^4/(b^2*c) - 4/15*sqrt(-b^2*c*x^2 + a^2*c)*C*a^2*f^3*x^2/(b^4*c) + A*e^3*
arcsin(b*x/a)/(b*sqrt(c)) - sqrt(-b^2*c*x^2 + a^2*c)*B*e^3/(b^2*c) - 3*sqrt(-b^2*c*x^2 + a^2*c)*A*e^2*f/(b^2*c
) - 8/15*sqrt(-b^2*c*x^2 + a^2*c)*C*a^4*f^3/(b^6*c) - 1/4*sqrt(-b^2*c*x^2 + a^2*c)*(3*C*e*f^2 + B*f^3)*x^3/(b^
2*c) - 1/3*sqrt(-b^2*c*x^2 + a^2*c)*(3*C*e^2*f + 3*B*e*f^2 + A*f^3)*x^2/(b^2*c) + 3/8*(3*C*e*f^2 + B*f^3)*a^4*
arcsin(b*x/a)/(b^5*sqrt(c)) + 1/2*(C*e^3 + 3*B*e^2*f + 3*A*e*f^2)*a^2*arcsin(b*x/a)/(b^3*sqrt(c)) - 3/8*sqrt(-
b^2*c*x^2 + a^2*c)*(3*C*e*f^2 + B*f^3)*a^2*x/(b^4*c) - 1/2*sqrt(-b^2*c*x^2 + a^2*c)*(C*e^3 + 3*B*e^2*f + 3*A*e
*f^2)*x/(b^2*c) - 2/3*sqrt(-b^2*c*x^2 + a^2*c)*(3*C*e^2*f + 3*B*e*f^2 + A*f^3)*a^2/(b^4*c)

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.14 \[ \int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {{\left ({\left (2 \, {\left (3 \, {\left (\frac {4 \, {\left (b x + a\right )} C f^{3}}{c} + \frac {15 \, C b c^{4} e f^{2} - 16 \, C a c^{4} f^{3} + 5 \, B b c^{4} f^{3}}{c^{5}}\right )} {\left (b x + a\right )} + \frac {60 \, C b^{2} c^{4} e^{2} f - 135 \, C a b c^{4} e f^{2} + 60 \, B b^{2} c^{4} e f^{2} + 88 \, C a^{2} c^{4} f^{3} - 45 \, B a b c^{4} f^{3} + 20 \, A b^{2} c^{4} f^{3}}{c^{5}}\right )} {\left (b x + a\right )} + \frac {5 \, {\left (12 \, C b^{3} c^{4} e^{3} - 48 \, C a b^{2} c^{4} e^{2} f + 36 \, B b^{3} c^{4} e^{2} f + 81 \, C a^{2} b c^{4} e f^{2} - 48 \, B a b^{2} c^{4} e f^{2} + 36 \, A b^{3} c^{4} e f^{2} - 32 \, C a^{3} c^{4} f^{3} + 27 \, B a^{2} b c^{4} f^{3} - 16 \, A a b^{2} c^{4} f^{3}\right )}}{c^{5}}\right )} {\left (b x + a\right )} - \frac {15 \, {\left (4 \, C a b^{3} c^{4} e^{3} - 8 \, B b^{4} c^{4} e^{3} - 24 \, C a^{2} b^{2} c^{4} e^{2} f + 12 \, B a b^{3} c^{4} e^{2} f - 24 \, A b^{4} c^{4} e^{2} f + 15 \, C a^{3} b c^{4} e f^{2} - 24 \, B a^{2} b^{2} c^{4} e f^{2} + 12 \, A a b^{3} c^{4} e f^{2} - 8 \, C a^{4} c^{4} f^{3} + 5 \, B a^{3} b c^{4} f^{3} - 8 \, A a^{2} b^{2} c^{4} f^{3}\right )}}{c^{5}}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a} + \frac {30 \, {\left (4 \, C a^{2} b^{3} e^{3} + 8 \, A b^{5} e^{3} + 12 \, B a^{2} b^{3} e^{2} f + 9 \, C a^{4} b e f^{2} + 12 \, A a^{2} b^{3} e f^{2} + 3 \, B a^{4} b f^{3}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}}}{120 \, b^{6}} \]

[In]

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

[Out]

-1/120*(((2*(3*(4*(b*x + a)*C*f^3/c + (15*C*b*c^4*e*f^2 - 16*C*a*c^4*f^3 + 5*B*b*c^4*f^3)/c^5)*(b*x + a) + (60
*C*b^2*c^4*e^2*f - 135*C*a*b*c^4*e*f^2 + 60*B*b^2*c^4*e*f^2 + 88*C*a^2*c^4*f^3 - 45*B*a*b*c^4*f^3 + 20*A*b^2*c
^4*f^3)/c^5)*(b*x + a) + 5*(12*C*b^3*c^4*e^3 - 48*C*a*b^2*c^4*e^2*f + 36*B*b^3*c^4*e^2*f + 81*C*a^2*b*c^4*e*f^
2 - 48*B*a*b^2*c^4*e*f^2 + 36*A*b^3*c^4*e*f^2 - 32*C*a^3*c^4*f^3 + 27*B*a^2*b*c^4*f^3 - 16*A*a*b^2*c^4*f^3)/c^
5)*(b*x + a) - 15*(4*C*a*b^3*c^4*e^3 - 8*B*b^4*c^4*e^3 - 24*C*a^2*b^2*c^4*e^2*f + 12*B*a*b^3*c^4*e^2*f - 24*A*
b^4*c^4*e^2*f + 15*C*a^3*b*c^4*e*f^2 - 24*B*a^2*b^2*c^4*e*f^2 + 12*A*a*b^3*c^4*e*f^2 - 8*C*a^4*c^4*f^3 + 5*B*a
^3*b*c^4*f^3 - 8*A*a^2*b^2*c^4*f^3)/c^5)*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a) + 30*(4*C*a^2*b^3*e^3 + 8*A*
b^5*e^3 + 12*B*a^2*b^3*e^2*f + 9*C*a^4*b*e*f^2 + 12*A*a^2*b^3*e*f^2 + 3*B*a^4*b*f^3)*log(abs(-sqrt(b*x + a)*sq
rt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c))/b^6

Mupad [B] (verification not implemented)

Time = 151.65 (sec) , antiderivative size = 4167, normalized size of antiderivative = 8.32 \[ \int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\text {Too large to display} \]

[In]

int(((e + f*x)^3*(A + B*x + C*x^2))/((a*c - b*c*x)^(1/2)*(a + b*x)^(1/2)),x)

[Out]

- ((((23*B*a^4*c*f^3)/2 - 18*B*a^2*b^2*c*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^13)/(b^5*((a + b*x)^(1/2)
- a^(1/2))^13) + (((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^15*((3*B*a^4*f^3)/2 + 6*B*a^2*b^2*e^2*f))/(b^5*((a + b*x
)^(1/2) - a^(1/2))^15) - (((3*B*a^4*c^7*f^3)/2 + 6*B*a^2*b^2*c^7*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(
b^5*((a + b*x)^(1/2) - a^(1/2))) - (((23*B*a^4*c^6*f^3)/2 - 18*B*a^2*b^2*c^6*e^2*f)*((a*c - b*c*x)^(1/2) - (a*
c)^(1/2))^3)/(b^5*((a + b*x)^(1/2) - a^(1/2))^3) + (((333*B*a^4*c^5*f^3)/2 + 90*B*a^2*b^2*c^5*e^2*f)*((a*c - b
*c*x)^(1/2) - (a*c)^(1/2))^5)/(b^5*((a + b*x)^(1/2) - a^(1/2))^5) - (((333*B*a^4*c^2*f^3)/2 + 90*B*a^2*b^2*c^2
*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^11)/(b^5*((a + b*x)^(1/2) - a^(1/2))^11) - (((671*B*a^4*c^4*f^3)/2
- 66*B*a^2*b^2*c^4*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/(b^5*((a + b*x)^(1/2) - a^(1/2))^7) + (((671
*B*a^4*c^3*f^3)/2 - 66*B*a^2*b^2*c^3*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^9)/(b^5*((a + b*x)^(1/2) - a^(
1/2))^9) + (a^(1/2)*(a*c)^(1/2)*(48*B*b^2*c^5*e^3 + 192*B*a^2*c^5*e*f^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4
)/(b^4*((a + b*x)^(1/2) - a^(1/2))^4) + (a^(1/2)*(a*c)^(1/2)*(160*B*b^2*c^3*e^3 + 128*B*a^2*c^3*e*f^2)*((a*c -
b*c*x)^(1/2) - (a*c)^(1/2))^8)/(b^4*((a + b*x)^(1/2) - a^(1/2))^8) + (a^(1/2)*(a*c)^(1/2)*(120*B*b^2*c^4*e^3
+ 256*B*a^2*c^4*e*f^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/(b^4*((a + b*x)^(1/2) - a^(1/2))^6) + (a^(1/2)*(
a*c)^(1/2)*(120*B*b^2*c^2*e^3 + 256*B*a^2*c^2*e*f^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^10)/(b^4*((a + b*x)^(
1/2) - a^(1/2))^10) + (a^(1/2)*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^12*(48*B*b^2*c*e^3 + 192*B*a^2*
c*e*f^2))/(b^4*((a + b*x)^(1/2) - a^(1/2))^12) + (8*B*a^(1/2)*e^3*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/
2))^14)/(b^2*((a + b*x)^(1/2) - a^(1/2))^14) + (8*B*a^(1/2)*c^6*e^3*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(
1/2))^2)/(b^2*((a + b*x)^(1/2) - a^(1/2))^2))/(((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^16/((a + b*x)^(1/2) - a^(1/
2))^16 + c^8 + (8*c*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^14)/((a + b*x)^(1/2) - a^(1/2))^14 + (8*c^7*((a*c - b*
c*x)^(1/2) - (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/2))^2 + (28*c^6*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/((
a + b*x)^(1/2) - a^(1/2))^4 + (56*c^5*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/((a + b*x)^(1/2) - a^(1/2))^6 + (
70*c^4*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^8)/((a + b*x)^(1/2) - a^(1/2))^8 + (56*c^3*((a*c - b*c*x)^(1/2) - (
a*c)^(1/2))^10)/((a + b*x)^(1/2) - a^(1/2))^10 + (28*c^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^12)/((a + b*x)^(1
/2) - a^(1/2))^12) - ((a^(1/2)*(a*c)^(1/2)*(64*A*a^2*c^3*f^3 + 96*A*b^2*c^3*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c
)^(1/2))^4)/(b^4*((a + b*x)^(1/2) - a^(1/2))^4) - (a^(1/2)*(a*c)^(1/2)*((128*A*a^2*c^2*f^3)/3 - 144*A*b^2*c^2*
e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/(b^4*((a + b*x)^(1/2) - a^(1/2))^6) + (a^(1/2)*(a*c)^(1/2)*((a*c
- b*c*x)^(1/2) - (a*c)^(1/2))^8*(64*A*a^2*c*f^3 + 96*A*b^2*c*e^2*f))/(b^4*((a + b*x)^(1/2) - a^(1/2))^8) + (6
*A*a^2*e*f^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^11)/(b^3*((a + b*x)^(1/2) - a^(1/2))^11) - (6*A*a^2*c^5*e*f^2
*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(b^3*((a + b*x)^(1/2) - a^(1/2))) - (30*A*a^2*c*e*f^2*((a*c - b*c*x)^(1/
2) - (a*c)^(1/2))^9)/(b^3*((a + b*x)^(1/2) - a^(1/2))^9) + (24*A*a^(1/2)*e^2*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2
) - (a*c)^(1/2))^10)/(b^2*((a + b*x)^(1/2) - a^(1/2))^10) + (30*A*a^2*c^4*e*f^2*((a*c - b*c*x)^(1/2) - (a*c)^(
1/2))^3)/(b^3*((a + b*x)^(1/2) - a^(1/2))^3) + (36*A*a^2*c^3*e*f^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^5)/(b^3
*((a + b*x)^(1/2) - a^(1/2))^5) - (36*A*a^2*c^2*e*f^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/(b^3*((a + b*x)^(
1/2) - a^(1/2))^7) + (24*A*a^(1/2)*c^4*e^2*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(b^2*((a + b*x
)^(1/2) - a^(1/2))^2))/(((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^12/((a + b*x)^(1/2) - a^(1/2))^12 + c^6 + (6*c*((a
*c - b*c*x)^(1/2) - (a*c)^(1/2))^10)/((a + b*x)^(1/2) - a^(1/2))^10 + (6*c^5*((a*c - b*c*x)^(1/2) - (a*c)^(1/2
))^2)/((a + b*x)^(1/2) - a^(1/2))^2 + (15*c^4*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/((a + b*x)^(1/2) - a^(1/2
))^4 + (20*c^3*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/((a + b*x)^(1/2) - a^(1/2))^6 + (15*c^2*((a*c - b*c*x)^(
1/2) - (a*c)^(1/2))^8)/((a + b*x)^(1/2) - a^(1/2))^8) - ((((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^19*((9*C*a^4*e*f
^2)/2 + 2*C*a^2*b^2*e^3))/(b^5*((a + b*x)^(1/2) - a^(1/2))^19) - ((2*C*a^2*b^2*c*e^3 - (87*C*a^4*c*e*f^2)/2)*(
(a*c - b*c*x)^(1/2) - (a*c)^(1/2))^17)/(b^5*((a + b*x)^(1/2) - a^(1/2))^17) - (((9*C*a^4*c^9*e*f^2)/2 + 2*C*a^
2*b^2*c^9*e^3)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(b^5*((a + b*x)^(1/2) - a^(1/2))) - (((87*C*a^4*c^8*e*f^2)
/2 - 2*C*a^2*b^2*c^8*e^3)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3)/(b^5*((a + b*x)^(1/2) - a^(1/2))^3) - ((42*C*
a^4*c^6*e*f^2 - 88*C*a^2*b^2*c^6*e^3)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/(b^5*((a + b*x)^(1/2) - a^(1/2))^
7) + ((42*C*a^4*c^3*e*f^2 - 88*C*a^2*b^2*c^3*e^3)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^13)/(b^5*((a + b*x)^(1/2
) - a^(1/2))^13) + ((426*C*a^4*c^7*e*f^2 + 40*C*a^2*b^2*c^7*e^3)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^5)/(b^5*(
(a + b*x)^(1/2) - a^(1/2))^5) - ((426*C*a^4*c^2*e*f^2 + 40*C*a^2*b^2*c^2*e^3)*((a*c - b*c*x)^(1/2) - (a*c)^(1/
2))^15)/(b^5*((a + b*x)^(1/2) - a^(1/2))^15) - ((507*C*a^4*c^5*e*f^2 - 52*C*a^2*b^2*c^5*e^3)*((a*c - b*c*x)^(1
/2) - (a*c)^(1/2))^9)/(b^5*((a + b*x)^(1/2) - a^(1/2))^9) + ((507*C*a^4*c^4*e*f^2 - 52*C*a^2*b^2*c^4*e^3)*((a*
c - b*c*x)^(1/2) - (a*c)^(1/2))^11)/(b^5*((a + b*x)^(1/2) - a^(1/2))^11) + (a^(1/2)*(a*c)^(1/2)*((2048*C*a^4*c
^6*f^3)/3 + 640*C*a^2*b^2*c^6*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/(b^6*((a + b*x)^(1/2) - a^(1/2))^6
) + (a^(1/2)*(a*c)^(1/2)*((2048*C*a^4*c^2*f^3)/3 + 640*C*a^2*b^2*c^2*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)
)^14)/(b^6*((a + b*x)^(1/2) - a^(1/2))^14) - (a^(1/2)*(a*c)^(1/2)*((4096*C*a^4*c^5*f^3)/3 - 832*C*a^2*b^2*c^5*
e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^8)/(b^6*((a + b*x)^(1/2) - a^(1/2))^8) - (a^(1/2)*(a*c)^(1/2)*((409
6*C*a^4*c^3*f^3)/3 - 832*C*a^2*b^2*c^3*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^12)/(b^6*((a + b*x)^(1/2) -
a^(1/2))^12) + (a^(1/2)*(a*c)^(1/2)*((12288*C*a^4*c^4*f^3)/5 + 768*C*a^2*b^2*c^4*e^2*f)*((a*c - b*c*x)^(1/2) -
(a*c)^(1/2))^10)/(b^6*((a + b*x)^(1/2) - a^(1/2))^10) + (192*C*a^(5/2)*c*e^2*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/
2) - (a*c)^(1/2))^16)/(b^4*((a + b*x)^(1/2) - a^(1/2))^16) + (192*C*a^(5/2)*c^7*e^2*f*(a*c)^(1/2)*((a*c - b*c*
x)^(1/2) - (a*c)^(1/2))^4)/(b^4*((a + b*x)^(1/2) - a^(1/2))^4))/(((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^20/((a +
b*x)^(1/2) - a^(1/2))^20 + c^10 + (10*c*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^18)/((a + b*x)^(1/2) - a^(1/2))^18
+ (10*c^9*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/2))^2 + (45*c^8*((a*c - b*c*x)^(1/2)
- (a*c)^(1/2))^4)/((a + b*x)^(1/2) - a^(1/2))^4 + (120*c^7*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/((a + b*x)^
(1/2) - a^(1/2))^6 + (210*c^6*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^8)/((a + b*x)^(1/2) - a^(1/2))^8 + (252*c^5*
((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^10)/((a + b*x)^(1/2) - a^(1/2))^10 + (210*c^4*((a*c - b*c*x)^(1/2) - (a*c)
^(1/2))^12)/((a + b*x)^(1/2) - a^(1/2))^12 + (120*c^3*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^14)/((a + b*x)^(1/2)
- a^(1/2))^14 + (45*c^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^16)/((a + b*x)^(1/2) - a^(1/2))^16) - (2*A*e*atan
((A*e*(3*a^2*f^2 + 2*b^2*e^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(c^(1/2)*(2*A*b^2*e^3 + 3*A*a^2*e*f^2)*((a
+ b*x)^(1/2) - a^(1/2))))*(3*a^2*f^2 + 2*b^2*e^2))/(b^3*c^(1/2)) - (3*B*a^2*f*atan((B*a^2*f*(a^2*f^2 + 4*b^2*e
^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(c^(1/2)*(B*a^4*f^3 + 4*B*a^2*b^2*e^2*f)*((a + b*x)^(1/2) - a^(1/2)))
)*(a^2*f^2 + 4*b^2*e^2))/(2*b^5*c^(1/2)) - (C*a^2*e*atan((C*a^2*e*(9*a^2*f^2 + 4*b^2*e^2)*((a*c - b*c*x)^(1/2)
- (a*c)^(1/2)))/(c^(1/2)*(9*C*a^4*e*f^2 + 4*C*a^2*b^2*e^3)*((a + b*x)^(1/2) - a^(1/2))))*(9*a^2*f^2 + 4*b^2*e
^2))/(2*b^5*c^(1/2))