Integrand size = 38, antiderivative size = 246 \[ \int \frac {(e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {C (e+f x)^2 \left (a^2-b^2 x^2\right )}{3 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left (2 \left (2 a^2 C f^2-b^2 \left (C e^2-3 f (B e+A f)\right )\right )-b^2 f (C e-3 B f) x\right ) \left (a^2-b^2 x^2\right )}{6 b^4 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (2 A b^2 e+a^2 (C e+B f)\right ) \sqrt {a^2 c-b^2 c x^2} \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{2 b^3 \sqrt {c} \sqrt {a+b x} \sqrt {a c-b c x}} \]
-1/3*C*(f*x+e)^2*(-b^2*x^2+a^2)/b^2/f/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-1/6 *(4*a^2*C*f^2-2*b^2*(C*e^2-3*f*(A*f+B*e))-b^2*f*(-3*B*f+C*e)*x)*(-b^2*x^2+ a^2)/b^4/f/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)+1/2*(2*A*b^2*e+a^2*(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^3/ c^(1/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)
Time = 0.23 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.52 \[ \int \frac {(e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {-\left ((a-b x) \sqrt {a+b x} \left (4 a^2 C f+b^2 \left (6 B e+6 A f+3 C e x+3 B f x+2 C f x^2\right )\right )\right )+6 b \left (2 A b^2 e+a^2 (C e+B f)\right ) \sqrt {a-b x} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{6 b^4 \sqrt {c (a-b x)}} \]
(-((a - b*x)*Sqrt[a + b*x]*(4*a^2*C*f + b^2*(6*B*e + 6*A*f + 3*C*e*x + 3*B *f*x + 2*C*f*x^2))) + 6*b*(2*A*b^2*e + a^2*(C*e + B*f))*Sqrt[a - b*x]*ArcT an[Sqrt[a + b*x]/Sqrt[a - b*x]])/(6*b^4*Sqrt[c*(a - b*x)])
Time = 0.56 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {2113, 2185, 25, 27, 676, 224, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx\) |
| \(\Big \downarrow \) 2113 |
| \(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \int \frac {(e+f x) \left (C x^2+B x+A\right )}{\sqrt {a^2 c-b^2 c x^2}}dx}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (-\frac {\int -\frac {c f (e+f x) \left (\left (2 C a^2+3 A b^2\right ) f-b^2 (C e-3 B f) x\right )}{\sqrt {a^2 c-b^2 c x^2}}dx}{3 b^2 c f^2}-\frac {C (e+f x)^2 \sqrt {a^2 c-b^2 c x^2}}{3 b^2 c f}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\int \frac {c f (e+f x) \left (\left (2 C a^2+3 A b^2\right ) f-b^2 (C e-3 B f) x\right )}{\sqrt {a^2 c-b^2 c x^2}}dx}{3 b^2 c f^2}-\frac {C (e+f x)^2 \sqrt {a^2 c-b^2 c x^2}}{3 b^2 c f}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\int \frac {(e+f x) \left (\left (2 C a^2+3 A b^2\right ) f-b^2 (C e-3 B f) x\right )}{\sqrt {a^2 c-b^2 c x^2}}dx}{3 b^2 f}-\frac {C (e+f x)^2 \sqrt {a^2 c-b^2 c x^2}}{3 b^2 c f}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 676 |
| \(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {3}{2} f \left (a^2 (B f+C e)+2 A b^2 e\right ) \int \frac {1}{\sqrt {a^2 c-b^2 c x^2}}dx-\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 a^2 C f^2-b^2 \left (C e^2-3 f (A f+B e)\right )\right )}{b^2 c}+\frac {f x \sqrt {a^2 c-b^2 c x^2} (C e-3 B f)}{2 c}}{3 b^2 f}-\frac {C (e+f x)^2 \sqrt {a^2 c-b^2 c x^2}}{3 b^2 c f}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {3}{2} f \left (a^2 (B f+C e)+2 A b^2 e\right ) \int \frac {1}{\frac {b^2 c x^2}{a^2 c-b^2 c x^2}+1}d\frac {x}{\sqrt {a^2 c-b^2 c x^2}}-\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 a^2 C f^2-b^2 \left (C e^2-3 f (A f+B e)\right )\right )}{b^2 c}+\frac {f x \sqrt {a^2 c-b^2 c x^2} (C e-3 B f)}{2 c}}{3 b^2 f}-\frac {C (e+f x)^2 \sqrt {a^2 c-b^2 c x^2}}{3 b^2 c f}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {3 f \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right ) \left (a^2 (B f+C e)+2 A b^2 e\right )}{2 b \sqrt {c}}-\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 a^2 C f^2-b^2 \left (C e^2-3 f (A f+B e)\right )\right )}{b^2 c}+\frac {f x \sqrt {a^2 c-b^2 c x^2} (C e-3 B f)}{2 c}}{3 b^2 f}-\frac {C (e+f x)^2 \sqrt {a^2 c-b^2 c x^2}}{3 b^2 c f}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
(Sqrt[a^2*c - b^2*c*x^2]*(-1/3*(C*(e + f*x)^2*Sqrt[a^2*c - b^2*c*x^2])/(b^ 2*c*f) + (-(((2*a^2*C*f^2 - b^2*(C*e^2 - 3*f*(B*e + A*f)))*Sqrt[a^2*c - b^ 2*c*x^2])/(b^2*c)) + (f*(C*e - 3*B*f)*x*Sqrt[a^2*c - b^2*c*x^2])/(2*c) + ( 3*f*(2*A*b^2*e + a^2*(C*e + B*f))*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c* x^2]])/(2*b*Sqrt[c]))/(3*b^2*f)))/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])
3.1.29.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(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]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_. )*(x_))^(p_.), x_Symbol] :> Simp[(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] && !IntegerQ[m]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(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 + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 1.67 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.71
| method | result | size |
| risch | \(-\frac {\left (2 C f \,x^{2} b^{2}+3 B \,b^{2} f x +3 C \,b^{2} e x +6 A \,b^{2} f +6 B \,b^{2} e +4 a^{2} C f \right ) \sqrt {b x +a}\, \left (-b x +a \right )}{6 b^{4} \sqrt {-c \left (b x -a \right )}}+\frac {\left (2 A \,b^{2} e +B \,a^{2} f +C \,a^{2} e \right ) \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}}{2 b^{2} \sqrt {b^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\) | \(175\) |
| default | \(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (6 A \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) b^{4} c e +3 B \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) a^{2} b^{2} c f +3 C \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) a^{2} b^{2} c e -2 C \,b^{2} f \,x^{2} \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}-3 B \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, b^{2} f x -3 C \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, b^{2} e x -6 A \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, b^{2} f -6 B \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, b^{2} e -4 C \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, a^{2} f \right )}{6 c \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, b^{4} \sqrt {b^{2} c}}\) | \(343\) |
-1/6*(2*C*b^2*f*x^2+3*B*b^2*f*x+3*C*b^2*e*x+6*A*b^2*f+6*B*b^2*e+4*C*a^2*f) *(b*x+a)^(1/2)/b^4*(-b*x+a)/(-c*(b*x-a))^(1/2)+1/2*(2*A*b^2*e+B*a^2*f+C*a^ 2*e)/b^2/(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)
Time = 0.31 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.23 \[ \int \frac {(e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\left [-\frac {3 \, {\left (B a^{2} b f + {\left (C a^{2} b + 2 \, A b^{3}\right )} e\right )} \sqrt {-c} \log \left (2 \, b^{2} c x^{2} - 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {-c} x - a^{2} c\right ) + 2 \, {\left (2 \, C b^{2} f x^{2} + 6 \, B b^{2} e + 2 \, {\left (2 \, C a^{2} + 3 \, A b^{2}\right )} f + 3 \, {\left (C b^{2} e + B b^{2} f\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{12 \, b^{4} c}, -\frac {3 \, {\left (B a^{2} b f + {\left (C a^{2} b + 2 \, A b^{3}\right )} e\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {c} x}{b^{2} c x^{2} - a^{2} c}\right ) + {\left (2 \, C b^{2} f x^{2} + 6 \, B b^{2} e + 2 \, {\left (2 \, C a^{2} + 3 \, A b^{2}\right )} f + 3 \, {\left (C b^{2} e + B b^{2} f\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{6 \, b^{4} c}\right ] \]
[-1/12*(3*(B*a^2*b*f + (C*a^2*b + 2*A*b^3)*e)*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*(2*C*b^2*f*x^2 + 6*B*b^2*e + 2*(2*C*a^2 + 3*A*b^2)*f + 3*(C*b^2*e + B*b^2*f)*x)*sqrt(-b* c*x + a*c)*sqrt(b*x + a))/(b^4*c), -1/6*(3*(B*a^2*b*f + (C*a^2*b + 2*A*b^3 )*e)*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*C*b^2*f*x^2 + 6*B*b^2*e + 2*(2*C*a^2 + 3*A*b^2)*f + 3*(C* b^2*e + B*b^2*f)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/(b^4*c)]
Timed out. \[ \int \frac {(e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.77 \[ \int \frac {(e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {\sqrt {-b^{2} c x^{2} + a^{2} c} C f x^{2}}{3 \, b^{2} c} + \frac {A e \arcsin \left (\frac {b x}{a}\right )}{b \sqrt {c}} + \frac {{\left (C e + B f\right )} a^{2} \arcsin \left (\frac {b x}{a}\right )}{2 \, b^{3} \sqrt {c}} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} B e}{b^{2} c} - \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} C a^{2} f}{3 \, b^{4} c} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} A f}{b^{2} c} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (C e + B f\right )} x}{2 \, b^{2} c} \]
-1/3*sqrt(-b^2*c*x^2 + a^2*c)*C*f*x^2/(b^2*c) + A*e*arcsin(b*x/a)/(b*sqrt( c)) + 1/2*(C*e + B*f)*a^2*arcsin(b*x/a)/(b^3*sqrt(c)) - sqrt(-b^2*c*x^2 + a^2*c)*B*e/(b^2*c) - 2/3*sqrt(-b^2*c*x^2 + a^2*c)*C*a^2*f/(b^4*c) - sqrt(- b^2*c*x^2 + a^2*c)*A*f/(b^2*c) - 1/2*sqrt(-b^2*c*x^2 + a^2*c)*(C*e + B*f)* x/(b^2*c)
Time = 0.33 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.78 \[ \int \frac {(e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {{\left ({\left (\frac {2 \, {\left (b x + a\right )} C f}{c} + \frac {3 \, C b c^{2} e - 4 \, C a c^{2} f + 3 \, B b c^{2} f}{c^{3}}\right )} {\left (b x + a\right )} - \frac {3 \, {\left (C a b c^{2} e - 2 \, B b^{2} c^{2} e - 2 \, C a^{2} c^{2} f + B a b c^{2} f - 2 \, A b^{2} c^{2} f\right )}}{c^{3}}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a} + \frac {6 \, {\left (C a^{2} b e + 2 \, A b^{3} e + B a^{2} b f\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}}}{6 \, b^{4}} \]
-1/6*(((2*(b*x + a)*C*f/c + (3*C*b*c^2*e - 4*C*a*c^2*f + 3*B*b*c^2*f)/c^3) *(b*x + a) - 3*(C*a*b*c^2*e - 2*B*b^2*c^2*e - 2*C*a^2*c^2*f + B*a*b*c^2*f - 2*A*b^2*c^2*f)/c^3)*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a) + 6*(C*a^2* b*e + 2*A*b^3*e + B*a^2*b*f)*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c))/b^4
Time = 37.95 (sec) , antiderivative size = 1011, normalized size of antiderivative = 4.11 \[ \int \frac {(e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {\frac {2\,B\,a^2\,f\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^7}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7}-\frac {2\,B\,a^2\,c^3\,f\,\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}{\sqrt {a+b\,x}-\sqrt {a}}-\frac {14\,B\,a^2\,c\,f\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^5}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5}+\frac {14\,B\,a^2\,c^2\,f\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^3}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}}{b^3\,c^4+\frac {b^3\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^8}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}+\frac {4\,b^3\,c^3\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^2}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}+\frac {6\,b^3\,c^2\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^4}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}+\frac {4\,b^3\,c\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^6}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}}-\frac {\frac {2\,C\,a^2\,e\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^7}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7}-\frac {2\,C\,a^2\,c^3\,e\,\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}{\sqrt {a+b\,x}-\sqrt {a}}-\frac {14\,C\,a^2\,c\,e\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^5}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5}+\frac {14\,C\,a^2\,c^2\,e\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^3}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}}{b^3\,c^4+\frac {b^3\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^8}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}+\frac {4\,b^3\,c^3\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^2}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}+\frac {6\,b^3\,c^2\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^4}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}+\frac {4\,b^3\,c\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^6}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}}-\frac {\sqrt {a\,c-b\,c\,x}\,\left (\frac {2\,C\,a^3\,f}{3\,b^4\,c}+\frac {C\,f\,x^3}{3\,b\,c}+\frac {C\,a\,f\,x^2}{3\,b^2\,c}+\frac {2\,C\,a^2\,f\,x}{3\,b^3\,c}\right )}{\sqrt {a+b\,x}}-\frac {4\,A\,e\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}{\sqrt {b^2\,c}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}\right )}{\sqrt {b^2\,c}}-\frac {A\,f\,\sqrt {a\,c-b\,c\,x}\,\sqrt {a+b\,x}}{b^2\,c}-\frac {B\,e\,\sqrt {a\,c-b\,c\,x}\,\sqrt {a+b\,x}}{b^2\,c}-\frac {2\,B\,a^2\,f\,\mathrm {atan}\left (\frac {\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}}{\sqrt {c}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}\right )}{b^3\,\sqrt {c}}-\frac {2\,C\,a^2\,e\,\mathrm {atan}\left (\frac {\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}}{\sqrt {c}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}\right )}{b^3\,\sqrt {c}} \]
- ((2*B*a^2*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/((a + b*x)^(1/2) - a^ (1/2))^7 - (2*B*a^2*c^3*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/((a + b*x)^ (1/2) - a^(1/2)) - (14*B*a^2*c*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^5)/(( a + b*x)^(1/2) - a^(1/2))^5 + (14*B*a^2*c^2*f*((a*c - b*c*x)^(1/2) - (a*c) ^(1/2))^3)/((a + b*x)^(1/2) - a^(1/2))^3)/(b^3*c^4 + (b^3*((a*c - b*c*x)^( 1/2) - (a*c)^(1/2))^8)/((a + b*x)^(1/2) - a^(1/2))^8 + (4*b^3*c^3*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/2))^2 + (6*b^3*c^2* ((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/((a + b*x)^(1/2) - a^(1/2))^4 + (4* b^3*c*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/((a + b*x)^(1/2) - a^(1/2))^6 ) - ((2*C*a^2*e*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/((a + b*x)^(1/2) - a^(1/2))^7 - (2*C*a^2*c^3*e*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/((a + b*x )^(1/2) - a^(1/2)) - (14*C*a^2*c*e*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^5)/ ((a + b*x)^(1/2) - a^(1/2))^5 + (14*C*a^2*c^2*e*((a*c - b*c*x)^(1/2) - (a* c)^(1/2))^3)/((a + b*x)^(1/2) - a^(1/2))^3)/(b^3*c^4 + (b^3*((a*c - b*c*x) ^(1/2) - (a*c)^(1/2))^8)/((a + b*x)^(1/2) - a^(1/2))^8 + (4*b^3*c^3*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/2))^2 + (6*b^3*c^ 2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/((a + b*x)^(1/2) - a^(1/2))^4 + ( 4*b^3*c*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/((a + b*x)^(1/2) - a^(1/2)) ^6) - ((a*c - b*c*x)^(1/2)*((2*C*a^3*f)/(3*b^4*c) + (C*f*x^3)/(3*b*c) + (C *a*f*x^2)/(3*b^2*c) + (2*C*a^2*f*x)/(3*b^3*c)))/(a + b*x)^(1/2) - (4*A*...