Integrand size = 40, antiderivative size = 322 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2} \, dx=\frac {f \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (a^2-b^2 x^2\right )}{\left (b^2 e^2-a^2 f^2\right ) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)}+\frac {C \sqrt {a^2 c-b^2 c x^2} \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{b \sqrt {c} f^2 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (a^2 f^2 (2 C e-B f)-b^2 \left (C e^3-A e f^2\right )\right ) \sqrt {a^2 c-b^2 c x^2} \arctan \left (\frac {\sqrt {c} \left (a^2 f+b^2 e x\right )}{\sqrt {b^2 e^2-a^2 f^2} \sqrt {a^2 c-b^2 c x^2}}\right )}{\sqrt {c} f^2 \left (b^2 e^2-a^2 f^2\right )^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}} \]
f*(A+e*(-B*f+C*e)/f^2)*(-b^2*x^2+a^2)/(-a^2*f^2+b^2*e^2)/(f*x+e)/(b*x+a)^( 1/2)/(-b*c*x+a*c)^(1/2)+C*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/f^2/c^(1/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)+(a^2* f^2*(-B*f+2*C*e)-b^2*(-A*e*f^2+C*e^3))*arctan((b^2*e*x+a^2*f)*c^(1/2)/(-a^ 2*f^2+b^2*e^2)^(1/2)/(-b^2*c*x^2+a^2*c)^(1/2))*(-b^2*c*x^2+a^2*c)^(1/2)/f^ 2/(-a^2*f^2+b^2*e^2)^(3/2)/c^(1/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)
Time = 0.76 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.71 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2} \, dx=\frac {2 \left (\frac {f \left (C e^2+f (-B e+A f)\right ) (-a+b x) \sqrt {a+b x}}{2 (-b e+a f) (b e+a f) (e+f x)}+\frac {C \sqrt {a-b x} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{b}-\frac {\left (a^2 f^2 (-2 C e+B f)+b^2 \left (C e^3-A e f^2\right )\right ) \sqrt {a-b x} \arctan \left (\frac {\sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a-b x}}\right )}{(b e-a f)^{3/2} (b e+a f)^{3/2}}\right )}{f^2 \sqrt {c (a-b x)}} \]
(2*((f*(C*e^2 + f*(-(B*e) + A*f))*(-a + b*x)*Sqrt[a + b*x])/(2*(-(b*e) + a *f)*(b*e + a*f)*(e + f*x)) + (C*Sqrt[a - b*x]*ArcTan[Sqrt[a + b*x]/Sqrt[a - b*x]])/b - ((a^2*f^2*(-2*C*e + B*f) + b^2*(C*e^3 - A*e*f^2))*Sqrt[a - b* x]*ArcTan[(Sqrt[b*e + a*f]*Sqrt[a + b*x])/(Sqrt[b*e - a*f]*Sqrt[a - b*x])] )/((b*e - a*f)^(3/2)*(b*e + a*f)^(3/2))))/(f^2*Sqrt[c*(a - b*x)])
Time = 0.70 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2113, 2182, 27, 719, 224, 216, 488, 217}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {A+B x+C x^2}{\sqrt {a+b x} (e+f x)^2 \sqrt {a c-b c x}} \, dx\) |
| \(\Big \downarrow \) 2113 |
| \(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \int \frac {C x^2+B x+A}{(e+f x)^2 \sqrt {a^2 c-b^2 c x^2}}dx}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\int \frac {c \left ((C e-B f) a^2+A b^2 e+C \left (\frac {b^2 e^2}{f}-a^2 f\right ) x\right )}{(e+f x) \sqrt {a^2 c-b^2 c x^2}}dx}{c \left (b^2 e^2-a^2 f^2\right )}+\frac {f \sqrt {a^2 c-b^2 c x^2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{c (e+f x) \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\int \frac {(C e-B f) a^2+A b^2 e+C \left (\frac {b^2 e^2}{f}-a^2 f\right ) x}{(e+f x) \sqrt {a^2 c-b^2 c x^2}}dx}{b^2 e^2-a^2 f^2}+\frac {f \sqrt {a^2 c-b^2 c x^2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{c (e+f x) \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {\left (a^2 f^2 (2 C e-B f)-b^2 \left (C e^3-A e f^2\right )\right ) \int \frac {1}{(e+f x) \sqrt {a^2 c-b^2 c x^2}}dx}{f^2}+\frac {C \left (b^2 e^2-a^2 f^2\right ) \int \frac {1}{\sqrt {a^2 c-b^2 c x^2}}dx}{f^2}}{b^2 e^2-a^2 f^2}+\frac {f \sqrt {a^2 c-b^2 c x^2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{c (e+f x) \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {\left (a^2 f^2 (2 C e-B f)-b^2 \left (C e^3-A e f^2\right )\right ) \int \frac {1}{(e+f x) \sqrt {a^2 c-b^2 c x^2}}dx}{f^2}+\frac {C \left (b^2 e^2-a^2 f^2\right ) \int \frac {1}{\frac {b^2 c x^2}{a^2 c-b^2 c x^2}+1}d\frac {x}{\sqrt {a^2 c-b^2 c x^2}}}{f^2}}{b^2 e^2-a^2 f^2}+\frac {f \sqrt {a^2 c-b^2 c x^2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{c (e+f x) \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {\left (a^2 f^2 (2 C e-B f)-b^2 \left (C e^3-A e f^2\right )\right ) \int \frac {1}{(e+f x) \sqrt {a^2 c-b^2 c x^2}}dx}{f^2}+\frac {C \left (b^2 e^2-a^2 f^2\right ) \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{b \sqrt {c} f^2}}{b^2 e^2-a^2 f^2}+\frac {f \sqrt {a^2 c-b^2 c x^2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{c (e+f x) \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {C \left (b^2 e^2-a^2 f^2\right ) \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{b \sqrt {c} f^2}-\frac {\left (a^2 f^2 (2 C e-B f)-b^2 \left (C e^3-A e f^2\right )\right ) \int \frac {1}{-b^2 c e^2+a^2 c f^2-\frac {\left (c f a^2+b^2 c e x\right )^2}{a^2 c-b^2 c x^2}}d\frac {c f a^2+b^2 c e x}{\sqrt {a^2 c-b^2 c x^2}}}{f^2}}{b^2 e^2-a^2 f^2}+\frac {f \sqrt {a^2 c-b^2 c x^2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{c (e+f x) \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {\left (a^2 f^2 (2 C e-B f)-b^2 \left (C e^3-A e f^2\right )\right ) \arctan \left (\frac {a^2 c f+b^2 c e x}{\sqrt {c} \sqrt {a^2 c-b^2 c x^2} \sqrt {b^2 e^2-a^2 f^2}}\right )}{\sqrt {c} f^2 \sqrt {b^2 e^2-a^2 f^2}}+\frac {C \left (b^2 e^2-a^2 f^2\right ) \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{b \sqrt {c} f^2}}{b^2 e^2-a^2 f^2}+\frac {f \sqrt {a^2 c-b^2 c x^2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{c (e+f x) \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
(Sqrt[a^2*c - b^2*c*x^2]*((f*(A + (e*(C*e - B*f))/f^2)*Sqrt[a^2*c - b^2*c* x^2])/(c*(b^2*e^2 - a^2*f^2)*(e + f*x)) + ((C*(b^2*e^2 - a^2*f^2)*ArcTan[( b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(b*Sqrt[c]*f^2) + ((a^2*f^2*(2*C*e - B*f) - b^2*(C*e^3 - A*e*f^2))*ArcTan[(a^2*c*f + b^2*c*e*x)/(Sqrt[c]*Sqrt [b^2*e^2 - a^2*f^2]*Sqrt[a^2*c - b^2*c*x^2])])/(Sqrt[c]*f^2*Sqrt[b^2*e^2 - a^2*f^2]))/(b^2*e^2 - a^2*f^2)))/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])
3.1.25.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[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[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
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[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b *e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1165\) vs. \(2(290)=580\).
Time = 1.70 (sec) , antiderivative size = 1166, normalized size of antiderivative = 3.62
(A*ln(2*(b^2*c*e*x+a^2*c*f+(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*(c*(-b^2*x^2+a^ 2))^(1/2)*f)/(f*x+e))*b^2*c*e*f^3*x*(b^2*c)^(1/2)-B*ln(2*(b^2*c*e*x+a^2*c* f+(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*(c*(-b^2*x^2+a^2))^(1/2)*f)/(f*x+e))*a^2 *c*f^4*x*(b^2*c)^(1/2)+C*arctan((b^2*c)^(1/2)*x/(c*(-b^2*x^2+a^2))^(1/2))* a^2*c*f^4*x*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)-C*arctan((b^2*c)^(1/2)*x/(c*(- b^2*x^2+a^2))^(1/2))*b^2*c*e^2*f^2*x*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)+2*C*l n(2*(b^2*c*e*x+a^2*c*f+(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*(c*(-b^2*x^2+a^2))^ (1/2)*f)/(f*x+e))*a^2*c*e*f^3*x*(b^2*c)^(1/2)-C*ln(2*(b^2*c*e*x+a^2*c*f+(c *(a^2*f^2-b^2*e^2)/f^2)^(1/2)*(c*(-b^2*x^2+a^2))^(1/2)*f)/(f*x+e))*b^2*c*e ^3*f*x*(b^2*c)^(1/2)+A*ln(2*(b^2*c*e*x+a^2*c*f+(c*(a^2*f^2-b^2*e^2)/f^2)^( 1/2)*(c*(-b^2*x^2+a^2))^(1/2)*f)/(f*x+e))*b^2*c*e^2*f^2*(b^2*c)^(1/2)-B*ln (2*(b^2*c*e*x+a^2*c*f+(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*(c*(-b^2*x^2+a^2))^( 1/2)*f)/(f*x+e))*a^2*c*e*f^3*(b^2*c)^(1/2)+C*arctan((b^2*c)^(1/2)*x/(c*(-b ^2*x^2+a^2))^(1/2))*a^2*c*e*f^3*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)-C*arctan(( b^2*c)^(1/2)*x/(c*(-b^2*x^2+a^2))^(1/2))*b^2*c*e^3*f*(c*(a^2*f^2-b^2*e^2)/ f^2)^(1/2)+2*C*ln(2*(b^2*c*e*x+a^2*c*f+(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*(c* (-b^2*x^2+a^2))^(1/2)*f)/(f*x+e))*a^2*c*e^2*f^2*(b^2*c)^(1/2)-C*ln(2*(b^2* c*e*x+a^2*c*f+(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*(c*(-b^2*x^2+a^2))^(1/2)*f)/ (f*x+e))*b^2*c*e^4*(b^2*c)^(1/2)-A*f^4*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*(c* (-b^2*x^2+a^2))^(1/2)*(b^2*c)^(1/2)+B*e*f^3*(c*(a^2*f^2-b^2*e^2)/f^2)^(...
Timed out. \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2} \, dx=\int \frac {A + B x + C x^{2}}{\sqrt {- c \left (- a + b x\right )} \sqrt {a + b x} \left (e + f x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((4*b^2*c>0)', see `assume?` for more detai
Time = 0.46 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.63 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2} \, dx=\frac {\frac {2 \, {\left (C b^{3} \sqrt {-c} e^{3} - 2 \, C a^{2} b \sqrt {-c} e f^{2} - A b^{3} \sqrt {-c} e f^{2} + B a^{2} b \sqrt {-c} f^{3}\right )} \arctan \left (-\frac {2 \, b c e - {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} f}{2 \, \sqrt {-b^{2} e^{2} + a^{2} f^{2}} c}\right )}{{\left (b^{2} e^{2} f^{2} - a^{2} f^{4}\right )} \sqrt {-b^{2} e^{2} + a^{2} f^{2}} c} - \frac {C \log \left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2}\right )}{\sqrt {-c} f^{2}} + \frac {4 \, {\left (C b^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} \sqrt {-c} e^{3} - B b^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} \sqrt {-c} e^{2} f - 2 \, C a^{2} b^{2} \sqrt {-c} c e^{2} f + A b^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} \sqrt {-c} e f^{2} + 2 \, B a^{2} b^{2} \sqrt {-c} c e f^{2} - 2 \, A a^{2} b^{2} \sqrt {-c} c f^{3}\right )}}{{\left (b^{2} e^{2} f^{2} - a^{2} f^{4}\right )} {\left (4 \, b {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} c e - {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} f - 4 \, a^{2} c^{2} f\right )}}}{b} \]
(2*(C*b^3*sqrt(-c)*e^3 - 2*C*a^2*b*sqrt(-c)*e*f^2 - A*b^3*sqrt(-c)*e*f^2 + B*a^2*b*sqrt(-c)*f^3)*arctan(-1/2*(2*b*c*e - (sqrt(b*x + a)*sqrt(-c) - sq rt(-(b*x + a)*c + 2*a*c))^2*f)/(sqrt(-b^2*e^2 + a^2*f^2)*c))/((b^2*e^2*f^2 - a^2*f^4)*sqrt(-b^2*e^2 + a^2*f^2)*c) - C*log((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2)/(sqrt(-c)*f^2) + 4*(C*b^3*(sqrt(b*x + a)*sq rt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*sqrt(-c)*e^3 - B*b^3*(sqrt(b*x + a) *sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*sqrt(-c)*e^2*f - 2*C*a^2*b^2*sqr t(-c)*c*e^2*f + A*b^3*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c) )^2*sqrt(-c)*e*f^2 + 2*B*a^2*b^2*sqrt(-c)*c*e*f^2 - 2*A*a^2*b^2*sqrt(-c)*c *f^3)/((b^2*e^2*f^2 - a^2*f^4)*(4*b*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*c*e - (sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a* c))^4*f - 4*a^2*c^2*f)))/b
Timed out. \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2} \, dx=\text {Hanged} \]