Integrand size = 40, antiderivative size = 363 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3} \, dx=\frac {f \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (a^2-b^2 x^2\right )}{2 \left (b^2 e^2-a^2 f^2\right ) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2}+\frac {\left (2 a^2 f^2 (2 C e-B f)-b^2 e \left (C e^2+f (B e-3 A f)\right )\right ) \left (a^2-b^2 x^2\right )}{2 f \left (b^2 e^2-a^2 f^2\right )^2 \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)}+\frac {\left (A \left (2 b^4 e^2+a^2 b^2 f^2\right )+a^2 \left (2 a^2 C f^2+b^2 e (C e-3 B f)\right )\right ) \sqrt {a^2 c-b^2 c x^2} \arctan \left (\frac {\sqrt {c} \left (a^2 f+b^2 e x\right )}{\sqrt {b^2 e^2-a^2 f^2} \sqrt {a^2 c-b^2 c x^2}}\right )}{2 \sqrt {c} \left (b^2 e^2-a^2 f^2\right )^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}} \]
1/2*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)^2/(b* x+a)^(1/2)/(-b*c*x+a*c)^(1/2)+1/2*(2*a^2*f^2*(-B*f+2*C*e)-b^2*e*(C*e^2+f*( -3*A*f+B*e)))*(-b^2*x^2+a^2)/f/(-a^2*f^2+b^2*e^2)^2/(f*x+e)/(b*x+a)^(1/2)/ (-b*c*x+a*c)^(1/2)+1/2*(A*(a^2*b^2*f^2+2*b^4*e^2)+a^2*(2*a^2*C*f^2+b^2*e*( -3*B*f+C*e)))*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)/(-a^2*f^2+b^2*e^2)^(5/2)/c^ (1/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)
Time = 1.03 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3} \, dx=\frac {\frac {(-a+b x) \sqrt {a+b x} \left (b^2 e \left (C e^2 x+B e (2 e+f x)-A f (4 e+3 f x)\right )+a^2 f (-C e (3 e+4 f x)+f (A f+B (e+2 f x)))\right )}{2 (b e-a f)^2 (b e+a f)^2 (e+f x)^2}+\frac {\left (2 a^4 C f^2+a^2 b^2 e (C e-3 B f)+A \left (2 b^4 e^2+a^2 b^2 f^2\right )\right ) \sqrt {a-b x} \arctan \left (\frac {\sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a-b x}}\right )}{(b e-a f)^{5/2} (b e+a f)^{5/2}}}{\sqrt {c (a-b x)}} \]
(((-a + b*x)*Sqrt[a + b*x]*(b^2*e*(C*e^2*x + B*e*(2*e + f*x) - A*f*(4*e + 3*f*x)) + a^2*f*(-(C*e*(3*e + 4*f*x)) + f*(A*f + B*(e + 2*f*x)))))/(2*(b*e - a*f)^2*(b*e + a*f)^2*(e + f*x)^2) + ((2*a^4*C*f^2 + a^2*b^2*e*(C*e - 3* B*f) + A*(2*b^4*e^2 + a^2*b^2*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)^(5/2)*(b*e + a*f)^(5/2)))/Sqrt[c*(a - b*x)]
Time = 0.79 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2113, 2182, 27, 679, 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.
\(\displaystyle \int \frac {A+B x+C x^2}{\sqrt {a+b x} (e+f x)^3 \sqrt {a c-b c x}} \, dx\) |
\(\Big \downarrow \) 2113 |
\(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \int \frac {C x^2+B x+A}{(e+f x)^3 \sqrt {a^2 c-b^2 c x^2}}dx}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 2182 |
\(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\int \frac {c \left (2 \left ((C e-B f) a^2+A b^2 e\right )-\left (2 a^2 C f-b^2 \left (\frac {C e^2}{f}+B e-A f\right )\right ) x\right )}{(e+f x)^2 \sqrt {a^2 c-b^2 c x^2}}dx}{2 c \left (b^2 e^2-a^2 f^2\right )}+\frac {f \sqrt {a^2 c-b^2 c x^2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{2 c (e+f x)^2 \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\int \frac {2 \left ((C e-B f) a^2+A b^2 e\right )-\left (2 a^2 C f-b^2 \left (\frac {C e^2}{f}+B e-A f\right )\right ) x}{(e+f x)^2 \sqrt {a^2 c-b^2 c x^2}}dx}{2 \left (b^2 e^2-a^2 f^2\right )}+\frac {f \sqrt {a^2 c-b^2 c x^2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{2 c (e+f x)^2 \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 679 |
\(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {\left (2 a^4 C f^2+A \left (a^2 b^2 f^2+2 b^4 e^2\right )+a^2 b^2 e (C e-3 B f)\right ) \int \frac {1}{(e+f x) \sqrt {a^2 c-b^2 c x^2}}dx}{b^2 e^2-a^2 f^2}+\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 a^2 f^2 (2 C e-B f)-b^2 \left (e f (B e-3 A f)+C e^3\right )\right )}{c f (e+f x) \left (b^2 e^2-a^2 f^2\right )}}{2 \left (b^2 e^2-a^2 f^2\right )}+\frac {f \sqrt {a^2 c-b^2 c x^2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{2 c (e+f x)^2 \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 a^2 f^2 (2 C e-B f)-b^2 \left (e f (B e-3 A f)+C e^3\right )\right )}{c f (e+f x) \left (b^2 e^2-a^2 f^2\right )}-\frac {\left (2 a^4 C f^2+A \left (a^2 b^2 f^2+2 b^4 e^2\right )+a^2 b^2 e (C e-3 B f)\right ) \int \frac {1}{-b^2 c e^2+a^2 c f^2-\frac {\left (c f a^2+b^2 c e x\right )^2}{a^2 c-b^2 c x^2}}d\frac {c f a^2+b^2 c e x}{\sqrt {a^2 c-b^2 c x^2}}}{b^2 e^2-a^2 f^2}}{2 \left (b^2 e^2-a^2 f^2\right )}+\frac {f \sqrt {a^2 c-b^2 c x^2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{2 c (e+f x)^2 \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {f \sqrt {a^2 c-b^2 c x^2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{2 c (e+f x)^2 \left (b^2 e^2-a^2 f^2\right )}+\frac {\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 a^2 f^2 (2 C e-B f)-b^2 \left (e f (B e-3 A f)+C e^3\right )\right )}{c f (e+f x) \left (b^2 e^2-a^2 f^2\right )}+\frac {\left (2 a^4 C f^2+A \left (a^2 b^2 f^2+2 b^4 e^2\right )+a^2 b^2 e (C e-3 B f)\right ) \arctan \left (\frac {a^2 c f+b^2 c e x}{\sqrt {c} \sqrt {a^2 c-b^2 c x^2} \sqrt {b^2 e^2-a^2 f^2}}\right )}{\sqrt {c} \left (b^2 e^2-a^2 f^2\right )^{3/2}}}{2 \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\) |
(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])/(2*c*(b^2*e^2 - a^2*f^2)*(e + f*x)^2) + (((2*a^2*f^2*(2*C*e - B*f) - b^2*(C*e^3 + e*f*(B*e - 3*A*f)))*Sqrt[a^2*c - b^2*c*x^2])/(c*f*(b^2*e^2 - a^2*f^2)*(e + f*x)) + ((2*a^4*C*f^2 + a^2*b^2*e*(C*e - 3*B*f) + A*(2*b^4* e^2 + a^2*b^2*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]*(b^2*e^2 - a^2*f^2)^(3/2)))/(2 *(b^2*e^2 - a^2*f^2))))/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])
3.1.26.3.1 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[(-(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/(((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[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 )/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[Simplify[m + 2*p + 3], 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. \(1793\) vs. \(2(333)=666\).
Time = 1.67 (sec) , antiderivative size = 1794, normalized size of antiderivative = 4.94
-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))*a^2*b^2*c*f^4*x^2+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^4*c* e^2*f^2*x^2+4*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^4*c*e^3*f*x+4*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^ 4*c*e*f^3*x+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))*a^2*b^2*c*e^2*f^2-3*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*b^2*c*e^3*f+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^4*c*f^4*x^2+2*B*a^2*f^4*x*(c*(-b^ 2*x^2+a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)-4*A*b^2*e^2*f^2*(c*(-b^2 *x^2+a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)+A*a^2*f^4*(c*(-b^2*x^2+a^ 2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)+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^4*c*e^4 +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^4*c*e^2*f^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*b^2*c*e^4+B *a^2*e*f^3*(c*(-b^2*x^2+a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)+2*B*b^ 2*e^3*f*(c*(-b^2*x^2+a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)-3*C*a^...
Time = 41.19 (sec) , antiderivative size = 1355, normalized size of antiderivative = 3.73 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3} \, dx=\text {Too large to display} \]
[1/4*((3*B*a^2*b^2*e^3*f - (C*a^2*b^2 + 2*A*b^4)*e^4 - (2*C*a^4 + A*a^2*b^ 2)*e^2*f^2 + (3*B*a^2*b^2*e*f^3 - (C*a^2*b^2 + 2*A*b^4)*e^2*f^2 - (2*C*a^4 + A*a^2*b^2)*f^4)*x^2 + 2*(3*B*a^2*b^2*e^2*f^2 - (C*a^2*b^2 + 2*A*b^4)*e^ 3*f - (2*C*a^4 + A*a^2*b^2)*e*f^3)*x)*sqrt(-b^2*c*e^2 + a^2*c*f^2)*log((2* a^2*b^2*c*e*f*x - a^2*b^2*c*e^2 + 2*a^4*c*f^2 + (2*b^4*c*e^2 - a^2*b^2*c*f ^2)*x^2 - 2*sqrt(-b^2*c*e^2 + a^2*c*f^2)*(b^2*e*x + a^2*f)*sqrt(-b*c*x + a *c)*sqrt(b*x + a))/(f^2*x^2 + 2*e*f*x + e^2)) - 2*(2*B*b^4*e^5 - B*a^2*b^2 *e^3*f^2 - B*a^4*e*f^4 - A*a^4*f^5 - (3*C*a^2*b^2 + 4*A*b^4)*e^4*f + (3*C* a^4 + 5*A*a^2*b^2)*e^2*f^3 + (C*b^4*e^5 + B*b^4*e^4*f + B*a^2*b^2*e^2*f^3 - 2*B*a^4*f^5 - (5*C*a^2*b^2 + 3*A*b^4)*e^3*f^2 + (4*C*a^4 + 3*A*a^2*b^2)* e*f^4)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/(b^6*c*e^8 - 3*a^2*b^4*c*e^6*f ^2 + 3*a^4*b^2*c*e^4*f^4 - a^6*c*e^2*f^6 + (b^6*c*e^6*f^2 - 3*a^2*b^4*c*e^ 4*f^4 + 3*a^4*b^2*c*e^2*f^6 - a^6*c*f^8)*x^2 + 2*(b^6*c*e^7*f - 3*a^2*b^4* c*e^5*f^3 + 3*a^4*b^2*c*e^3*f^5 - a^6*c*e*f^7)*x), -1/2*((3*B*a^2*b^2*e^3* f - (C*a^2*b^2 + 2*A*b^4)*e^4 - (2*C*a^4 + A*a^2*b^2)*e^2*f^2 + (3*B*a^2*b ^2*e*f^3 - (C*a^2*b^2 + 2*A*b^4)*e^2*f^2 - (2*C*a^4 + A*a^2*b^2)*f^4)*x^2 + 2*(3*B*a^2*b^2*e^2*f^2 - (C*a^2*b^2 + 2*A*b^4)*e^3*f - (2*C*a^4 + A*a^2* b^2)*e*f^3)*x)*sqrt(b^2*c*e^2 - a^2*c*f^2)*arctan(sqrt(b^2*c*e^2 - a^2*c*f ^2)*(b^2*e*x + a^2*f)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)/(a^2*b^2*c*e^2 - a^ 4*c*f^2 - (b^4*c*e^2 - a^2*b^2*c*f^2)*x^2)) + (2*B*b^4*e^5 - B*a^2*b^2*...
Timed out. \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((a*f-b*e)>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 1425 vs. \(2 (335) = 670\).
Time = 0.71 (sec) , antiderivative size = 1425, normalized size of antiderivative = 3.93 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3} \, dx=\text {Too large to display} \]
-((C*a^2*b^3*sqrt(-c)*e^2 + 2*A*b^5*sqrt(-c)*e^2 - 3*B*a^2*b^3*sqrt(-c)*e* f + 2*C*a^4*b*sqrt(-c)*f^2 + A*a^2*b^3*sqrt(-c)*f^2)*arctan(-1/2*(2*b*c*e - (sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*f)/(sqrt(-b^2*e^ 2 + a^2*f^2)*c))/((b^4*e^4 - 2*a^2*b^2*e^2*f^2 + a^4*f^4)*sqrt(-b^2*e^2 + a^2*f^2)*c) + 2*(4*C*b^6*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a *c))^4*sqrt(-c)*c*e^5 - 2*C*b^5*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)* c + 2*a*c))^6*sqrt(-c)*e^4*f + 4*B*b^6*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b* x + a)*c + 2*a*c))^4*sqrt(-c)*c*e^4*f - 8*C*a^2*b^5*(sqrt(b*x + a)*sqrt(-c ) - sqrt(-(b*x + a)*c + 2*a*c))^2*sqrt(-c)*c^2*e^4*f - 14*C*a^2*b^4*(sqrt( b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4*sqrt(-c)*c*e^3*f^2 - 12* A*b^6*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4*sqrt(-c)*c*e ^3*f^2 - 16*B*a^2*b^5*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c) )^2*sqrt(-c)*c^2*e^3*f^2 + 8*C*a^4*b^4*sqrt(-c)*c^3*e^3*f^2 + 5*C*a^2*b^3* (sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^6*sqrt(-c)*e^2*f^3 + 2*A*b^5*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^6*sqrt(-c)* e^2*f^3 + 10*B*a^2*b^4*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c ))^4*sqrt(-c)*c*e^2*f^3 + 44*C*a^4*b^3*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b* x + a)*c + 2*a*c))^2*sqrt(-c)*c^2*e^2*f^3 + 40*A*a^2*b^5*(sqrt(b*x + a)*sq rt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*sqrt(-c)*c^2*e^2*f^3 + 8*B*a^4*b^4* sqrt(-c)*c^3*e^2*f^3 - 3*B*a^2*b^3*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x...
Time = 108.41 (sec) , antiderivative size = 9344, normalized size of antiderivative = 25.74 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3} \, dx=\text {Too large to display} \]
((((a*c - b*c*x)^(1/2) - (a*c)^(1/2))*(4*C*a^4*c^3*f^2 + 2*C*a^2*b^2*c^3*e ^2))/(((a + b*x)^(1/2) - a^(1/2))*(b^5*e^5 - 2*a^2*b^3*e^3*f^2 + a^4*b*e*f ^4)) + (((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3*(68*C*a^4*c^2*f^2 - 14*C*a^2 *b^2*c^2*e^2))/(((a + b*x)^(1/2) - a^(1/2))^3*(b^5*e^5 - 2*a^2*b^3*e^3*f^2 + a^4*b*e*f^4)) - ((68*C*a^4*c*f^2 - 14*C*a^2*b^2*c*e^2)*((a*c - b*c*x)^( 1/2) - (a*c)^(1/2))^5)/(((a + b*x)^(1/2) - a^(1/2))^5*(b^5*e^5 - 2*a^2*b^3 *e^3*f^2 + a^4*b*e*f^4)) - ((4*C*a^4*f^2 + 2*C*a^2*b^2*e^2)*((a*c - b*c*x) ^(1/2) - (a*c)^(1/2))^7)/(((a + b*x)^(1/2) - a^(1/2))^7*(b^5*e^5 - 2*a^2*b ^3*e^3*f^2 + a^4*b*e*f^4)) - (a^(1/2)*(a*c)^(1/2)*(48*C*a^4*c*f^3 - 24*C*a ^2*b^2*c*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/(((a + b*x)^(1/2) - a^(1/2))^4*(b^6*e^6 - 2*a^2*b^4*e^4*f^2 + a^4*b^2*e^2*f^4)) + (a^(1/2)*(a *c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6*(24*C*a^4*f^3 + 12*C*a^2*b ^2*e^2*f))/(((a + b*x)^(1/2) - a^(1/2))^6*(b^6*e^6 - 2*a^2*b^4*e^4*f^2 + a ^4*b^2*e^2*f^4)) + (a^(1/2)*(a*c)^(1/2)*(24*C*a^4*c^2*f^3 + 12*C*a^2*b^2*c ^2*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(((a + b*x)^(1/2) - a^(1/ 2))^2*(b^6*e^6 - 2*a^2*b^4*e^4*f^2 + a^4*b^2*e^2*f^4)))/(((a*c - b*c*x)^(1 /2) - (a*c)^(1/2))^8/((a + b*x)^(1/2) - a^(1/2))^8 + c^4 + (((a*c - b*c*x) ^(1/2) - (a*c)^(1/2))^6*(16*a^2*c*f^2 + 4*b^2*c*e^2))/(b^2*e^2*((a + b*x)^ (1/2) - a^(1/2))^6) + ((16*a^2*c^3*f^2 + 4*b^2*c^3*e^2)*((a*c - b*c*x)^(1/ 2) - (a*c)^(1/2))^2)/(b^2*e^2*((a + b*x)^(1/2) - a^(1/2))^2) - ((32*a^2...