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

Optimal. Leaf size=363 \[ \frac{\left (a^2-b^2 x^2\right ) \left (2 a^2 f^2 (2 C e-B f)-b^2 e \left (f (B e-3 A f)+C e^2\right )\right )}{2 f \sqrt{a+b x} (e+f x) \sqrt{a c-b c x} \left (b^2 e^2-a^2 f^2\right )^2}+\frac{f \left (a^2-b^2 x^2\right ) \left (A+\frac{e (C e-B f)}{f^2}\right )}{2 \sqrt{a+b x} (e+f x)^2 \sqrt{a c-b c x} \left (b^2 e^2-a^2 f^2\right )}+\frac{\sqrt{a^2 c-b^2 c x^2} \left (A \left (a^2 b^2 f^2+2 b^4 e^2\right )+a^2 \left (2 a^2 C f^2+b^2 e (C e-3 B f)\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c} \left (a^2 f+b^2 e x\right )}{\sqrt{a^2 c-b^2 c x^2} \sqrt{b^2 e^2-a^2 f^2}}\right )}{2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \left (b^2 e^2-a^2 f^2\right )^{5/2}} \]

[Out]

(f*(A + (e*(C*e - B*f))/f^2)*(a^2 - b^2*x^2))/(2*(b^2*e^2 - a^2*f^2)*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)
^2) + ((2*a^2*f^2*(2*C*e - B*f) - b^2*e*(C*e^2 + f*(B*e - 3*A*f)))*(a^2 - b^2*x^2))/(2*f*(b^2*e^2 - a^2*f^2)^2
*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)) + ((A*(2*b^4*e^2 + a^2*b^2*f^2) + a^2*(2*a^2*C*f^2 + b^2*e*(C*e -
3*B*f)))*Sqrt[a^2*c - b^2*c*x^2]*ArcTan[(Sqrt[c]*(a^2*f + b^2*e*x))/(Sqrt[b^2*e^2 - a^2*f^2]*Sqrt[a^2*c - b^2*
c*x^2])])/(2*Sqrt[c]*(b^2*e^2 - a^2*f^2)^(5/2)*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])

________________________________________________________________________________________

Rubi [A]  time = 0.676915, antiderivative size = 361, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1610, 1651, 807, 725, 204} \[ \frac{\left (a^2-b^2 x^2\right ) \left (2 a^2 f^2 (2 C e-B f)-b^2 \left (e f (B e-3 A f)+C e^3\right )\right )}{2 f \sqrt{a+b x} (e+f x) \sqrt{a c-b c x} \left (b^2 e^2-a^2 f^2\right )^2}+\frac{f \left (a^2-b^2 x^2\right ) \left (A+\frac{e (C e-B f)}{f^2}\right )}{2 \sqrt{a+b x} (e+f x)^2 \sqrt{a c-b c x} \left (b^2 e^2-a^2 f^2\right )}+\frac{\sqrt{a^2 c-b^2 c x^2} \left (A \left (a^2 b^2 f^2+2 b^4 e^2\right )+a^2 b^2 e (C e-3 B f)+2 a^4 C f^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} \left (a^2 f+b^2 e x\right )}{\sqrt{a^2 c-b^2 c x^2} \sqrt{b^2 e^2-a^2 f^2}}\right )}{2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \left (b^2 e^2-a^2 f^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f*(A + (e*(C*e - B*f))/f^2)*(a^2 - b^2*x^2))/(2*(b^2*e^2 - a^2*f^2)*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(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)))*(a^2 - b^2*x^2))/(2*f*(b^2*e^2 - a^2*f^2)^2
*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(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))*Sqrt[a^2*c - b^2*c*x^2]*ArcTan[(Sqrt[c]*(a^2*f + b^2*e*x))/(Sqrt[b^2*e^2 - a^2*f^2]*Sqrt[a^2*c - b^2*c*
x^2])])/(2*Sqrt[c]*(b^2*e^2 - a^2*f^2)^(5/2)*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])

Rule 1610

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 1651

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 807

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] + Dist[(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] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{A+B x+C x^2}{\sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^3} \, dx &=\frac{\sqrt{a^2 c-b^2 c x^2} \int \frac{A+B x+C x^2}{(e+f x)^3 \sqrt{a^2 c-b^2 c x^2}} \, dx}{\sqrt{a+b x} \sqrt{a c-b c x}}\\ &=\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{\sqrt{a^2 c-b^2 c x^2} \int \frac{2 c \left (A b^2 e+a^2 (C e-B f)\right )-c \left (2 a^2 C f-b^2 \left (B e+\frac{C e^2}{f}-A f\right )\right ) x}{(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 ) \sqrt{a+b x} \sqrt{a c-b c x}}\\ &=\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 \left (C e^3+e 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 (\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^2 c-b^2 c x^2}\right ) \int \frac{1}{(e+f x) \sqrt{a^2 c-b^2 c x^2}} \, dx}{2 \left (b^2 e^2-a^2 f^2\right )^2 \sqrt{a+b x} \sqrt{a c-b c x}}\\ &=\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 \left (C e^3+e 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 (\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^2 c-b^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-b^2 c e^2+a^2 c f^2-x^2} \, dx,x,\frac{a^2 c f+b^2 c e x}{\sqrt{a^2 c-b^2 c x^2}}\right )}{2 \left (b^2 e^2-a^2 f^2\right )^2 \sqrt{a+b x} \sqrt{a c-b c x}}\\ &=\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 \left (C e^3+e 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 (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^2 c-b^2 c x^2} \tan ^{-1}\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}}\\ \end{align*}

Mathematica [A]  time = 1.91553, size = 492, normalized size = 1.36 \[ \frac{\frac{b^2 \sqrt{a-b x} \left (f (A f-B e)+C e^2\right ) \left (2 (e+f x) \left (a^2 f^2+2 b^2 e^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b x} \sqrt{a f-b e}}{\sqrt{a+b x} \sqrt{-a f-b e}}\right )+3 e f \sqrt{a-b x} \sqrt{a+b x} \sqrt{-a f-b e} \sqrt{a f-b e}\right )}{(e+f x) (-a f-b e)^{5/2} (a f-b e)^{5/2}}+\frac{2 f (b x-a) \sqrt{a+b x} (B f-2 C e)}{(e+f x) \left (a^2 f^2-b^2 e^2\right )}+\frac{f (b x-a) \sqrt{a+b x} \left (f (A f-B e)+C e^2\right )}{(e+f x)^2 (a f-b e) (a f+b e)}-\frac{4 b^2 e \sqrt{a-b x} (2 C e-B f) \tan ^{-1}\left (\frac{\sqrt{a-b x} \sqrt{a f-b e}}{\sqrt{a+b x} \sqrt{-a f-b e}}\right )}{(-a f-b e)^{3/2} (a f-b e)^{3/2}}+\frac{4 C \sqrt{a-b x} \tan ^{-1}\left (\frac{\sqrt{a-b x} \sqrt{a f-b e}}{\sqrt{a+b x} \sqrt{-a f-b e}}\right )}{\sqrt{-a f-b e} \sqrt{a f-b e}}}{2 f^2 \sqrt{c (a-b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((f*(C*e^2 + f*(-(B*e) + A*f))*(-a + b*x)*Sqrt[a + b*x])/((-(b*e) + a*f)*(b*e + a*f)*(e + f*x)^2) + (2*f*(-2*C
*e + B*f)*(-a + b*x)*Sqrt[a + b*x])/((-(b^2*e^2) + a^2*f^2)*(e + f*x)) + (4*C*Sqrt[a - b*x]*ArcTan[(Sqrt[-(b*e
) + a*f]*Sqrt[a - b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])])/(Sqrt[-(b*e) - a*f]*Sqrt[-(b*e) + a*f]) - (4*b^2*
e*(2*C*e - B*f)*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)) + (b^2*(C*e^2 + f*(-(B*e) + A*f))*Sqrt[a - b*x]*(3*e*f*Sqrt[-(b*e)
- a*f]*Sqrt[-(b*e) + a*f]*Sqrt[a - b*x]*Sqrt[a + b*x] + 2*(2*b^2*e^2 + a^2*f^2)*(e + f*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)*(e + f*
x)))/(2*f^2*Sqrt[c*(a - b*x)])

________________________________________________________________________________________

Maple [B]  time = 0.052, size = 1848, normalized size = 5.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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

[Out]

Timed out

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Sympy [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)/(f*x+e)**3/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)

[Out]

Exception raised: ValueError

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Giac [B]  time = 16.9052, size = 2238, normalized size = 6.17 \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)/(f*x+e)^3/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="giac")

[Out]

-(2*C*a^4*sqrt(-c)*c^2*f^2 + A*a^2*b^2*sqrt(-c)*c^2*f^2 - 3*B*a^2*b^2*sqrt(-c)*c^2*f*e + C*a^2*b^2*sqrt(-c)*c^
2*e^2 + 2*A*b^4*sqrt(-c)*c^2*e^2)*arctan(1/2*(2*b*c^2*e + (sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x
- a*c)*c))^2*f)/(sqrt(a^2*f^2 - b^2*e^2)*c^2))/((a^4*f^4*abs(c) - 2*a^2*b^2*f^2*abs(c)*e^2 + b^4*abs(c)*e^4)*
sqrt(a^2*f^2 - b^2*e^2)*c^2) + 2*(16*B*a^6*b*sqrt(-c)*c^8*f^5 - 32*C*a^6*b*sqrt(-c)*c^8*f^4*e - 24*A*a^4*b^3*s
qrt(-c)*c^8*f^4*e + 4*A*a^4*b^2*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^2*sqrt(-c)*c^6
*f^5 + 8*B*a^4*b^3*sqrt(-c)*c^8*f^3*e^2 + 20*B*a^4*b^2*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x -
a*c)*c))^2*sqrt(-c)*c^6*f^4*e + 4*B*a^4*b*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^4*sq
rt(-c)*c^4*f^5 + 8*C*a^4*b^3*sqrt(-c)*c^8*f^2*e^3 - 44*C*a^4*b^2*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 +
(b*c*x - a*c)*c))^2*sqrt(-c)*c^6*f^3*e^2 - 40*A*a^2*b^4*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x
- a*c)*c))^2*sqrt(-c)*c^6*f^3*e^2 - 8*C*a^4*b*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^
4*sqrt(-c)*c^4*f^4*e - 6*A*a^2*b^3*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^4*sqrt(-c)*
c^4*f^4*e - A*a^2*b^2*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^6*sqrt(-c)*c^2*f^5 + 16*
B*a^2*b^4*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^2*sqrt(-c)*c^6*f^2*e^3 + 10*B*a^2*b^
3*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^4*sqrt(-c)*c^4*f^3*e^2 + 3*B*a^2*b^2*(sqrt(-
b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^6*sqrt(-c)*c^2*f^4*e + 8*C*a^2*b^4*(sqrt(-b*c*x + a*c
)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^2*sqrt(-c)*c^6*f*e^4 - 14*C*a^2*b^3*(sqrt(-b*c*x + a*c)*sqrt(-c)
- sqrt(2*a*c^2 + (b*c*x - a*c)*c))^4*sqrt(-c)*c^4*f^2*e^3 - 12*A*b^5*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*
c^2 + (b*c*x - a*c)*c))^4*sqrt(-c)*c^4*f^2*e^3 - 5*C*a^2*b^2*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*
c*x - a*c)*c))^6*sqrt(-c)*c^2*f^3*e^2 - 2*A*b^4*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c)
)^6*sqrt(-c)*c^2*f^3*e^2 + 4*B*b^5*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^4*sqrt(-c)*
c^4*f*e^4 + 4*C*b^5*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^4*sqrt(-c)*c^4*e^5 + 2*C*b
^4*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^6*sqrt(-c)*c^2*f*e^4)/((a^4*f^6*abs(c) - 2*
a^2*b^2*f^4*abs(c)*e^2 + b^4*f^2*abs(c)*e^4)*(4*a^2*c^4*f + 4*b*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 +
(b*c*x - a*c)*c))^2*c^2*e + (sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^4*f)^2)