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

Optimal. Leaf size=246 \[ -\frac{\left (a^2-b^2 x^2\right ) \left (2 \left (2 a^2 C f^2-b^2 \left (C e^2-3 f (A f+B e)\right )\right )-b^2 f x (C e-3 B f)\right )}{6 b^4 f \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right ) \left (a^2 (B f+C e)+2 A b^2 e\right )}{2 b^3 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C \left (a^2-b^2 x^2\right ) (e+f x)^2}{3 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}} \]

[Out]

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

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Rubi [A]  time = 0.400279, antiderivative size = 249, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {1610, 1654, 780, 217, 203} \[ -\frac{\left (a^2-b^2 x^2\right ) \left (2 \left (2 a^2 C f^2-\frac{1}{2} b^2 \left (2 C e^2-6 f (A f+B e)\right )\right )-b^2 f x (C e-3 B f)\right )}{6 b^4 f \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right ) \left (a^2 (B f+C e)+2 A b^2 e\right )}{2 b^3 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C \left (a^2-b^2 x^2\right ) (e+f x)^2}{3 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(C*(e + f*x)^2*(a^2 - b^2*x^2))/(3*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - ((2*(2*a^2*C*f^2 - (b^2*(2*C*e^2
- 6*f*(B*e + A*f)))/2) - b^2*f*(C*e - 3*B*f)*x)*(a^2 - b^2*x^2))/(6*b^4*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + (
(2*A*b^2*e + a^2*(C*e + B*f))*Sqrt[a^2*c - b^2*c*x^2]*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(2*b^3*Sq
rt[c]*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 1654

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]))

Rule 780

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 217

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 203

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

Rubi steps

\begin{align*} \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{a^2 c-b^2 c x^2} \int \frac{(e+f x) \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)^2 \left (a^2-b^2 x^2\right )}{3 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 (-c \left (3 A b^2+2 a^2 C\right ) f^2+b^2 c f (C e-3 B f) x\right )}{\sqrt{a^2 c-b^2 c x^2}} \, dx}{3 b^2 c f^2 \sqrt{a+b x} \sqrt{a c-b c x}}\\ &=-\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-\frac{1}{2} b^2 \left (2 C e^2-6 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 (\left (2 A b^2 e+a^2 (C e+B f)\right ) \sqrt{a^2 c-b^2 c x^2}\right ) \int \frac{1}{\sqrt{a^2 c-b^2 c x^2}} \, dx}{2 b^2 \sqrt{a+b x} \sqrt{a c-b c x}}\\ &=-\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-\frac{1}{2} b^2 \left (2 C e^2-6 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 (\left (2 A b^2 e+a^2 (C e+B f)\right ) \sqrt{a^2 c-b^2 c x^2}\right ) \operatorname{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 )}{2 b^2 \sqrt{a+b x} \sqrt{a c-b c x}}\\ &=-\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-\frac{1}{2} b^2 \left (2 C e^2-6 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} \tan ^{-1}\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}}\\ \end{align*}

Mathematica [A]  time = 1.62425, size = 390, normalized size = 1.59 \[ -\frac{6 \sqrt{a-b x} \sqrt{a+b x} \left (\sqrt{a-b x} \sqrt{\frac{b x}{a}+1}+2 \sqrt{a} \sin ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right )\right ) \left (3 a^2 C f-2 a b (B f+C e)+b^2 (A f+B e)\right )+C f \sqrt{a+b x} \left ((a-b x) \sqrt{\frac{b x}{a}+1} \left (22 a^2+9 a b x+2 b^2 x^2\right )+30 a^{5/2} \sqrt{a-b x} \sin ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right )\right )+3 \sqrt{a-b x} \sqrt{a+b x} \left (6 a^{3/2} \sin ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right )+\sqrt{a-b x} (4 a+b x) \sqrt{\frac{b x}{a}+1}\right ) (-3 a C f+b B f+b C e)+12 \sqrt{a-b x} \sqrt{\frac{b x}{a}+1} (b e-a f) \tan ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{a+b x}}\right ) \left (a (a C-b B)+A b^2\right )}{6 b^4 \sqrt{\frac{b x}{a}+1} \sqrt{c (a-b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(6*(3*a^2*C*f + b^2*(B*e + A*f) - 2*a*b*(C*e + B*f))*Sqrt[a - b*x]*Sqrt[a + b*x]*(Sqrt[a - b*x]*Sqrt[1 + (b*x
)/a] + 2*Sqrt[a]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) + 3*(b*C*e + b*B*f - 3*a*C*f)*Sqrt[a - b*x]*Sqrt[a +
b*x]*(Sqrt[a - b*x]*(4*a + b*x)*Sqrt[1 + (b*x)/a] + 6*a^(3/2)*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) + C*f*
Sqrt[a + b*x]*((a - b*x)*Sqrt[1 + (b*x)/a]*(22*a^2 + 9*a*b*x + 2*b^2*x^2) + 30*a^(5/2)*Sqrt[a - b*x]*ArcSin[Sq
rt[a - b*x]/(Sqrt[2]*Sqrt[a])]) + 12*(A*b^2 + a*(-(b*B) + a*C))*(b*e - a*f)*Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*Ar
cTan[Sqrt[a - b*x]/Sqrt[a + b*x]])/(6*b^4*Sqrt[c*(a - b*x)]*Sqrt[1 + (b*x)/a])

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Maple [A]  time = 0.022, size = 365, normalized size = 1.5 \begin{align*}{\frac{1}{6\,c{b}^{4}}\sqrt{bx+a}\sqrt{-c \left ( bx-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}ce+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}cf+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}ce-2\,C{x}^{2}{b}^{2}f\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }\sqrt{{b}^{2}c}-3\,B\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }x{b}^{2}f-3\,C\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }x{b}^{2}e-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 ){\frac{1}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}}{\frac{1}{\sqrt{{b}^{2}c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.70071, size = 689, normalized size = 2.8 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-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)]

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Sympy [C]  time = 138.497, size = 736, normalized size = 2.99 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*A*a*f*meijerg(((-1/4, 1/4), (0, 0, 1/2, 1)), ((-1/2, -1/4, 0, 1/4, 1/2, 0), ()), a**2/(b**2*x**2))/(4*pi**(
3/2)*b**2*sqrt(c)) - A*a*f*meijerg(((-1, -3/4, -1/2, -1/4, 0, 1), ()), ((-3/4, -1/4), (-1, -1/2, -1/2, 0)), a*
*2*exp_polar(-2*I*pi)/(b**2*x**2))/(4*pi**(3/2)*b**2*sqrt(c)) - I*A*e*meijerg(((1/4, 3/4), (1/2, 1/2, 1, 1)),
((0, 1/4, 1/2, 3/4, 1, 0), ()), a**2/(b**2*x**2))/(4*pi**(3/2)*b*sqrt(c)) + A*e*meijerg(((-1/2, -1/4, 0, 1/4,
1/2, 1), ()), ((-1/4, 1/4), (-1/2, 0, 0, 0)), a**2*exp_polar(-2*I*pi)/(b**2*x**2))/(4*pi**(3/2)*b*sqrt(c)) - I
*B*a**2*f*meijerg(((-3/4, -1/4), (-1/2, -1/2, 0, 1)), ((-1, -3/4, -1/2, -1/4, 0, 0), ()), a**2/(b**2*x**2))/(4
*pi**(3/2)*b**3*sqrt(c)) + B*a**2*f*meijerg(((-3/2, -5/4, -1, -3/4, -1/2, 1), ()), ((-5/4, -3/4), (-3/2, -1, -
1, 0)), a**2*exp_polar(-2*I*pi)/(b**2*x**2))/(4*pi**(3/2)*b**3*sqrt(c)) - I*B*a*e*meijerg(((-1/4, 1/4), (0, 0,
1/2, 1)), ((-1/2, -1/4, 0, 1/4, 1/2, 0), ()), a**2/(b**2*x**2))/(4*pi**(3/2)*b**2*sqrt(c)) - B*a*e*meijerg(((
-1, -3/4, -1/2, -1/4, 0, 1), ()), ((-3/4, -1/4), (-1, -1/2, -1/2, 0)), a**2*exp_polar(-2*I*pi)/(b**2*x**2))/(4
*pi**(3/2)*b**2*sqrt(c)) - I*C*a**3*f*meijerg(((-5/4, -3/4), (-1, -1, -1/2, 1)), ((-3/2, -5/4, -1, -3/4, -1/2,
0), ()), a**2/(b**2*x**2))/(4*pi**(3/2)*b**4*sqrt(c)) - C*a**3*f*meijerg(((-2, -7/4, -3/2, -5/4, -1, 1), ()),
((-7/4, -5/4), (-2, -3/2, -3/2, 0)), a**2*exp_polar(-2*I*pi)/(b**2*x**2))/(4*pi**(3/2)*b**4*sqrt(c)) - I*C*a*
*2*e*meijerg(((-3/4, -1/4), (-1/2, -1/2, 0, 1)), ((-1, -3/4, -1/2, -1/4, 0, 0), ()), a**2/(b**2*x**2))/(4*pi**
(3/2)*b**3*sqrt(c)) + C*a**2*e*meijerg(((-3/2, -5/4, -1, -3/4, -1/2, 1), ()), ((-5/4, -3/4), (-3/2, -1, -1, 0)
), a**2*exp_polar(-2*I*pi)/(b**2*x**2))/(4*pi**(3/2)*b**3*sqrt(c))

________________________________________________________________________________________

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

[Out]

Timed out