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

Optimal. Leaf size=368 \[ -\frac{\left (a^2-b^2 x^2\right ) \left (f x \left (9 a^2 C f^2-b^2 \left (2 C e^2-4 f (3 A f+2 B e)\right )\right )+4 \left (4 a^2 f^2 (B f+2 C e)-b^2 e \left (C e^2-4 f (3 A f+B e)\right )\right )\right )}{24 b^4 f \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right ) \left (4 A \left (a^2 b^2 f^2+2 b^4 e^2\right )+a^2 \left (3 a^2 C f^2+4 b^2 e (2 B f+C e)\right )\right )}{8 b^5 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\left (a^2-b^2 x^2\right ) (e+f x)^2 (C e-4 B f)}{12 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C \left (a^2-b^2 x^2\right ) (e+f x)^3}{4 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}} \]

[Out]

((C*e - 4*B*f)*(e + f*x)^2*(a^2 - b^2*x^2))/(12*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - (C*(e + f*x)^3*(a^2 -
b^2*x^2))/(4*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - ((4*(4*a^2*f^2*(2*C*e + B*f) - b^2*e*(C*e^2 - 4*f*(B*e
+ 3*A*f))) + f*(9*a^2*C*f^2 - b^2*(2*C*e^2 - 4*f*(2*B*e + 3*A*f)))*x)*(a^2 - b^2*x^2))/(24*b^4*f*Sqrt[a + b*x]
*Sqrt[a*c - b*c*x]) + ((4*A*(2*b^4*e^2 + a^2*b^2*f^2) + a^2*(3*a^2*C*f^2 + 4*b^2*e*(C*e + 2*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]])/(8*b^5*Sqrt[c]*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])

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Rubi [A]  time = 0.875056, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1610, 1654, 833, 780, 217, 203} \[ -\frac{\left (a^2-b^2 x^2\right ) \left (f x \left (9 a^2 C f^2-b^2 \left (2 C e^2-4 f (3 A f+2 B e)\right )\right )+4 \left (4 a^2 f^2 (B f+2 C e)-\frac{1}{4} b^2 \left (4 C e^3-16 e f (3 A f+B e)\right )\right )\right )}{24 b^4 f \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right ) \left (4 A \left (a^2 b^2 f^2+2 b^4 e^2\right )+4 a^2 b^2 e (2 B f+C e)+3 a^4 C f^2\right )}{8 b^5 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\left (a^2-b^2 x^2\right ) (e+f x)^2 (C e-4 B f)}{12 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C \left (a^2-b^2 x^2\right ) (e+f x)^3}{4 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((C*e - 4*B*f)*(e + f*x)^2*(a^2 - b^2*x^2))/(12*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - (C*(e + f*x)^3*(a^2 -
b^2*x^2))/(4*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - ((4*(4*a^2*f^2*(2*C*e + B*f) - (b^2*(4*C*e^3 - 16*e*f*(
B*e + 3*A*f)))/4) + f*(9*a^2*C*f^2 - b^2*(2*C*e^2 - 4*f*(2*B*e + 3*A*f)))*x)*(a^2 - b^2*x^2))/(24*b^4*f*Sqrt[a
+ b*x]*Sqrt[a*c - b*c*x]) + ((3*a^4*C*f^2 + 4*a^2*b^2*e*(C*e + 2*B*f) + 4*A*(2*b^4*e^2 + a^2*b^2*f^2))*Sqrt[a
^2*c - b^2*c*x^2]*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(8*b^5*Sqrt[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 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 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)^2 \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)^2 \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)^3 \left (a^2-b^2 x^2\right )}{4 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)^2 \left (-c \left (4 A b^2+3 a^2 C\right ) f^2+b^2 c f (C e-4 B f) x\right )}{\sqrt{a^2 c-b^2 c x^2}} \, dx}{4 b^2 c f^2 \sqrt{a+b x} \sqrt{a c-b c x}}\\ &=\frac{(C e-4 B f) (e+f x)^2 \left (a^2-b^2 x^2\right )}{12 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C (e+f x)^3 \left (a^2-b^2 x^2\right )}{4 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 (b^2 c^2 f^2 \left (12 A b^2 e+a^2 (7 C e+8 B f)\right )+b^2 c^2 f \left (9 a^2 C f^2-2 b^2 \left (C e^2-2 f (2 B e+3 A f)\right )\right ) x\right )}{\sqrt{a^2 c-b^2 c x^2}} \, dx}{12 b^4 c^2 f^2 \sqrt{a+b x} \sqrt{a c-b c x}}\\ &=\frac{(C e-4 B f) (e+f x)^2 \left (a^2-b^2 x^2\right )}{12 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C (e+f x)^3 \left (a^2-b^2 x^2\right )}{4 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{\left (4 \left (4 a^2 f^2 (2 C e+B f)-\frac{1}{4} b^2 \left (4 C e^3-16 e f (B e+3 A f)\right )\right )+f \left (9 a^2 C f^2-b^2 \left (2 C e^2-4 f (2 B e+3 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{24 b^4 f \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\left (\left (3 a^4 C f^2+4 a^2 b^2 e (C e+2 B f)+4 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}{\sqrt{a^2 c-b^2 c x^2}} \, dx}{8 b^4 \sqrt{a+b x} \sqrt{a c-b c x}}\\ &=\frac{(C e-4 B f) (e+f x)^2 \left (a^2-b^2 x^2\right )}{12 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C (e+f x)^3 \left (a^2-b^2 x^2\right )}{4 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{\left (4 \left (4 a^2 f^2 (2 C e+B f)-\frac{1}{4} b^2 \left (4 C e^3-16 e f (B e+3 A f)\right )\right )+f \left (9 a^2 C f^2-b^2 \left (2 C e^2-4 f (2 B e+3 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{24 b^4 f \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\left (\left (3 a^4 C f^2+4 a^2 b^2 e (C e+2 B f)+4 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}{1+b^2 c x^2} \, dx,x,\frac{x}{\sqrt{a^2 c-b^2 c x^2}}\right )}{8 b^4 \sqrt{a+b x} \sqrt{a c-b c x}}\\ &=\frac{(C e-4 B f) (e+f x)^2 \left (a^2-b^2 x^2\right )}{12 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C (e+f x)^3 \left (a^2-b^2 x^2\right )}{4 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{\left (4 \left (4 a^2 f^2 (2 C e+B f)-\frac{1}{4} b^2 \left (4 C e^3-16 e f (B e+3 A f)\right )\right )+f \left (9 a^2 C f^2-b^2 \left (2 C e^2-4 f (2 B e+3 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{24 b^4 f \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\left (3 a^4 C f^2+4 a^2 b^2 e (C e+2 B f)+4 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{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right )}{8 b^5 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x}}\\ \end{align*}

Mathematica [A]  time = 3.81353, size = 555, normalized size = 1.51 \[ \frac{-12 \sqrt{a-b x} \sqrt{a+b x} \left (6 a^{3/2} \sin ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right )+\sqrt{a-b x} (4 a+b x) \sqrt{\frac{b x}{a}+1}\right ) \left (6 a^2 C f^2-3 a b f (B f+2 C e)+b^2 \left (f (A f+2 B e)+C e^2\right )\right )-24 \sqrt{a-b x} \sqrt{a+b x} (b e-a f) \left (\sqrt{a-b x} \sqrt{\frac{b x}{a}+1}+2 \sqrt{a} \sin ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right )\right ) \left (4 a^2 C f-a b (3 B f+2 C e)+b^2 (2 A f+B e)\right )-4 f \sqrt{a-b x} \sqrt{a+b x} \left (\sqrt{a-b x} \sqrt{\frac{b x}{a}+1} \left (22 a^2+9 a b x+2 b^2 x^2\right )+30 a^{5/2} \sin ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right )\right ) (-4 a C f+b B f+2 b C e)-C f^2 \sqrt{a+b x} \left ((a-b x) \sqrt{\frac{b x}{a}+1} \left (81 a^2 b x+160 a^3+32 a b^2 x^2+6 b^3 x^3\right )+210 a^{7/2} \sqrt{a-b x} \sin ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right )\right )-48 \sqrt{a-b x} \sqrt{\frac{b x}{a}+1} (b e-a f)^2 \tan ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{a+b x}}\right ) \left (a (a C-b B)+A b^2\right )}{24 b^5 \sqrt{\frac{b x}{a}+1} \sqrt{c (a-b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-24*(b*e - a*f)*(4*a^2*C*f + b^2*(B*e + 2*A*f) - a*b*(2*C*e + 3*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])]) - 12*(6*a^2*C*f^2 - 3*a*b*f*(2*C*e
+ B*f) + b^2*(C*e^2 + f*(2*B*e + A*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])]) - 4*f*(2*b*C*e + b*B*f - 4*a*C*f)*Sqrt[a - b*x]*Sqrt[a
+ b*x]*(Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*(22*a^2 + 9*a*b*x + 2*b^2*x^2) + 30*a^(5/2)*ArcSin[Sqrt[a - b*x]/(Sqr
t[2]*Sqrt[a])]) - C*f^2*Sqrt[a + b*x]*((a - b*x)*Sqrt[1 + (b*x)/a]*(160*a^3 + 81*a^2*b*x + 32*a*b^2*x^2 + 6*b^
3*x^3) + 210*a^(7/2)*Sqrt[a - b*x]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) - 48*(A*b^2 + a*(-(b*B) + a*C))*(b
*e - a*f)^2*Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*ArcTan[Sqrt[a - b*x]/Sqrt[a + b*x]])/(24*b^5*Sqrt[c*(a - b*x)]*Sqr
t[1 + (b*x)/a])

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Maple [A]  time = 0.025, size = 635, normalized size = 1.7 \begin{align*}{\frac{1}{24\,c{b}^{4}}\sqrt{bx+a}\sqrt{-c \left ( bx-a \right ) } \left ( -6\,C{x}^{3}{b}^{2}{f}^{2}\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }+12\,A\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}^{2}+24\,A\arctan \left ({\frac{\sqrt{{b}^{2}c}x}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}} \right ){b}^{4}c{e}^{2}+24\,B\arctan \left ({\frac{\sqrt{{b}^{2}c}x}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}} \right ){a}^{2}{b}^{2}cef-8\,B{x}^{2}{b}^{2}{f}^{2}\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }+9\,C\arctan \left ({\frac{\sqrt{{b}^{2}c}x}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}} \right ){a}^{4}c{f}^{2}+12\,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}-16\,C{x}^{2}{b}^{2}ef\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }-12\,A\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }x{b}^{2}{f}^{2}-24\,B\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }x{b}^{2}ef-9\,C\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }x{a}^{2}{f}^{2}-12\,C\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }x{b}^{2}{e}^{2}-48\,A\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }{b}^{2}ef-16\,B\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }{a}^{2}{f}^{2}-24\,B\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }{b}^{2}{e}^{2}-32\,C\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }{a}^{2}ef \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)^2*(C*x^2+B*x+A)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x)

[Out]

1/24*(b*x+a)^(1/2)*(-c*(b*x-a))^(1/2)/c*(-6*C*x^3*b^2*f^2*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)+12*A*arctan((
b^2*c)^(1/2)*x/(-c*(b^2*x^2-a^2))^(1/2))*a^2*b^2*c*f^2+24*A*arctan((b^2*c)^(1/2)*x/(-c*(b^2*x^2-a^2))^(1/2))*b
^4*c*e^2+24*B*arctan((b^2*c)^(1/2)*x/(-c*(b^2*x^2-a^2))^(1/2))*a^2*b^2*c*e*f-8*B*x^2*b^2*f^2*(b^2*c)^(1/2)*(-c
*(b^2*x^2-a^2))^(1/2)+9*C*arctan((b^2*c)^(1/2)*x/(-c*(b^2*x^2-a^2))^(1/2))*a^4*c*f^2+12*C*arctan((b^2*c)^(1/2)
*x/(-c*(b^2*x^2-a^2))^(1/2))*a^2*b^2*c*e^2-16*C*x^2*b^2*e*f*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)-12*A*(b^2*c
)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)*x*b^2*f^2-24*B*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)*x*b^2*e*f-9*C*(b^2*c)^(
1/2)*(-c*(b^2*x^2-a^2))^(1/2)*x*a^2*f^2-12*C*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)*x*b^2*e^2-48*A*(b^2*c)^(1/
2)*(-c*(b^2*x^2-a^2))^(1/2)*b^2*e*f-16*B*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)*a^2*f^2-24*B*(b^2*c)^(1/2)*(-c
*(b^2*x^2-a^2))^(1/2)*b^2*e^2-32*C*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)*a^2*e*f)/b^4/(-c*(b^2*x^2-a^2))^(1/2
)/(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)^2*(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 = 1.9133, size = 1065, normalized size = 2.89 \begin{align*} \left [-\frac{3 \,{\left (8 \, B a^{2} b^{2} e f + 4 \,{\left (C a^{2} b^{2} + 2 \, A b^{4}\right )} e^{2} +{\left (3 \, C a^{4} + 4 \, A a^{2} b^{2}\right )} f^{2}\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 (6 \, C b^{3} f^{2} x^{3} + 24 \, B b^{3} e^{2} + 16 \, B a^{2} b f^{2} + 16 \,{\left (2 \, C a^{2} b + 3 \, A b^{3}\right )} e f + 8 \,{\left (2 \, C b^{3} e f + B b^{3} f^{2}\right )} x^{2} + 3 \,{\left (4 \, C b^{3} e^{2} + 8 \, B b^{3} e f +{\left (3 \, C a^{2} b + 4 \, A b^{3}\right )} f^{2}\right )} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{48 \, b^{5} c}, -\frac{3 \,{\left (8 \, B a^{2} b^{2} e f + 4 \,{\left (C a^{2} b^{2} + 2 \, A b^{4}\right )} e^{2} +{\left (3 \, C a^{4} + 4 \, A a^{2} b^{2}\right )} f^{2}\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 (6 \, C b^{3} f^{2} x^{3} + 24 \, B b^{3} e^{2} + 16 \, B a^{2} b f^{2} + 16 \,{\left (2 \, C a^{2} b + 3 \, A b^{3}\right )} e f + 8 \,{\left (2 \, C b^{3} e f + B b^{3} f^{2}\right )} x^{2} + 3 \,{\left (4 \, C b^{3} e^{2} + 8 \, B b^{3} e f +{\left (3 \, C a^{2} b + 4 \, A b^{3}\right )} f^{2}\right )} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{24 \, b^{5} c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/48*(3*(8*B*a^2*b^2*e*f + 4*(C*a^2*b^2 + 2*A*b^4)*e^2 + (3*C*a^4 + 4*A*a^2*b^2)*f^2)*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*(6*C*b^3*f^2*x^3 + 24*B*b^3*e^2 + 16*B*a^2*b
*f^2 + 16*(2*C*a^2*b + 3*A*b^3)*e*f + 8*(2*C*b^3*e*f + B*b^3*f^2)*x^2 + 3*(4*C*b^3*e^2 + 8*B*b^3*e*f + (3*C*a^
2*b + 4*A*b^3)*f^2)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/(b^5*c), -1/24*(3*(8*B*a^2*b^2*e*f + 4*(C*a^2*b^2 + 2
*A*b^4)*e^2 + (3*C*a^4 + 4*A*a^2*b^2)*f^2)*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)) + (6*C*b^3*f^2*x^3 + 24*B*b^3*e^2 + 16*B*a^2*b*f^2 + 16*(2*C*a^2*b + 3*A*b^3)*e*f + 8*(2*C*b^3*e
*f + B*b^3*f^2)*x^2 + 3*(4*C*b^3*e^2 + 8*B*b^3*e*f + (3*C*a^2*b + 4*A*b^3)*f^2)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x
+ a))/(b^5*c)]

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Sympy [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)**2*(C*x**2+B*x+A)/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)

[Out]

Timed out

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

[Out]

Timed out