3.57 \(\int \sqrt{a+b x} \sqrt{a c-b c x} \left (A+B x+C x^2\right ) \, dx\)

Optimal. Leaf size=221 \[ \frac{1}{8} x \sqrt{a+b x} \left (\frac{a^2 C}{b^2}+4 A\right ) \sqrt{a c-b c x}+\frac{a^2 \sqrt{c} \sqrt{a+b x} \left (a^2 C+4 A b^2\right ) \sqrt{a c-b c x} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right )}{8 b^3 \sqrt{a^2 c-b^2 c x^2}}-\frac{B \sqrt{a+b x} \left (a^2-b^2 x^2\right ) \sqrt{a c-b c x}}{3 b^2}-\frac{C x \sqrt{a+b x} \left (a^2-b^2 x^2\right ) \sqrt{a c-b c x}}{4 b^2} \]

[Out]

((4*A + (a^2*C)/b^2)*x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/8 - (B*Sqrt[a + b*x]*Sqr
t[a*c - b*c*x]*(a^2 - b^2*x^2))/(3*b^2) - (C*x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(
a^2 - b^2*x^2))/(4*b^2) + (a^2*Sqrt[c]*(4*A*b^2 + a^2*C)*Sqrt[a + b*x]*Sqrt[a*c
- b*c*x]*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(8*b^3*Sqrt[a^2*c - b^2*
c*x^2])

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Rubi [A]  time = 0.313048, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{8} x \sqrt{a+b x} \left (\frac{a^2 C}{b^2}+4 A\right ) \sqrt{a c-b c x}+\frac{a^2 \sqrt{c} \sqrt{a+b x} \left (a^2 C+4 A b^2\right ) \sqrt{a c-b c x} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right )}{8 b^3 \sqrt{a^2 c-b^2 c x^2}}-\frac{B \sqrt{a+b x} \left (a^2-b^2 x^2\right ) \sqrt{a c-b c x}}{3 b^2}-\frac{C x \sqrt{a+b x} \left (a^2-b^2 x^2\right ) \sqrt{a c-b c x}}{4 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((4*A + (a^2*C)/b^2)*x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/8 - (B*Sqrt[a + b*x]*Sqr
t[a*c - b*c*x]*(a^2 - b^2*x^2))/(3*b^2) - (C*x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(
a^2 - b^2*x^2))/(4*b^2) + (a^2*Sqrt[c]*(4*A*b^2 + a^2*C)*Sqrt[a + b*x]*Sqrt[a*c
- b*c*x]*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(8*b^3*Sqrt[a^2*c - b^2*
c*x^2])

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Rubi in Sympy [A]  time = 35.242, size = 172, normalized size = 0.78 \[ \frac{a^{2} \sqrt{c} \sqrt{a + b x} \left (4 A b^{2} + C a^{2}\right ) \sqrt{a c - b c x} \operatorname{atan}{\left (\frac{b \sqrt{c} x}{\sqrt{a^{2} c - b^{2} c x^{2}}} \right )}}{8 b^{3} \sqrt{a^{2} c - b^{2} c x^{2}}} + \frac{x \sqrt{a + b x} \left (4 A b^{2} + C a^{2}\right ) \sqrt{a c - b c x}}{8 b^{2}} - \frac{\left (4 B + 3 C x\right ) \sqrt{a + b x} \left (a^{2} - b^{2} x^{2}\right ) \sqrt{a c - b c x}}{12 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*sqrt(c)*sqrt(a + b*x)*(4*A*b**2 + C*a**2)*sqrt(a*c - b*c*x)*atan(b*sqrt(c)*
x/sqrt(a**2*c - b**2*c*x**2))/(8*b**3*sqrt(a**2*c - b**2*c*x**2)) + x*sqrt(a + b
*x)*(4*A*b**2 + C*a**2)*sqrt(a*c - b*c*x)/(8*b**2) - (4*B + 3*C*x)*sqrt(a + b*x)
*(a**2 - b**2*x**2)*sqrt(a*c - b*c*x)/(12*b**2)

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Mathematica [A]  time = 0.28605, size = 125, normalized size = 0.57 \[ \frac{\sqrt{c (a-b x)} \left (b \sqrt{a-b x} \sqrt{a+b x} \left (2 b^2 x (6 A+x (4 B+3 C x))-a^2 (8 B+3 C x)\right )+3 a^2 \left (a^2 C+4 A b^2\right ) \tan ^{-1}\left (\frac{b x}{\sqrt{a-b x} \sqrt{a+b x}}\right )\right )}{24 b^3 \sqrt{a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c*(a - b*x)]*(b*Sqrt[a - b*x]*Sqrt[a + b*x]*(-(a^2*(8*B + 3*C*x)) + 2*b^2*
x*(6*A + x*(4*B + 3*C*x))) + 3*a^2*(4*A*b^2 + a^2*C)*ArcTan[(b*x)/(Sqrt[a - b*x]
*Sqrt[a + b*x])]))/(24*b^3*Sqrt[a - b*x])

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Maple [A]  time = 0.016, size = 287, normalized size = 1.3 \[{\frac{1}{24\,{b}^{2}}\sqrt{bx+a}\sqrt{-c \left ( bx-a \right ) } \left ( 6\,C{x}^{3}{b}^{2}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }\sqrt{{b}^{2}c}+12\,Ac{a}^{2}\arctan \left ({\frac{\sqrt{{b}^{2}c}x}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}} \right ){b}^{2}+8\,B{x}^{2}{b}^{2}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }\sqrt{{b}^{2}c}+3\,C{a}^{4}c\arctan \left ({\frac{\sqrt{{b}^{2}c}x}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}} \right ) +12\,Ax\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }\sqrt{{b}^{2}c}{b}^{2}-3\,C{a}^{2}x\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }\sqrt{{b}^{2}c}-8\,B{a}^{2}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }\sqrt{{b}^{2}c} \right ){\frac{1}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}}{\frac{1}{\sqrt{{b}^{2}c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((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)*(6*C*x^3*b^2*(-c*(b^2*x^2-a^2))^(1/2)*(b^2
*c)^(1/2)+12*A*c*a^2*arctan((b^2*c)^(1/2)*x/(-c*(b^2*x^2-a^2))^(1/2))*b^2+8*B*x^
2*b^2*(-c*(b^2*x^2-a^2))^(1/2)*(b^2*c)^(1/2)+3*C*a^4*c*arctan((b^2*c)^(1/2)*x/(-
c*(b^2*x^2-a^2))^(1/2))+12*A*x*(-c*(b^2*x^2-a^2))^(1/2)*(b^2*c)^(1/2)*b^2-3*C*a^
2*x*(-c*(b^2*x^2-a^2))^(1/2)*(b^2*c)^(1/2)-8*B*a^2*(-c*(b^2*x^2-a^2))^(1/2)*(b^2
*c)^(1/2))/(-c*(b^2*x^2-a^2))^(1/2)/(b^2*c)^(1/2)/b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b*c*x + a*c)*(C*x^2 + B*x + A)*sqrt(b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242365, size = 1, normalized size = 0. \[ \left [\frac{3 \,{\left (C a^{4} + 4 \, A a^{2} b^{2}\right )} \sqrt{-c} \log \left (2 \, b^{2} c x^{2} + 2 \, \sqrt{-b c x + a c} \sqrt{b x + a} b \sqrt{-c} x - a^{2} c\right ) + 2 \,{\left (6 \, C b^{3} x^{3} + 8 \, B b^{3} x^{2} - 8 \, B a^{2} b - 3 \,{\left (C a^{2} b - 4 \, A b^{3}\right )} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{48 \, b^{3}}, \frac{3 \,{\left (C a^{4} + 4 \, A a^{2} b^{2}\right )} \sqrt{c} \arctan \left (\frac{b \sqrt{c} x}{\sqrt{-b c x + a c} \sqrt{b x + a}}\right ) +{\left (6 \, C b^{3} x^{3} + 8 \, B b^{3} x^{2} - 8 \, B a^{2} b - 3 \,{\left (C a^{2} b - 4 \, A b^{3}\right )} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{24 \, b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b*c*x + a*c)*(C*x^2 + B*x + A)*sqrt(b*x + a),x, algorithm="fricas")

[Out]

[1/48*(3*(C*a^4 + 4*A*a^2*b^2)*sqrt(-c)*log(2*b^2*c*x^2 + 2*sqrt(-b*c*x + a*c)*s
qrt(b*x + a)*b*sqrt(-c)*x - a^2*c) + 2*(6*C*b^3*x^3 + 8*B*b^3*x^2 - 8*B*a^2*b -
3*(C*a^2*b - 4*A*b^3)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b^3, 1/24*(3*(C*a^4 +
4*A*a^2*b^2)*sqrt(c)*arctan(b*sqrt(c)*x/(sqrt(-b*c*x + a*c)*sqrt(b*x + a))) + (
6*C*b^3*x^3 + 8*B*b^3*x^2 - 8*B*a^2*b - 3*(C*a^2*b - 4*A*b^3)*x)*sqrt(-b*c*x + a
*c)*sqrt(b*x + a))/b^3]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- c \left (- a + b x\right )} \sqrt{a + b x} \left (A + B x + C x^{2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-c*(-a + b*x))*sqrt(a + b*x)*(A + B*x + C*x**2), x)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b*c*x + a*c)*(C*x^2 + B*x + A)*sqrt(b*x + a),x, algorithm="giac")

[Out]

Timed out