3.1147 \(\int \sqrt{a+b x} \sqrt{a c-b c x} \, dx\)

Optimal. Leaf size=69 \[ \frac{a^2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a c-b c x}}\right )}{b}+\frac{1}{2} x \sqrt{a+b x} \sqrt{a c-b c x} \]

[Out]

(x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/2 + (a^2*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[a + b*
x])/Sqrt[a*c - b*c*x]])/b

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Rubi [A]  time = 0.0715188, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{a^2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a c-b c x}}\right )}{b}+\frac{1}{2} x \sqrt{a+b x} \sqrt{a c-b c x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x]*Sqrt[a*c - b*c*x],x]

[Out]

(x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/2 + (a^2*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[a + b*
x])/Sqrt[a*c - b*c*x]])/b

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Rubi in Sympy [A]  time = 12.5883, size = 60, normalized size = 0.87 \[ \frac{a^{2} \sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a c - b c x}} \right )}}{b} + \frac{x \sqrt{a + b x} \sqrt{a c - b c x}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*sqrt(c)*atan(sqrt(c)*sqrt(a + b*x)/sqrt(a*c - b*c*x))/b + x*sqrt(a + b*x)*s
qrt(a*c - b*c*x)/2

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Mathematica [A]  time = 0.054807, size = 79, normalized size = 1.14 \[ \frac{\sqrt{c (a-b x)} \left (a^2 \tan ^{-1}\left (\frac{b x}{\sqrt{a-b x} \sqrt{a+b x}}\right )+b x \sqrt{a-b x} \sqrt{a+b x}\right )}{2 b \sqrt{a-b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x]*Sqrt[a*c - b*c*x],x]

[Out]

(Sqrt[c*(a - b*x)]*(b*x*Sqrt[a - b*x]*Sqrt[a + b*x] + a^2*ArcTan[(b*x)/(Sqrt[a -
b*x]*Sqrt[a + b*x])]))/(2*b*Sqrt[a - b*x])

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Maple [B]  time = 0.007, size = 127, normalized size = 1.8 \[ -{\frac{1}{2\,bc}\sqrt{bx+a} \left ( -bcx+ac \right ) ^{{\frac{3}{2}}}}+{\frac{a}{2\,b}\sqrt{bx+a}\sqrt{-bcx+ac}}+{\frac{{a}^{2}c}{2}\sqrt{ \left ( bx+a \right ) \left ( -bcx+ac \right ) }\arctan \left ({x\sqrt{{b}^{2}c}{\frac{1}{\sqrt{-{b}^{2}c{x}^{2}+{a}^{2}c}}}} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{-bcx+ac}}}{\frac{1}{\sqrt{{b}^{2}c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2),x)

[Out]

-1/2/b/c*(b*x+a)^(1/2)*(-b*c*x+a*c)^(3/2)+1/2*a/b*(-b*c*x+a*c)^(1/2)*(b*x+a)^(1/
2)+1/2*a^2*c*((b*x+a)*(-b*c*x+a*c))^(1/2)/(-b*c*x+a*c)^(1/2)/(b*x+a)^(1/2)/(b^2*
c)^(1/2)*arctan((b^2*c)^(1/2)*x/(-b^2*c*x^2+a^2*c)^(1/2))

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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)*sqrt(b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227085, size = 1, normalized size = 0.01 \[ \left [\frac{a^{2} \sqrt{-c} \log \left (2 \, b^{2} c x^{2} + 2 \, \sqrt{-b c x + a c} \sqrt{b x + a} b \sqrt{-c} x - a^{2} c\right ) + 2 \, \sqrt{-b c x + a c} \sqrt{b x + a} b x}{4 \, b}, \frac{a^{2} \sqrt{c} \arctan \left (\frac{b \sqrt{c} x}{\sqrt{-b c x + a c} \sqrt{b x + a}}\right ) + \sqrt{-b c x + a c} \sqrt{b x + a} b x}{2 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b*c*x + a*c)*sqrt(b*x + a),x, algorithm="fricas")

[Out]

[1/4*(a^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*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*x)/b, 1/2*(a^2*sqrt(c)*arct
an(b*sqrt(c)*x/(sqrt(-b*c*x + a*c)*sqrt(b*x + a))) + sqrt(-b*c*x + a*c)*sqrt(b*x
+ a)*b*x)/b]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-c*(-a + b*x))*sqrt(a + b*x), 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)*sqrt(b*x + a),x, algorithm="giac")

[Out]

Timed out