3.769 \(\int (a+b x) \sqrt{a^2-b^2 x^2} \, dx\)

Optimal. Leaf size=76 \[ \frac{1}{2} a x \sqrt{a^2-b^2 x^2}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{3 b}+\frac{a^3 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{2 b} \]

[Out]

(a*x*Sqrt[a^2 - b^2*x^2])/2 - (a^2 - b^2*x^2)^(3/2)/(3*b) + (a^3*ArcTan[(b*x)/Sq
rt[a^2 - b^2*x^2]])/(2*b)

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Rubi [A]  time = 0.061853, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{2} a x \sqrt{a^2-b^2 x^2}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{3 b}+\frac{a^3 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

(a*x*Sqrt[a^2 - b^2*x^2])/2 - (a^2 - b^2*x^2)^(3/2)/(3*b) + (a^3*ArcTan[(b*x)/Sq
rt[a^2 - b^2*x^2]])/(2*b)

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Rubi in Sympy [A]  time = 13.9893, size = 60, normalized size = 0.79 \[ \frac{a^{3} \operatorname{atan}{\left (\frac{b x}{\sqrt{a^{2} - b^{2} x^{2}}} \right )}}{2 b} + \frac{a x \sqrt{a^{2} - b^{2} x^{2}}}{2} - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*atan(b*x/sqrt(a**2 - b**2*x**2))/(2*b) + a*x*sqrt(a**2 - b**2*x**2)/2 - (a*
*2 - b**2*x**2)**(3/2)/(3*b)

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Mathematica [A]  time = 0.0637109, size = 69, normalized size = 0.91 \[ \frac{\sqrt{a^2-b^2 x^2} \left (-2 a^2+3 a b x+2 b^2 x^2\right )+3 a^3 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a^2 - b^2*x^2]*(-2*a^2 + 3*a*b*x + 2*b^2*x^2) + 3*a^3*ArcTan[(b*x)/Sqrt[a^
2 - b^2*x^2]])/(6*b)

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Maple [A]  time = 0.01, size = 71, normalized size = 0.9 \[{\frac{ax}{2}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}+{\frac{{a}^{3}}{2}\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{1}{3\,b} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.763777, size = 85, normalized size = 1.12 \[ \frac{a^{3} \arcsin \left (\frac{b^{2} x}{\sqrt{a^{2} b^{2}}}\right )}{2 \, \sqrt{b^{2}}} + \frac{1}{2} \, \sqrt{-b^{2} x^{2} + a^{2}} a x - \frac{{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a^3*arcsin(b^2*x/sqrt(a^2*b^2))/sqrt(b^2) + 1/2*sqrt(-b^2*x^2 + a^2)*a*x - 1
/3*(-b^2*x^2 + a^2)^(3/2)/b

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Fricas [A]  time = 0.222354, size = 325, normalized size = 4.28 \[ \frac{2 \, b^{6} x^{6} + 3 \, a b^{5} x^{5} - 12 \, a^{2} b^{4} x^{4} - 15 \, a^{3} b^{3} x^{3} + 12 \, a^{4} b^{2} x^{2} + 12 \, a^{5} b x - 6 \,{\left (3 \, a^{4} b^{2} x^{2} - 4 \, a^{6} -{\left (a^{3} b^{2} x^{2} - 4 \, a^{5}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) + 3 \,{\left (2 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{3} - 4 \, a^{3} b^{2} x^{2} - 4 \, a^{4} b x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{6 \,{\left (3 \, a b^{3} x^{2} - 4 \, a^{3} b -{\left (b^{3} x^{2} - 4 \, a^{2} b\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*b^6*x^6 + 3*a*b^5*x^5 - 12*a^2*b^4*x^4 - 15*a^3*b^3*x^3 + 12*a^4*b^2*x^2
+ 12*a^5*b*x - 6*(3*a^4*b^2*x^2 - 4*a^6 - (a^3*b^2*x^2 - 4*a^5)*sqrt(-b^2*x^2 +
a^2))*arctan(-(a - sqrt(-b^2*x^2 + a^2))/(b*x)) + 3*(2*a*b^4*x^4 + 3*a^2*b^3*x^3
- 4*a^3*b^2*x^2 - 4*a^4*b*x)*sqrt(-b^2*x^2 + a^2))/(3*a*b^3*x^2 - 4*a^3*b - (b^
3*x^2 - 4*a^2*b)*sqrt(-b^2*x^2 + a^2))

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Sympy [A]  time = 8.44562, size = 144, normalized size = 1.89 \[ a \left (\begin{cases} - \frac{i a^{2} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{2 b} - \frac{i a x}{2 \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{i b^{2} x^{3}}{2 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \left |{\frac{b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac{a^{2} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{2 b} + \frac{a x \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}}{2} & \text{otherwise} \end{cases}\right ) + b \left (\begin{cases} \frac{x^{2} \sqrt{a^{2}}}{2} & \text{for}\: b^{2} = 0 \\- \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{3 b^{2}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*Piecewise((-I*a**2*acosh(b*x/a)/(2*b) - I*a*x/(2*sqrt(-1 + b**2*x**2/a**2)) +
I*b**2*x**3/(2*a*sqrt(-1 + b**2*x**2/a**2)), Abs(b**2*x**2/a**2) > 1), (a**2*asi
n(b*x/a)/(2*b) + a*x*sqrt(1 - b**2*x**2/a**2)/2, True)) + b*Piecewise((x**2*sqrt
(a**2)/2, Eq(b**2, 0)), (-(a**2 - b**2*x**2)**(3/2)/(3*b**2), True))

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GIAC/XCAS [A]  time = 0.226497, size = 76, normalized size = 1. \[ \frac{a^{3} \arcsin \left (\frac{b x}{a}\right ){\rm sign}\left (a\right ){\rm sign}\left (b\right )}{2 \,{\left | b \right |}} + \frac{1}{6} \, \sqrt{-b^{2} x^{2} + a^{2}}{\left ({\left (2 \, b x + 3 \, a\right )} x - \frac{2 \, a^{2}}{b}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a^3*arcsin(b*x/a)*sign(a)*sign(b)/abs(b) + 1/6*sqrt(-b^2*x^2 + a^2)*((2*b*x
+ 3*a)*x - 2*a^2/b)