3.1105 \(\int \frac{(1+a x) \sqrt{1-a^2 x^2}}{1-a x} \, dx\)

Optimal. Leaf size=62 \[ -\frac{\left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}-\frac{3 \sqrt{1-a^2 x^2}}{2 a}+\frac{3 \sin ^{-1}(a x)}{2 a} \]

[Out]

(-3*Sqrt[1 - a^2*x^2])/(2*a) - (1 - a^2*x^2)^(3/2)/(2*a*(1 - a*x)) + (3*ArcSin[a
*x])/(2*a)

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Rubi [A]  time = 0.109323, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ -\frac{\left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}-\frac{3 \sqrt{1-a^2 x^2}}{2 a}+\frac{3 \sin ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]  Int[((1 + a*x)*Sqrt[1 - a^2*x^2])/(1 - a*x),x]

[Out]

(-3*Sqrt[1 - a^2*x^2])/(2*a) - (1 - a^2*x^2)^(3/2)/(2*a*(1 - a*x)) + (3*ArcSin[a
*x])/(2*a)

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Rubi in Sympy [A]  time = 12.8075, size = 46, normalized size = 0.74 \[ - \frac{3 \sqrt{- a^{2} x^{2} + 1}}{2 a} + \frac{3 \operatorname{asin}{\left (a x \right )}}{2 a} - \frac{\left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{2 a \left (- a x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(-a**2*x**2 + 1)/(2*a) + 3*asin(a*x)/(2*a) - (-a**2*x**2 + 1)**(3/2)/(2*a
*(-a*x + 1))

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Mathematica [A]  time = 0.10661, size = 71, normalized size = 1.15 \[ \frac{\sqrt{1-a^2 x^2} \left (-a x+\frac{6 \log \left (\sqrt{-a x-1}+\sqrt{1-a x}\right )}{\sqrt{-a x-1} \sqrt{1-a x}}-4\right )}{2 a} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 + a*x)*Sqrt[1 - a^2*x^2])/(1 - a*x),x]

[Out]

(Sqrt[1 - a^2*x^2]*(-4 - a*x + (6*Log[Sqrt[-1 - a*x] + Sqrt[1 - a*x]])/(Sqrt[-1
- a*x]*Sqrt[1 - a*x])))/(2*a)

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Maple [B]  time = 0.02, size = 118, normalized size = 1.9 \[ -{\frac{x}{2}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{1}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-2\,{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\, \left ( x-{a}^{-1} \right ) a}}+2\,{\frac{1}{\sqrt{{a}^{2}}}\arctan \left ({\sqrt{{a}^{2}}x{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\, \left ( x-{a}^{-1} \right ) a}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.47933, size = 57, normalized size = 0.92 \[ -\frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} x + \frac{3 \, \arcsin \left (a x\right )}{2 \, a} - \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(-a^2*x^2 + 1)*(a*x + 1)/(a*x - 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.213965, size = 178, normalized size = 2.87 \[ \frac{2 \, a^{3} x^{3} + 4 \, a^{2} x^{2} - 2 \, a x - 6 \,{\left (a^{2} x^{2} + 2 \, \sqrt{-a^{2} x^{2} + 1} - 2\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (a^{3} x^{3} + 4 \, a^{2} x^{2} - 2 \, a x\right )} \sqrt{-a^{2} x^{2} + 1}}{2 \,{\left (a^{3} x^{2} + 2 \, \sqrt{-a^{2} x^{2} + 1} a - 2 \, a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(-a^2*x^2 + 1)*(a*x + 1)/(a*x - 1),x, algorithm="fricas")

[Out]

1/2*(2*a^3*x^3 + 4*a^2*x^2 - 2*a*x - 6*(a^2*x^2 + 2*sqrt(-a^2*x^2 + 1) - 2)*arct
an((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (a^3*x^3 + 4*a^2*x^2 - 2*a*x)*sqrt(-a^2*x^2
+ 1))/(a^3*x^2 + 2*sqrt(-a^2*x^2 + 1)*a - 2*a)

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Sympy [A]  time = 10.247, size = 76, normalized size = 1.23 \[ - \begin{cases} - \frac{- \sqrt{- a^{2} x^{2} + 1} + \operatorname{asin}{\left (a x \right )}}{a} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases} - \begin{cases} - \frac{- \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} - \sqrt{- a^{2} x^{2} + 1} + \frac{\operatorname{asin}{\left (a x \right )}}{2}}{a} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Piecewise((-(-sqrt(-a**2*x**2 + 1) + asin(a*x))/a, (a*x > -1) & (a*x < 1))) - P
iecewise((-(-a*x*sqrt(-a**2*x**2 + 1)/2 - sqrt(-a**2*x**2 + 1) + asin(a*x)/2)/a,
(a*x > -1) & (a*x < 1)))

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GIAC/XCAS [A]  time = 0.215474, size = 46, normalized size = 0.74 \[ -\frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1}{\left (x + \frac{4}{a}\right )} + \frac{3 \, \arcsin \left (a x\right ){\rm sign}\left (a\right )}{2 \,{\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(-a^2*x^2 + 1)*(a*x + 1)/(a*x - 1),x, algorithm="giac")

[Out]

-1/2*sqrt(-a^2*x^2 + 1)*(x + 4/a) + 3/2*arcsin(a*x)*sign(a)/abs(a)