3.804 \(\int \frac{\sqrt{a+b x}}{\sqrt{-a-b x}} \, dx\)

Optimal. Leaf size=23 \[ \frac{x \sqrt{a+b x}}{\sqrt{-a-b x}} \]

[Out]

(x*Sqrt[a + b*x])/Sqrt[-a - b*x]

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Rubi [A]  time = 0.0102039, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x \sqrt{a+b x}}{\sqrt{-a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[a + b*x])/Sqrt[-a - b*x]

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Rubi in Sympy [A]  time = 3.35592, size = 19, normalized size = 0.83 \[ \frac{x \sqrt{a + b x}}{\sqrt{- a - b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(a + b*x)/sqrt(-a - b*x)

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Mathematica [A]  time = 0.00579617, size = 23, normalized size = 1. \[ \frac{x \sqrt{a+b x}}{\sqrt{-a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[a + b*x])/Sqrt[-a - b*x]

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Maple [A]  time = 0.003, size = 23, normalized size = 1. \[ -{\frac{1}{b}\sqrt{bx+a}\sqrt{-bx-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.233949, size = 1, normalized size = 0.04 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

0

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Sympy [A]  time = 6.01976, size = 48, normalized size = 2.09 \[ \begin{cases} - i \left (\frac{a}{b} + x\right ) & \text{for}\: \left (\frac{a}{b} + x > -1 \wedge \frac{a}{b} + x < 1\right ) \vee \frac{a}{b} + x > 1 \vee \frac{a}{b} + x < -1 \\- i{G_{2, 2}^{1, 1}\left (\begin{matrix} 1 & 2 \\1 & 0 \end{matrix} \middle |{\frac{a}{b} + x} \right )} - i{G_{2, 2}^{0, 2}\left (\begin{matrix} 2, 1 & \\ & 1, 0 \end{matrix} \middle |{\frac{a}{b} + x} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-I*(a/b + x), (a/b + x > 1) | (a/b + x < -1) | ((a/b + x > -1) & (a/b
+ x < 1))), (-I*meijerg(((1,), (2,)), ((1,), (0,)), a/b + x) - I*meijerg(((2, 1
), ()), ((), (1, 0)), a/b + x), True))

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GIAC/XCAS [A]  time = 0.220148, size = 14, normalized size = 0.61 \[ -\frac{i \,{\left (b x + a\right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-I*(b*x + a)/b