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

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 = 0.0026331, 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, Rules used = {23, 8} \[ \frac{x \sqrt{a+b x}}{\sqrt{-a-b x}} \]

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]

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b x}}{\sqrt{-a-b x}} \, dx &=\frac{\sqrt{a+b x} \int 1 \, dx}{\sqrt{-a-b x}}\\ &=\frac{x \sqrt{a+b x}}{\sqrt{-a-b x}}\\ \end{align*}

Mathematica [A]  time = 0.0033936, 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 = 0.002, size = 23, normalized size = 1. \begin{align*} -{\frac{1}{b}\sqrt{bx+a}\sqrt{-bx-a}} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/(-b*x-a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.75712, size = 4, normalized size = 0.17 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/(-b*x-a)^(1/2),x, algorithm="fricas")

[Out]

0

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Sympy [C]  time = 2.49574, size = 37, normalized size = 1.61 \begin{align*} \begin{cases} - i \left (\frac{a}{b} + x\right ) & \text{for}\: \left |{\frac{a}{b} + x}\right | > 1 \vee \left |{\frac{a}{b} + x}\right | < 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} \end{align*}

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), (Abs(a/b + x) > 1) | (Abs(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 [C]  time = 1.57455, size = 14, normalized size = 0.61 \begin{align*} -\frac{i \,{\left (b x + a\right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/(-b*x-a)^(1/2),x, algorithm="giac")

[Out]

-I*(b*x + a)/b