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

Optimal. Leaf size=26 \[ -\frac{\sqrt{a+b x}}{x \sqrt{-a-b x}} \]

[Out]

-(Sqrt[a + b*x]/(x*Sqrt[-a - b*x]))

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Rubi [A]  time = 0.0033106, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {23, 30} \[ -\frac{\sqrt{a+b x}}{x \sqrt{-a-b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[a + b*x]/(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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b x}}{x^2 \sqrt{-a-b x}} \, dx &=\frac{\sqrt{a+b x} \int \frac{1}{x^2} \, dx}{\sqrt{-a-b x}}\\ &=-\frac{\sqrt{a+b x}}{x \sqrt{-a-b x}}\\ \end{align*}

Mathematica [A]  time = 0.004671, size = 26, normalized size = 1. \[ -\frac{\sqrt{a+b x}}{x \sqrt{-a-b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[a + b*x]/(x*Sqrt[-a - b*x]))

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Maple [A]  time = 0., size = 23, normalized size = 0.9 \begin{align*} -{\frac{1}{x}\sqrt{bx+a}{\frac{1}{\sqrt{-bx-a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.81379, size = 4, normalized size = 0.15 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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Sympy [C]  time = 1.86493, size = 20, normalized size = 0.77 \begin{align*} \frac{i b^{2} \left (\frac{a}{b} + x\right )}{- a^{2} + a b \left (\frac{a}{b} + x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*b**2*(a/b + x)/(-a**2 + a*b*(a/b + x))

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Giac [C]  time = 1.36119, size = 24, normalized size = 0.92 \begin{align*} \frac{i \,{\left (\frac{b^{2}}{a} + \frac{b}{x}\right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*(b^2/a + b/x)/b