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]

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

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Rubi [A]  time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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*}

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Mathematica [A]  time = 0.00, size = 26, normalized size = 1.00 \[ -\frac {\sqrt {a+b x}}{x \sqrt {-a-b x}} \]

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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fricas [A]  time = 0.86, size = 1, normalized size = 0.04 \[ 0 \]

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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giac [C]  time = 1.20, size = 5, normalized size = 0.19 \[ \frac {i}{x} \]

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/x

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maple [A]  time = 0.00, size = 23, normalized size = 0.88 \[ -\frac {\sqrt {b x +a}}{\sqrt {-b x -a}\, x} \]

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 [A]  time = 0.91, size = 28, normalized size = 1.08 \[ \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2}}}{a x} \]

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]

sqrt(-b^2*x^2 - 2*a*b*x - a^2)/(a*x)

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mupad [B]  time = 1.14, size = 21, normalized size = 0.81 \[ \frac {\sqrt {-a-b\,x}}{x\,\sqrt {a+b\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(- a - b*x)^(1/2)/(x*(a + b*x)^(1/2))

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sympy [C]  time = 1.51, size = 20, normalized size = 0.77 \[ \frac {i b^{2} \left (\frac {a}{b} + x\right )}{- a^{2} + a b \left (\frac {a}{b} + x\right )} \]

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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