3.819 \(\int \frac{\sqrt{a+b x}}{x^3 \sqrt{-a-b x}} \, dx\)

Optimal. Leaf size=28 \[ -\frac{\sqrt{a+b x}}{2 x^2 \sqrt{-a-b x}} \]

[Out]

-Sqrt[a + b*x]/(2*x^2*Sqrt[-a - b*x])

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Rubi [A]  time = 0.0033396, antiderivative size = 28, 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} \[ -\frac{\sqrt{a+b x}}{2 x^2 \sqrt{-a-b x}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*x]/(x^3*Sqrt[-a - b*x]),x]

[Out]

-Sqrt[a + b*x]/(2*x^2*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^3 \sqrt{-a-b x}} \, dx &=\frac{\sqrt{a+b x} \int \frac{1}{x^3} \, dx}{\sqrt{-a-b x}}\\ &=-\frac{\sqrt{a+b x}}{2 x^2 \sqrt{-a-b x}}\\ \end{align*}

Mathematica [A]  time = 0.0042085, size = 28, normalized size = 1. \[ -\frac{\sqrt{a+b x}}{2 x^2 \sqrt{-a-b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*x]/(x^3*Sqrt[-a - b*x]),x]

[Out]

-Sqrt[a + b*x]/(2*x^2*Sqrt[-a - b*x])

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Maple [A]  time = 0.001, size = 23, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,{x}^{2}}\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^3/(-b*x-a)^(1/2),x)

[Out]

-1/2*(b*x+a)^(1/2)/x^2/(-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^3/(-b*x-a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.86342, size = 4, normalized size = 0.14 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/x^3/(-b*x-a)^(1/2),x, algorithm="fricas")

[Out]

0

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Sympy [C]  time = 4.08265, size = 88, normalized size = 3.14 \begin{align*} \frac{2 i a b^{3} \left (\frac{a}{b} + x\right )}{2 a^{4} - 4 a^{3} b \left (\frac{a}{b} + x\right ) + 2 a^{2} b^{2} \left (\frac{a}{b} + x\right )^{2}} - \frac{i b^{4} \left (\frac{a}{b} + x\right )^{2}}{2 a^{4} - 4 a^{3} b \left (\frac{a}{b} + x\right ) + 2 a^{2} b^{2} \left (\frac{a}{b} + x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(1/2)/x**3/(-b*x-a)**(1/2),x)

[Out]

2*I*a*b**3*(a/b + x)/(2*a**4 - 4*a**3*b*(a/b + x) + 2*a**2*b**2*(a/b + x)**2) - I*b**4*(a/b + x)**2/(2*a**4 -
4*a**3*b*(a/b + x) + 2*a**2*b**2*(a/b + x)**2)

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Giac [C]  time = 1.41103, size = 26, normalized size = 0.93 \begin{align*} -\frac{i \,{\left (\frac{b^{3}}{a^{2}} - \frac{b}{x^{2}}\right )}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/x^3/(-b*x-a)^(1/2),x, algorithm="giac")

[Out]

-1/2*I*(b^3/a^2 - b/x^2)/b