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

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

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Rubi [A]  time = 0.00, antiderivative size = 28, 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} \begin {gather*} -\frac {\sqrt {a+b x}}{2 x^2 \sqrt {-a-b x}} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a+b x}}{2 x^2 \sqrt {-a-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a+b x}}{x^3 \sqrt {-a-b x}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.54, size = 1, normalized size = 0.04 \begin {gather*} 0 \end {gather*}

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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giac [C]  time = 1.21, size = 5, normalized size = 0.18 \begin {gather*} \frac {i}{2 \, x^{2}} \end {gather*}

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/x^2

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maple [A]  time = 0.00, size = 23, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {b x +a}}{2 \sqrt {-b x -a}\, x^{2}} \end {gather*}

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 [B]  time = 0.88, size = 60, normalized size = 2.14 \begin {gather*} -\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2}} b}{2 \, a^{2} x} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2}}}{2 \, a x^{2}} \end {gather*}

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]

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

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mupad [B]  time = 1.12, size = 22, normalized size = 0.79 \begin {gather*} \frac {\sqrt {-a-b\,x}}{2\,x^2\,\sqrt {a+b\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 1.80, size = 88, normalized size = 3.14 \begin {gather*} - \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 {gather*}

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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