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

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 = 0.011282, 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 \[ -\frac{\sqrt{a+b x}}{2 x^2 \sqrt{-a-b x}} \]

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])

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Rubi in Sympy [A]  time = 3.55278, size = 24, normalized size = 0.86 \[ - \frac{\sqrt{a + b x}}{2 x^{2} \sqrt{- a - b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/2)/x**3/(-b*x-a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.00751544, 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 = 0.004, size = 23, normalized size = 0.8 \[ -{\frac{1}{2\,{x}^{2}}\sqrt{bx+a}{\frac{1}{\sqrt{-bx-a}}}} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(sqrt(-b*x - a)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236231, size = 1, normalized size = 0.04 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(sqrt(-b*x - a)*x^3),x, algorithm="fricas")

[Out]

0

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Sympy [A]  time = 6.91313, size = 88, normalized size = 3.14 \[ \frac{2 i a b^{3} \left (\frac{a}{b} + x\right )}{2 a^{4} - 4 a^{3} b \left (\frac{a}{b} + x\right ) + 2 a^{2} b^{2} \left (\frac{a}{b} + x\right )^{2}} - \frac{i b^{4} \left (\frac{a}{b} + x\right )^{2}}{2 a^{4} - 4 a^{3} b \left (\frac{a}{b} + x\right ) + 2 a^{2} b^{2} \left (\frac{a}{b} + x\right )^{2}} \]

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/XCAS [A]  time = 0.221677, size = 26, normalized size = 0.93 \[ -\frac{i \,{\left (\frac{b^{3}}{a^{2}} - \frac{b}{x^{2}}\right )}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(sqrt(-b*x - a)*x^3),x, algorithm="giac")

[Out]

-1/2*I*(b^3/a^2 - b/x^2)/b