3.805 \(\int \frac{\sqrt{a+b x}}{x \sqrt{-a-b x}} \, dx\)

Optimal. Leaf size=24 \[ \frac{\log (x) \sqrt{a+b x}}{\sqrt{-a-b x}} \]

[Out]

(Sqrt[a + b*x]*Log[x])/Sqrt[-a - b*x]

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Rubi [A]  time = 0.0164548, antiderivative size = 24, 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 \[ \frac{\log (x) \sqrt{a+b x}}{\sqrt{-a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Log[x])/Sqrt[-a - b*x]

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Rubi in Sympy [A]  time = 3.5038, size = 20, normalized size = 0.83 \[ \frac{\sqrt{a + b x} \log{\left (x \right )}}{\sqrt{- a - b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a + b*x)*log(x)/sqrt(-a - b*x)

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Mathematica [A]  time = 0.00705403, size = 24, normalized size = 1. \[ \frac{\log (x) \sqrt{a+b x}}{\sqrt{-a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Log[x])/Sqrt[-a - b*x]

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Maple [A]  time = 0.014, size = 22, normalized size = 0.9 \[ -{\ln \left ( x \right ) \sqrt{-bx-a}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234828, 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),x, algorithm="fricas")

[Out]

0

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Sympy [A]  time = 5.55302, size = 37, normalized size = 1.54 \[ \begin{cases} - i \log{\left (-1 + \frac{b \left (\frac{a}{b} + x\right )}{a} \right )} & \text{for}\: \left |{\frac{b \left (\frac{a}{b} + x\right )}{a}}\right | > 1 \\- i \log{\left (1 - \frac{b \left (\frac{a}{b} + x\right )}{a} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-I*log(-1 + b*(a/b + x)/a), Abs(b*(a/b + x)/a) > 1), (-I*log(1 - b*(a
/b + x)/a), True))

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GIAC/XCAS [A]  time = 0.219548, size = 18, normalized size = 0.75 \[ -i \,{\rm ln}\left ({\left | b x \right |}\right ) + i \,{\rm ln}\left ({\left | a \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-I*ln(abs(b*x)) + I*ln(abs(a))