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

Optimal. Leaf size=36 \[ \frac{x^{1-m} \sqrt{a+b x}}{(1-m) \sqrt{-a-b x}} \]

[Out]

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

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Rubi [A]  time = 0.0183763, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{x^{1-m} \sqrt{a+b x}}{(1-m) \sqrt{-a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.26393, size = 26, normalized size = 0.72 \[ \frac{x^{- m + 1} \sqrt{a + b x}}{\sqrt{- a - b x} \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(-m + 1)*sqrt(a + b*x)/(sqrt(-a - b*x)*(-m + 1))

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Mathematica [A]  time = 0.0207019, size = 36, normalized size = 1. \[ \frac{x^{1-m} \sqrt{a+b x}}{(1-m) \sqrt{-a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 31, normalized size = 0.9 \[ -{\frac{x}{ \left ( -1+m \right ){x}^{m}}\sqrt{bx+a}{\frac{1}{\sqrt{-bx-a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.3627, size = 20, normalized size = 0.56 \[ -\frac{x x^{-m}}{i \, m - i} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*x^(-m)/(I*m - I)

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Fricas [A]  time = 0.250254, size = 57, normalized size = 1.58 \[ \frac{\sqrt{b x + a} \sqrt{-b x - a} x}{{\left (a m +{\left (b m - b\right )} x - a\right )} x^{m}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*x + a)*sqrt(-b*x - a)*x/((a*m + (b*m - b)*x - a)*x^m)

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Sympy [A]  time = 14.2645, size = 143, normalized size = 3.97 \[ \begin{cases} - \frac{i a a^{- m} b^{m} \left (-1 + \frac{b \left (\frac{a}{b} + x\right )}{a}\right )^{- m}}{b \left (m - 1\right )} + \frac{i a^{- m} b^{m} \left (-1 + \frac{b \left (\frac{a}{b} + x\right )}{a}\right )^{- m} \left (\frac{a}{b} + x\right )}{m - 1} & \text{for}\: \left |{\frac{b \left (\frac{a}{b} + x\right )}{a}}\right | > 1 \\- \frac{i a a^{- m} b^{m} \left (1 - \frac{b \left (\frac{a}{b} + x\right )}{a}\right )^{- m}}{b \left (m e^{i \pi m} - e^{i \pi m}\right )} + \frac{i a^{- m} b^{m} \left (1 - \frac{b \left (\frac{a}{b} + x\right )}{a}\right )^{- m} \left (\frac{a}{b} + x\right )}{m e^{i \pi m} - e^{i \pi m}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-I*a*a**(-m)*b**m*(-1 + b*(a/b + x)/a)**(-m)/(b*(m - 1)) + I*a**(-m)*
b**m*(-1 + b*(a/b + x)/a)**(-m)*(a/b + x)/(m - 1), Abs(b*(a/b + x)/a) > 1), (-I*
a*a**(-m)*b**m*(1 - b*(a/b + x)/a)**(-m)/(b*(m*exp(I*pi*m) - exp(I*pi*m))) + I*a
**(-m)*b**m*(1 - b*(a/b + x)/a)**(-m)*(a/b + x)/(m*exp(I*pi*m) - exp(I*pi*m)), T
rue))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x + a}}{\sqrt{-b x - a} x^{m}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(b*x + a)/(sqrt(-b*x - a)*x^m), x)