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