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 = 0.006489, 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, Rules used = {23, 30} \[ \frac{x^{1-m} \sqrt{a+b x}}{(1-m) \sqrt{-a-b x}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 30
Rubi steps
\begin{align*} \int \frac{x^{-m} \sqrt{a+b x}}{\sqrt{-a-b x}} \, dx &=\frac{\sqrt{a+b x} \int x^{-m} \, dx}{\sqrt{-a-b x}}\\ &=\frac{x^{1-m} \sqrt{a+b x}}{(1-m) \sqrt{-a-b x}}\\ \end{align*}
Mathematica [A] time = 0.0113113, size = 36, normalized size = 1. \[ \frac{x^{1-m} \sqrt{a+b x}}{(1-m) \sqrt{-a-b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 31, normalized size = 0.9 \begin{align*} -{\frac{x}{ \left ( -1+m \right ){x}^{m}}\sqrt{bx+a}{\frac{1}{\sqrt{-bx-a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.31847, size = 20, normalized size = 0.56 \begin{align*} -\frac{x}{{\left (i \, m - i\right )} x^{m}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7015, size = 84, normalized size = 2.33 \begin{align*} \frac{\sqrt{b x + a} \sqrt{-b x - a} x}{{\left (a m +{\left (b m - b\right )} x - a\right )} x^{m}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 6.19846, size = 144, normalized size = 4. \begin{align*} \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}\: \frac{\left |{b \left (\frac{a}{b} + x\right )}\right |}{\left |{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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b x + a}}{\sqrt{-b x - a} x^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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