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 = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x^{1-m} \sqrt {a+b x}}{(1-m) \sqrt {-a-b x}} \end {gather*}
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*}
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Mathematica [A] time = 0.01, size = 36, normalized size = 1.00 \begin {gather*} \frac {x^{1-m} \sqrt {a+b x}}{(1-m) \sqrt {-a-b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{-m} \sqrt {a+b x}}{\sqrt {-a-b x}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.82, size = 42, normalized size = 1.17 \begin {gather*} \frac {\sqrt {b x + a} \sqrt {-b x - a} x}{{\left (a m + {\left (b m - b\right )} x - a\right )} x^{m}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.21, size = 14, normalized size = 0.39 \begin {gather*} \frac {i \, x^{-m + 1}}{m - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {b x +a}\, x \,x^{-m}}{\left (m -1\right ) \sqrt {-b x -a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.86, size = 15, normalized size = 0.42 \begin {gather*} -\frac {x}{{\left (i \, m - i\right )} x^{m}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 31, normalized size = 0.86 \begin {gather*} -\frac {x^{1-m}\,\sqrt {a+b\,x}}{\left (m-1\right )\,\sqrt {-a-b\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.83, size = 146, normalized size = 4.06 \begin {gather*} \begin {cases} \frac {i a a^{- m} b^{m} \left (-1 + \frac {b \left (\frac {a}{b} + x\right )}{a}\right )^{- m}}{b \left (1 - 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 )}{1 - m} & \text {for}\: \left |{\frac {b \left (\frac {a}{b} + x\right )}{a}}\right | > 1 \\- \frac {a a^{- m} b^{m} \left (1 - \frac {b \left (\frac {a}{b} + x\right )}{a}\right )^{- m}}{b \left (- i m e^{i \pi m} + i e^{i \pi m}\right )} + \frac {a^{- m} b^{m} \left (1 - \frac {b \left (\frac {a}{b} + x\right )}{a}\right )^{- m} \left (\frac {a}{b} + x\right )}{- i m e^{i \pi m} + i e^{i \pi m}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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