Optimal. Leaf size=13 \[ \frac{x^m}{\sqrt{a+b x}} \]
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Rubi [A] time = 0.0062557, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {12, 74} \[ \frac{x^m}{\sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 74
Rubi steps
\begin{align*} \int \frac{x^{-1+m} (2 a m+b (-1+2 m) x)}{2 (a+b x)^{3/2}} \, dx &=\frac{1}{2} \int \frac{x^{-1+m} (2 a m+b (-1+2 m) x)}{(a+b x)^{3/2}} \, dx\\ &=\frac{x^m}{\sqrt{a+b x}}\\ \end{align*}
Mathematica [A] time = 0.0793584, size = 13, normalized size = 1. \[ \frac{x^m}{\sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 12, normalized size = 0.9 \begin{align*}{{x}^{m}{\frac{1}{\sqrt{bx+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.25709, size = 15, normalized size = 1.15 \begin{align*} \frac{x^{m}}{\sqrt{b x + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96447, size = 36, normalized size = 2.77 \begin{align*} \frac{x x^{m - 1}}{\sqrt{b x + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{{\left (b{\left (2 \, m - 1\right )} x + 2 \, a m\right )} x^{m - 1}}{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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