Optimal. Leaf size=83 \[ a x \left (a^2-b^2 x^2\right )^p \left (1-\frac{b^2 x^2}{a^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{b^2 x^2}{a^2}\right )-\frac{\left (a^2-b^2 x^2\right )^{p+1}}{2 b (p+1)} \]
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Rubi [A] time = 0.022037, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {641, 246, 245} \[ a x \left (a^2-b^2 x^2\right )^p \left (1-\frac{b^2 x^2}{a^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{b^2 x^2}{a^2}\right )-\frac{\left (a^2-b^2 x^2\right )^{p+1}}{2 b (p+1)} \]
Antiderivative was successfully verified.
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Rule 641
Rule 246
Rule 245
Rubi steps
\begin{align*} \int (a+b x) \left (a^2-b^2 x^2\right )^p \, dx &=-\frac{\left (a^2-b^2 x^2\right )^{1+p}}{2 b (1+p)}+a \int \left (a^2-b^2 x^2\right )^p \, dx\\ &=-\frac{\left (a^2-b^2 x^2\right )^{1+p}}{2 b (1+p)}+\left (a \left (a^2-b^2 x^2\right )^p \left (1-\frac{b^2 x^2}{a^2}\right )^{-p}\right ) \int \left (1-\frac{b^2 x^2}{a^2}\right )^p \, dx\\ &=-\frac{\left (a^2-b^2 x^2\right )^{1+p}}{2 b (1+p)}+a x \left (a^2-b^2 x^2\right )^p \left (1-\frac{b^2 x^2}{a^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{b^2 x^2}{a^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0463599, size = 83, normalized size = 1. \[ a x \left (a^2-b^2 x^2\right )^p \left (1-\frac{b^2 x^2}{a^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{b^2 x^2}{a^2}\right )-\frac{\left (a^2-b^2 x^2\right )^{p+1}}{2 b (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.381, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x + a\right )}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.94537, size = 82, normalized size = 0.99 \begin{align*} a a^{2 p} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{b^{2} x^{2} e^{2 i \pi }}{a^{2}}} \right )} + b \left (\begin{cases} \frac{x^{2} \left (a^{2}\right )^{p}}{2} & \text{for}\: b^{2} = 0 \\- \frac{\begin{cases} \frac{\left (a^{2} - b^{2} x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a^{2} - b^{2} x^{2} \right )} & \text{otherwise} \end{cases}}{2 b^{2}} & \text{otherwise} \end{cases}\right ) \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{\left (b x + a\right )}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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