Optimal. Leaf size=28 \[ -\frac{\left (a^2-b^2 x^2\right )^{p+1}}{2 b (p+1)} \]
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Rubi [A] time = 0.0413844, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079, Rules used = {246, 245, 641} \[ -\frac{\left (a^2-b^2 x^2\right )^{p+1}}{2 b (p+1)} \]
Antiderivative was successfully verified.
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Rule 246
Rule 245
Rule 641
Rubi steps
\begin{align*} \int \left (-a \left (a^2-b^2 x^2\right )^p+(a+b x) \left (a^2-b^2 x^2\right )^p\right ) \, dx &=-\left (a \int \left (a^2-b^2 x^2\right )^p \, dx\right )+\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-\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 )+\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)}\\ \end{align*}
Mathematica [A] time = 0.0051315, size = 28, normalized size = 1. \[ -\frac{\left (a^2-b^2 x^2\right )^{p+1}}{2 b (p+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 36, normalized size = 1.3 \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( -bx+a \right ) \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{p}}{2\,b \left ( 1+p \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.39302, size = 57, normalized size = 2.04 \begin{align*} \frac{{\left (b^{2} x^{2} - a^{2}\right )} e^{\left (p \log \left (b x + a\right ) + p \log \left (-b x + a\right )\right )}}{2 \, b{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97114, size = 68, normalized size = 2.43 \begin{align*} \frac{{\left (b^{2} x^{2} - a^{2}\right )}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}}{2 \,{\left (b p + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.82176, size = 49, normalized size = 1.75 \begin{align*} 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 [A] time = 1.2646, size = 35, normalized size = 1.25 \begin{align*} -\frac{{\left (-b^{2} x^{2} + a^{2}\right )}^{p + 1}}{2 \, b{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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