Optimal. Leaf size=28 \[ -\frac {\left (a^2-b^2 x^2\right )^{1+p}}{2 b (1+p)} \]
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Rubi [A]
time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 = {252, 251, 655}
\begin {gather*} -\frac {\left (a^2-b^2 x^2\right )^{p+1}}{2 b (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rule 655
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*}
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Mathematica [A]
time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} -\frac {\left (a^2-b^2 x^2\right )^{1+p}}{2 b (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 36, normalized size = 1.29
method | result | size |
gosper | \(-\frac {\left (b x +a \right ) \left (-b x +a \right ) \left (-b^{2} x^{2}+a^{2}\right )^{p}}{2 b \left (1+p \right )}\) | \(36\) |
risch | \(-\frac {\left (-b^{2} x^{2}+a^{2}\right ) \left (-b^{2} x^{2}+a^{2}\right )^{p}}{2 b \left (1+p \right )}\) | \(37\) |
norman | \(-\frac {a^{2} {\mathrm e}^{p \ln \left (-b^{2} x^{2}+a^{2}\right )}}{2 b \left (1+p \right )}+\frac {b \,x^{2} {\mathrm e}^{p \ln \left (-b^{2} x^{2}+a^{2}\right )}}{2 p +2}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 42, normalized size = 1.50 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.54, size = 36, normalized size = 1.29 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.72, size = 49, normalized size = 1.75 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.87, size = 26, normalized size = 0.93 \begin {gather*} -\frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{p + 1}}{2 \, b {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.46, size = 26, normalized size = 0.93 \begin {gather*} -\frac {{\left (a^2-b^2\,x^2\right )}^{p+1}}{2\,b\,\left (p+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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