Optimal. Leaf size=40 \[ \frac{2 a (a+b x)^{m+2}}{b (m+2)}-\frac{(a+b x)^{m+3}}{b (m+3)} \]
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Rubi [A] time = 0.0165849, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {627, 43} \[ \frac{2 a (a+b x)^{m+2}}{b (m+2)}-\frac{(a+b x)^{m+3}}{b (m+3)} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int (a+b x)^m \left (a^2-b^2 x^2\right ) \, dx &=\int (a-b x) (a+b x)^{1+m} \, dx\\ &=\int \left (2 a (a+b x)^{1+m}-(a+b x)^{2+m}\right ) \, dx\\ &=\frac{2 a (a+b x)^{2+m}}{b (2+m)}-\frac{(a+b x)^{3+m}}{b (3+m)}\\ \end{align*}
Mathematica [A] time = 0.0284567, size = 36, normalized size = 0.9 \[ \frac{(a+b x)^{m+2} (a (m+4)-b (m+2) x)}{b (m+2) (m+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 40, normalized size = 1. \begin{align*}{\frac{ \left ( bx+a \right ) ^{2+m} \left ( -bmx+am-2\,bx+4\,a \right ) }{b \left ({m}^{2}+5\,m+6 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81246, size = 151, normalized size = 3.78 \begin{align*} -\frac{{\left (a b^{2} m x^{2} - a^{3} m +{\left (b^{3} m + 2 \, b^{3}\right )} x^{3} - 4 \, a^{3} -{\left (a^{2} b m + 6 \, a^{2} b\right )} x\right )}{\left (b x + a\right )}^{m}}{b m^{2} + 5 \, b m + 6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.05682, size = 267, normalized size = 6.68 \begin{align*} \begin{cases} a^{2} a^{m} x & \text{for}\: b = 0 \\- \frac{a \log{\left (\frac{a}{b} + x \right )}}{a b + b^{2} x} - \frac{2 a}{a b + b^{2} x} - \frac{b x \log{\left (\frac{a}{b} + x \right )}}{a b + b^{2} x} & \text{for}\: m = -3 \\\frac{2 a \log{\left (\frac{a}{b} + x \right )}}{b} - x & \text{for}\: m = -2 \\\frac{a^{3} m \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} + \frac{4 a^{3} \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} + \frac{a^{2} b m x \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} + \frac{6 a^{2} b x \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} - \frac{a b^{2} m x^{2} \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} - \frac{b^{3} m x^{3} \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} - \frac{2 b^{3} x^{3} \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2403, size = 159, normalized size = 3.98 \begin{align*} -\frac{{\left (b x + a\right )}^{m} b^{3} m x^{3} +{\left (b x + a\right )}^{m} a b^{2} m x^{2} + 2 \,{\left (b x + a\right )}^{m} b^{3} x^{3} -{\left (b x + a\right )}^{m} a^{2} b m x -{\left (b x + a\right )}^{m} a^{3} m - 6 \,{\left (b x + a\right )}^{m} a^{2} b x - 4 \,{\left (b x + a\right )}^{m} a^{3}}{b m^{2} + 5 \, b m + 6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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