Optimal. Leaf size=57 \[ x (a+b x)^m \left (1-\frac{b^2 x^2}{a^2}\right )^{-m} (a c-b c x)^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{b^2 x^2}{a^2}\right ) \]
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Rubi [A] time = 0.0179656, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {42, 246, 245} \[ x (a+b x)^m \left (1-\frac{b^2 x^2}{a^2}\right )^{-m} (a c-b c x)^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{b^2 x^2}{a^2}\right ) \]
Antiderivative was successfully verified.
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Rule 42
Rule 246
Rule 245
Rubi steps
\begin{align*} \int (a+b x)^m (a c-b c x)^m \, dx &=\left ((a+b x)^m (a c-b c x)^m \left (a^2 c-b^2 c x^2\right )^{-m}\right ) \int \left (a^2 c-b^2 c x^2\right )^m \, dx\\ &=\left ((a+b x)^m (a c-b c x)^m \left (1-\frac{b^2 x^2}{a^2}\right )^{-m}\right ) \int \left (1-\frac{b^2 x^2}{a^2}\right )^m \, dx\\ &=x (a+b x)^m (a c-b c x)^m \left (1-\frac{b^2 x^2}{a^2}\right )^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{b^2 x^2}{a^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0239115, size = 72, normalized size = 1.26 \[ -\frac{2^m (a-b x) (a+b x)^m \left (\frac{a+b x}{a}\right )^{-m} (c (a-b x))^m \, _2F_1\left (-m,m+1;m+2;\frac{a-b x}{2 a}\right )}{b (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.084, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{m} \left ( -bcx+ac \right ) ^{m}\, 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 c x + a c\right )}^{m}{\left (b x + a\right )}^{m}\,{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 c x + a c\right )}^{m}{\left (b x + a\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 5.12157, size = 146, normalized size = 2.56 \begin{align*} \frac{a a^{2 m} c^{m}{G_{6, 6}^{5, 3}\left (\begin{matrix} - \frac{m}{2}, \frac{1}{2} - \frac{m}{2}, 1 & \frac{1}{2}, - m, \frac{1}{2} - m \\- m - \frac{1}{2}, - m, - \frac{m}{2}, \frac{1}{2} - m, \frac{1}{2} - \frac{m}{2} & 0 \end{matrix} \middle |{\frac{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )} e^{- i \pi m}}{4 \pi b \Gamma \left (- m\right )} - \frac{a a^{2 m} c^{m}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, 0, \frac{1}{2}, - \frac{m}{2} - \frac{1}{2}, - \frac{m}{2}, 1 & \\- \frac{m}{2} - \frac{1}{2}, - \frac{m}{2} & - \frac{1}{2}, 0, - m - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi b \Gamma \left (- m\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 c x + a c\right )}^{m}{\left (b x + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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