Optimal. Leaf size=57 \[ x (a+b x)^m (a c-b c x)^m \left (1-\frac {b^2 x^2}{a^2}\right )^{-m} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};\frac {b^2 x^2}{a^2}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, 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, 252, 251}
\begin {gather*} 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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 42
Rule 251
Rule 252
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*}
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Mathematica [A]
time = 0.04, size = 72, normalized size = 1.26 \begin {gather*} -\frac {2^m (a-b x) (c (a-b x))^m (a+b x)^m \left (\frac {a+b x}{a}\right )^{-m} \, _2F_1\left (-m,1+m;2+m;\frac {a-b x}{2 a}\right )}{b (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{m} \left (-b c x +a c \right )^{m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 2.81, size = 146, normalized size = 2.56 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (a\,c-b\,c\,x\right )}^m\,{\left (a+b\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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