Optimal. Leaf size=38 \[ \frac{(a+b x)^m \, _2F_1\left (1,m;m+1;\frac{a+b x}{2 a}\right )}{2 a b m} \]
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Rubi [A] time = 0.0133872, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {627, 68} \[ \frac{(a+b x)^m \, _2F_1\left (1,m;m+1;\frac{a+b x}{2 a}\right )}{2 a b m} \]
Antiderivative was successfully verified.
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Rule 627
Rule 68
Rubi steps
\begin{align*} \int \frac{(a+b x)^m}{a^2-b^2 x^2} \, dx &=\int \frac{(a+b x)^{-1+m}}{a-b x} \, dx\\ &=\frac{(a+b x)^m \, _2F_1\left (1,m;1+m;\frac{a+b x}{2 a}\right )}{2 a b m}\\ \end{align*}
Mathematica [A] time = 0.0421651, size = 59, normalized size = 1.55 \[ \frac{(a+b x)^m \left (m (a+b x) \, _2F_1\left (1,m+1;m+2;\frac{a+b x}{2 a}\right )+2 a (m+1)\right )}{4 a^2 b m (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.522, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m}}{-{b}^{2}{x}^{2}+{a}^{2}}}\, 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 \frac{{\left (b x + a\right )}^{m}}{b^{2} x^{2} - a^{2}}\,{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 (-\frac{{\left (b x + a\right )}^{m}}{b^{2} x^{2} - a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\left (a + b x\right )^{m}}{- a^{2} + b^{2} x^{2}}\, dx \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 -\frac{{\left (b x + a\right )}^{m}}{b^{2} x^{2} - a^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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