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 = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 68
Rule 627
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*}
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Mathematica [A] time = 0.04, 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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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b x + a\right )}^{m}}{b^{2} x^{2} - a^{2}}, x\right ) \]
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 \[ \int -\frac {{\left (b x + a\right )}^{m}}{b^{2} x^{2} - a^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.79, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{m}}{-b^{2} x^{2}+a^{2}}\, dx \]
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 \[ -\int \frac {{\left (b x + a\right )}^{m}}{b^{2} x^{2} - a^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (a+b\,x\right )}^m}{a^2-b^2\,x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\left (a + b x\right )^{m}}{- a^{2} + b^{2} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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