Integrand size = 22, antiderivative size = 63 \[ \int (a+b x)^m \left (a^2-b^2 x^2\right )^p \, dx=\frac {(a+b x)^m \left (a^2-b^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,2+m+2 p,2+m+p,\frac {a+b x}{2 a}\right )}{2 a b (1+m+p)} \]
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Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.35, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {694, 692, 71} \[ \int (a+b x)^m \left (a^2-b^2 x^2\right )^p \, dx=-\frac {2^{m+p} (a+b x)^m \left (a^2-b^2 x^2\right )^{p+1} \left (\frac {b x}{a}+1\right )^{-m-p-1} \operatorname {Hypergeometric2F1}\left (-m-p,p+1,p+2,\frac {a-b x}{2 a}\right )}{a b (p+1)} \]
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Rule 71
Rule 692
Rule 694
Rubi steps \begin{align*} \text {integral}& = \left ((a+b x)^m \left (1+\frac {b x}{a}\right )^{-m}\right ) \int \left (1+\frac {b x}{a}\right )^m \left (a^2-b^2 x^2\right )^p \, dx \\ & = \left ((a+b x)^m \left (1+\frac {b x}{a}\right )^{-1-m-p} \left (a^2-a b x\right )^{-1-p} \left (a^2-b^2 x^2\right )^{1+p}\right ) \int \left (1+\frac {b x}{a}\right )^{m+p} \left (a^2-a b x\right )^p \, dx \\ & = -\frac {2^{m+p} (a+b x)^m \left (1+\frac {b x}{a}\right )^{-1-m-p} \left (a^2-b^2 x^2\right )^{1+p} \, _2F_1\left (-m-p,1+p;2+p;\frac {a-b x}{2 a}\right )}{a b (1+p)} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.35 \[ \int (a+b x)^m \left (a^2-b^2 x^2\right )^p \, dx=\frac {2^{m+p} (-a+b x) (a+b x)^m \left (1+\frac {b x}{a}\right )^{-m-p} \left (a^2-b^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (-m-p,1+p,2+p,\frac {a-b x}{2 a}\right )}{b (1+p)} \]
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\[\int \left (b x +a \right )^{m} \left (-b^{2} x^{2}+a^{2}\right )^{p}d x\]
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\[ \int (a+b x)^m \left (a^2-b^2 x^2\right )^p \, dx=\int { {\left (-b^{2} x^{2} + a^{2}\right )}^{p} {\left (b x + a\right )}^{m} \,d x } \]
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\[ \int (a+b x)^m \left (a^2-b^2 x^2\right )^p \, dx=\int \left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{p} \left (a + b x\right )^{m}\, dx \]
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\[ \int (a+b x)^m \left (a^2-b^2 x^2\right )^p \, dx=\int { {\left (-b^{2} x^{2} + a^{2}\right )}^{p} {\left (b x + a\right )}^{m} \,d x } \]
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\[ \int (a+b x)^m \left (a^2-b^2 x^2\right )^p \, dx=\int { {\left (-b^{2} x^{2} + a^{2}\right )}^{p} {\left (b x + a\right )}^{m} \,d x } \]
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Timed out. \[ \int (a+b x)^m \left (a^2-b^2 x^2\right )^p \, dx=\int {\left (a^2-b^2\,x^2\right )}^p\,{\left (a+b\,x\right )}^m \,d x \]
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