Integrand size = 24, antiderivative size = 88 \[ \int \sqrt {a+b x} \left (a^2-b^2 x^2\right )^p \, dx=-\frac {2^{\frac {1}{2}+p} \sqrt {a+b x} \left (1+\frac {b x}{a}\right )^{-\frac {3}{2}-p} \left (a^2-b^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,1+p,2+p,\frac {a-b x}{2 a}\right )}{a b (1+p)} \]
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Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {694, 692, 71} \[ \int \sqrt {a+b x} \left (a^2-b^2 x^2\right )^p \, dx=-\frac {2^{p+\frac {1}{2}} \sqrt {a+b x} \left (\frac {b x}{a}+1\right )^{-p-\frac {3}{2}} \left (a^2-b^2 x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p-\frac {1}{2},p+1,p+2,\frac {a-b x}{2 a}\right )}{a b (p+1)} \]
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Rule 71
Rule 692
Rule 694
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a+b x} \int \sqrt {1+\frac {b x}{a}} \left (a^2-b^2 x^2\right )^p \, dx}{\sqrt {1+\frac {b x}{a}}} \\ & = \left (\sqrt {a+b x} \left (1+\frac {b x}{a}\right )^{-\frac {3}{2}-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 )^{\frac {1}{2}+p} \left (a^2-a b x\right )^p \, dx \\ & = -\frac {2^{\frac {1}{2}+p} \sqrt {a+b x} \left (1+\frac {b x}{a}\right )^{-\frac {3}{2}-p} \left (a^2-b^2 x^2\right )^{1+p} \, _2F_1\left (-\frac {1}{2}-p,1+p;2+p;\frac {a-b x}{2 a}\right )}{a b (1+p)} \\ \end{align*}
Time = 0.75 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.01 \[ \int \sqrt {a+b x} \left (a^2-b^2 x^2\right )^p \, dx=\frac {2^{\frac {1}{2}+p} (-a+b x) \sqrt {a+b x} \left (1+\frac {b x}{a}\right )^{-\frac {1}{2}-p} \left (a^2-b^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,1+p,2+p,\frac {a-b x}{2 a}\right )}{b (1+p)} \]
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\[\int \sqrt {b x +a}\, \left (-b^{2} x^{2}+a^{2}\right )^{p}d x\]
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\[ \int \sqrt {a+b x} \left (a^2-b^2 x^2\right )^p \, dx=\int { \sqrt {b x + a} {\left (-b^{2} x^{2} + a^{2}\right )}^{p} \,d x } \]
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\[ \int \sqrt {a+b x} \left (a^2-b^2 x^2\right )^p \, dx=\int \left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{p} \sqrt {a + b x}\, dx \]
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\[ \int \sqrt {a+b x} \left (a^2-b^2 x^2\right )^p \, dx=\int { \sqrt {b x + a} {\left (-b^{2} x^{2} + a^{2}\right )}^{p} \,d x } \]
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\[ \int \sqrt {a+b x} \left (a^2-b^2 x^2\right )^p \, dx=\int { \sqrt {b x + a} {\left (-b^{2} x^{2} + a^{2}\right )}^{p} \,d x } \]
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Timed out. \[ \int \sqrt {a+b x} \left (a^2-b^2 x^2\right )^p \, dx=\int {\left (a^2-b^2\,x^2\right )}^p\,\sqrt {a+b\,x} \,d x \]
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