Integrand size = 24, antiderivative size = 88 \[ \int \frac {\left (a^2-b^2 x^2\right )^p}{(a+b x)^{3/2}} \, dx=-\frac {2^{-\frac {3}{2}+p} \left (1+\frac {b x}{a}\right )^{-\frac {1}{2}-p} \left (a^2-b^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2}-p,1+p,2+p,\frac {a-b x}{2 a}\right )}{a^2 b (1+p) \sqrt {a+b x}} \]
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Time = 0.05 (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} \[ \int \frac {\left (a^2-b^2 x^2\right )^p}{(a+b x)^{3/2}} \, dx=-\frac {2^{p-\frac {3}{2}} \left (\frac {b x}{a}+1\right )^{-p-\frac {1}{2}} \left (a^2-b^2 x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (\frac {3}{2}-p,p+1,p+2,\frac {a-b x}{2 a}\right )}{a^2 b (p+1) \sqrt {a+b x}} \]
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Rule 71
Rule 692
Rule 694
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+\frac {b x}{a}} \int \frac {\left (a^2-b^2 x^2\right )^p}{\left (1+\frac {b x}{a}\right )^{3/2}} \, dx}{a \sqrt {a+b x}} \\ & = \frac {\left (\left (1+\frac {b x}{a}\right )^{-\frac {1}{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 {3}{2}+p} \left (a^2-a b x\right )^p \, dx}{a \sqrt {a+b x}} \\ & = -\frac {2^{-\frac {3}{2}+p} \left (1+\frac {b x}{a}\right )^{-\frac {1}{2}-p} \left (a^2-b^2 x^2\right )^{1+p} \, _2F_1\left (\frac {3}{2}-p,1+p;2+p;\frac {a-b x}{2 a}\right )}{a^2 b (1+p) \sqrt {a+b x}} \\ \end{align*}
Time = 1.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a^2-b^2 x^2\right )^p}{(a+b x)^{3/2}} \, dx=-\frac {2^{-\frac {3}{2}+p} (a-b x) \left (1+\frac {b x}{a}\right )^{\frac {1}{2}-p} \left (a^2-b^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {3}{2}-p,1+p,2+p,\frac {a-b x}{2 a}\right )}{a b (1+p) \sqrt {a+b x}} \]
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\[\int \frac {\left (-b^{2} x^{2}+a^{2}\right )^{p}}{\left (b x +a \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {\left (a^2-b^2 x^2\right )^p}{(a+b x)^{3/2}} \, dx=\int { \frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{p}}{{\left (b x + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\left (a^2-b^2 x^2\right )^p}{(a+b x)^{3/2}} \, dx=\int \frac {\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{p}}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\left (a^2-b^2 x^2\right )^p}{(a+b x)^{3/2}} \, dx=\int { \frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{p}}{{\left (b x + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\left (a^2-b^2 x^2\right )^p}{(a+b x)^{3/2}} \, dx=\int { \frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{p}}{{\left (b x + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a^2-b^2 x^2\right )^p}{(a+b x)^{3/2}} \, dx=\int \frac {{\left (a^2-b^2\,x^2\right )}^p}{{\left (a+b\,x\right )}^{3/2}} \,d x \]
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