Integrand size = 28, antiderivative size = 83 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {2 \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-2 p,\frac {5}{2},\frac {b (d+e x)}{b d-a e}\right )}{3 e} \]
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Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {660, 72, 71} \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {2 (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^p \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-2 p,\frac {5}{2},\frac {b (d+e x)}{b d-a e}\right )}{3 e} \]
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Rule 71
Rule 72
Rule 660
Rubi steps \begin{align*} \text {integral}& = \left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \left (a b+b^2 x\right )^{2 p} \sqrt {d+e x} \, dx \\ & = \left (\left (\frac {e \left (a b+b^2 x\right )}{-b^2 d+a b e}\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \sqrt {d+e x} \left (-\frac {a e}{b d-a e}-\frac {b e x}{b d-a e}\right )^{2 p} \, dx \\ & = \frac {2 \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (\frac {3}{2},-2 p;\frac {5}{2};\frac {b (d+e x)}{b d-a e}\right )}{3 e} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.88 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {2 \left (\frac {e (a+b x)}{-b d+a e}\right )^{-2 p} \left ((a+b x)^2\right )^p (d+e x)^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-2 p,\frac {5}{2},\frac {b (d+e x)}{b d-a e}\right )}{3 e} \]
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\[\int \sqrt {e x +d}\, \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}d x\]
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\[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\int { \sqrt {e x + d} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} \,d x } \]
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\[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\int \sqrt {d + e x} \left (\left (a + b x\right )^{2}\right )^{p}\, dx \]
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\[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\int { \sqrt {e x + d} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} \,d x } \]
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\[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\int { \sqrt {e x + d} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} \,d x } \]
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Timed out. \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\int \sqrt {d+e\,x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p \,d x \]
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