Optimal. Leaf size=83 \[ \frac{2 (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^p \left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (\frac{3}{2},-2 p;\frac{5}{2};\frac{b (d+e x)}{b d-a e}\right )}{3 e} \]
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Rubi [A] time = 0.038224, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {646, 70, 69} \[ \frac{2 (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^p \left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (\frac{3}{2},-2 p;\frac{5}{2};\frac{b (d+e x)}{b d-a e}\right )}{3 e} \]
Antiderivative was successfully verified.
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Rule 646
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx &=\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*}
Mathematica [A] time = 0.0224397, size = 73, normalized size = 0.88 \[ \frac{2 (d+e x)^{3/2} \left ((a+b x)^2\right )^p \left (\frac{e (a+b x)}{a e-b d}\right )^{-2 p} \, _2F_1\left (\frac{3}{2},-2 p;\frac{5}{2};\frac{b (d+e x)}{b d-a e}\right )}{3 e} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.162, size = 0, normalized size = 0. \begin{align*} \int \sqrt{ex+d} \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e x + d}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{e x + d}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e x + d}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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