Optimal. Leaf size=61 \[ \frac{2 x \sqrt{d x} \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+\frac{3}{2};p+\frac{5}{2};-\frac{c x}{b}\right )}{2 p+3} \]
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Rubi [A] time = 0.0236347, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {674, 66, 64} \[ \frac{2 x \sqrt{d x} \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+\frac{3}{2};p+\frac{5}{2};-\frac{c x}{b}\right )}{2 p+3} \]
Antiderivative was successfully verified.
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Rule 674
Rule 66
Rule 64
Rubi steps
\begin{align*} \int \sqrt{d x} \left (b x+c x^2\right )^p \, dx &=\left (x^{-\frac{1}{2}-p} \sqrt{d x} (b+c x)^{-p} \left (b x+c x^2\right )^p\right ) \int x^{\frac{1}{2}+p} (b+c x)^p \, dx\\ &=\left (x^{-\frac{1}{2}-p} \sqrt{d x} \left (1+\frac{c x}{b}\right )^{-p} \left (b x+c x^2\right )^p\right ) \int x^{\frac{1}{2}+p} \left (1+\frac{c x}{b}\right )^p \, dx\\ &=\frac{2 x \sqrt{d x} \left (1+\frac{c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,\frac{3}{2}+p;\frac{5}{2}+p;-\frac{c x}{b}\right )}{3+2 p}\\ \end{align*}
Mathematica [A] time = 0.0104214, size = 58, normalized size = 0.95 \[ \frac{x \sqrt{d x} (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (-p,p+\frac{3}{2};p+\frac{5}{2};-\frac{c x}{b}\right )}{p+\frac{3}{2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.355, size = 0, normalized size = 0. \begin{align*} \int \sqrt{dx} \left ( c{x}^{2}+bx \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{d x}{\left (c x^{2} + b x\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{d x}{\left (c x^{2} + b x\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d x} \left (x \left (b + c x\right )\right )^{p}\, dx \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{d x}{\left (c x^{2} + b x\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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