Optimal. Leaf size=61 \[ \frac{2 x \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+\frac{1}{2};p+\frac{3}{2};-\frac{c x}{b}\right )}{(2 p+1) \sqrt{d x}} \]
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Rubi [A] time = 0.0234973, 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} \[ \frac{2 x \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+\frac{1}{2};p+\frac{3}{2};-\frac{c x}{b}\right )}{(2 p+1) \sqrt{d x}} \]
Antiderivative was successfully verified.
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Rule 674
Rule 66
Rule 64
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^p}{\sqrt{d x}} \, dx &=\frac{\left (x^{\frac{1}{2}-p} (b+c x)^{-p} \left (b x+c x^2\right )^p\right ) \int x^{-\frac{1}{2}+p} (b+c x)^p \, dx}{\sqrt{d x}}\\ &=\frac{\left (x^{\frac{1}{2}-p} \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}{\sqrt{d x}}\\ &=\frac{2 x \left (1+\frac{c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,\frac{1}{2}+p;\frac{3}{2}+p;-\frac{c x}{b}\right )}{(1+2 p) \sqrt{d x}}\\ \end{align*}
Mathematica [A] time = 0.0120283, size = 58, normalized size = 0.95 \[ \frac{x (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (-p,p+\frac{1}{2};p+\frac{3}{2};-\frac{c x}{b}\right )}{\left (p+\frac{1}{2}\right ) \sqrt{d x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.361, size = 0, normalized size = 0. \begin{align*} \int{ \left ( c{x}^{2}+bx \right ) ^{p}{\frac{1}{\sqrt{dx}}}}\, 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 \frac{{\left (c x^{2} + b x\right )}^{p}}{\sqrt{d x}}\,{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 (\frac{\sqrt{d x}{\left (c x^{2} + b x\right )}^{p}}{d x}, 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 \frac{\left (x \left (b + c x\right )\right )^{p}}{\sqrt{d x}}\, 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 \frac{{\left (c x^{2} + b x\right )}^{p}}{\sqrt{d x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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