Optimal. Leaf size=61 \[ -\frac{2 x \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (p-\frac{1}{2},-p;p+\frac{1}{2};-\frac{c x}{b}\right )}{(1-2 p) (d x)^{3/2}} \]
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Rubi [A] time = 0.0265773, 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} \[ -\frac{2 x \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (p-\frac{1}{2},-p;p+\frac{1}{2};-\frac{c x}{b}\right )}{(1-2 p) (d x)^{3/2}} \]
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}{(d x)^{3/2}} \, dx &=\frac{\left (x^{\frac{3}{2}-p} (b+c x)^{-p} \left (b x+c x^2\right )^p\right ) \int x^{-\frac{3}{2}+p} (b+c x)^p \, dx}{(d x)^{3/2}}\\ &=\frac{\left (x^{\frac{3}{2}-p} \left (1+\frac{c x}{b}\right )^{-p} \left (b x+c x^2\right )^p\right ) \int x^{-\frac{3}{2}+p} \left (1+\frac{c x}{b}\right )^p \, dx}{(d x)^{3/2}}\\ &=-\frac{2 x \left (1+\frac{c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-\frac{1}{2}+p,-p;\frac{1}{2}+p;-\frac{c x}{b}\right )}{(1-2 p) (d x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0121797, size = 58, normalized size = 0.95 \[ \frac{x (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (p-\frac{1}{2},-p;p+\frac{1}{2};-\frac{c x}{b}\right )}{\left (p-\frac{1}{2}\right ) (d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.342, size = 0, normalized size = 0. \begin{align*} \int{ \left ( c{x}^{2}+bx \right ) ^{p} \left ( dx \right ) ^{-{\frac{3}{2}}}}\, 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}}{\left (d x\right )^{\frac{3}{2}}}\,{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^{2} x^{2}}, 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}}{\left (d x\right )^{\frac{3}{2}}}\, 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}}{\left (d x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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