Optimal. Leaf size=25 \[ \frac{2 \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}} \]
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Rubi [A] time = 0.0069109, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {648} \[ \frac{2 \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 648
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx &=\frac{2 \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0138718, size = 23, normalized size = 0.92 \[ \frac{2 (x (b+c x))^{5/2}}{5 c x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 25, normalized size = 1. \begin{align*}{\frac{2\,cx+2\,b}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.13455, size = 66, normalized size = 2.64 \begin{align*} \frac{2 \,{\left (5 \, b c x^{2} + 5 \, b^{2} x +{\left (3 \, c^{2} x^{2} + b c x - 2 \, b^{2}\right )} x\right )} \sqrt{c x + b}}{15 \, c x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97336, size = 82, normalized size = 3.28 \begin{align*} \frac{2 \,{\left (c^{2} x^{2} + 2 \, b c x + b^{2}\right )} \sqrt{c x^{2} + b x}}{5 \, c \sqrt{x}} \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 )^{\frac{3}{2}}}{x^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.32269, size = 81, normalized size = 3.24 \begin{align*} \frac{2}{15} \, c{\left (\frac{2 \, b^{\frac{5}{2}}}{c^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b}{c^{2}}\right )} + \frac{2}{3} \, b{\left (\frac{{\left (c x + b\right )}^{\frac{3}{2}}}{c} - \frac{b^{\frac{3}{2}}}{c}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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