Optimal. Leaf size=52 \[ \frac{2 \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}-\frac{4 b \left (b x+c x^2\right )^{5/2}}{35 c^2 x^{5/2}} \]
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Rubi [A] time = 0.0163527, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {656, 648} \[ \frac{2 \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}-\frac{4 b \left (b x+c x^2\right )^{5/2}}{35 c^2 x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^{3/2}}{\sqrt{x}} \, dx &=\frac{2 \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}-\frac{(2 b) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx}{7 c}\\ &=-\frac{4 b \left (b x+c x^2\right )^{5/2}}{35 c^2 x^{5/2}}+\frac{2 \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0198447, size = 31, normalized size = 0.6 \[ \frac{2 (x (b+c x))^{5/2} (5 c x-2 b)}{35 c^2 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 33, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -5\,cx+2\,b \right ) }{35\,{c}^{2}} \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 [A] time = 1.12821, size = 104, normalized size = 2. \begin{align*} \frac{2 \,{\left ({\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} x^{2} + 7 \,{\left (3 \, b c^{2} x^{3} + b^{2} c x^{2} - 2 \, b^{3} x\right )} x\right )} \sqrt{c x + b}}{105 \, c^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83903, size = 111, normalized size = 2.13 \begin{align*} \frac{2 \,{\left (5 \, c^{3} x^{3} + 8 \, b c^{2} x^{2} + b^{2} c x - 2 \, b^{3}\right )} \sqrt{c x^{2} + b x}}{35 \, c^{2} \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}}}{\sqrt{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27858, size = 116, normalized size = 2.23 \begin{align*} -\frac{2}{105} \, c{\left (\frac{8 \, b^{\frac{7}{2}}}{c^{3}} - \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}}{c^{3}}\right )} + \frac{2}{15} \, b{\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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