Optimal. Leaf size=80 \[ \frac{16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 x^{3/2}}-\frac{8 b \left (b x+c x^2\right )^{3/2}}{35 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}{7 c} \]
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Rubi [A] time = 0.0267062, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {656, 648} \[ \frac{16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 x^{3/2}}-\frac{8 b \left (b x+c x^2\right )^{3/2}}{35 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}{7 c} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int x^{3/2} \sqrt{b x+c x^2} \, dx &=\frac{2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}{7 c}-\frac{(4 b) \int \sqrt{x} \sqrt{b x+c x^2} \, dx}{7 c}\\ &=-\frac{8 b \left (b x+c x^2\right )^{3/2}}{35 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}{7 c}+\frac{\left (8 b^2\right ) \int \frac{\sqrt{b x+c x^2}}{\sqrt{x}} \, dx}{35 c^2}\\ &=\frac{16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 x^{3/2}}-\frac{8 b \left (b x+c x^2\right )^{3/2}}{35 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}{7 c}\\ \end{align*}
Mathematica [A] time = 0.0217325, size = 42, normalized size = 0.52 \[ \frac{2 (x (b+c x))^{3/2} \left (8 b^2-12 b c x+15 c^2 x^2\right )}{105 c^3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 44, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 15\,{c}^{2}{x}^{2}-12\,bcx+8\,{b}^{2} \right ) }{105\,{c}^{3}}\sqrt{c{x}^{2}+bx}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15948, size = 57, normalized size = 0.71 \begin{align*} \frac{2 \,{\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt{c x + b}}{105 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28884, size = 116, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt{c x^{2} + b x}}{105 \, c^{3} \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 x^{\frac{3}{2}} \sqrt{x \left (b + c x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16809, size = 62, normalized size = 0.78 \begin{align*} -\frac{16 \, b^{\frac{7}{2}}}{105 \, c^{3}} + \frac{2 \,{\left (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}\right )}}{105 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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