Optimal. Leaf size=108 \[ -\frac{32 b^3 \left (b x+c x^2\right )^{3/2}}{315 c^4 x^{3/2}}+\frac{16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt{x}}-\frac{4 b \sqrt{x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac{2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c} \]
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Rubi [A] time = 0.0402275, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {656, 648} \[ -\frac{32 b^3 \left (b x+c x^2\right )^{3/2}}{315 c^4 x^{3/2}}+\frac{16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt{x}}-\frac{4 b \sqrt{x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac{2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int x^{5/2} \sqrt{b x+c x^2} \, dx &=\frac{2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}-\frac{(2 b) \int x^{3/2} \sqrt{b x+c x^2} \, dx}{3 c}\\ &=-\frac{4 b \sqrt{x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac{2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}+\frac{\left (8 b^2\right ) \int \sqrt{x} \sqrt{b x+c x^2} \, dx}{21 c^2}\\ &=\frac{16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt{x}}-\frac{4 b \sqrt{x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac{2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}-\frac{\left (16 b^3\right ) \int \frac{\sqrt{b x+c x^2}}{\sqrt{x}} \, dx}{105 c^3}\\ &=-\frac{32 b^3 \left (b x+c x^2\right )^{3/2}}{315 c^4 x^{3/2}}+\frac{16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt{x}}-\frac{4 b \sqrt{x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac{2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}\\ \end{align*}
Mathematica [A] time = 0.028814, size = 53, normalized size = 0.49 \[ \frac{2 (x (b+c x))^{3/2} \left (24 b^2 c x-16 b^3-30 b c^2 x^2+35 c^3 x^3\right )}{315 c^4 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 55, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -35\,{x}^{3}{c}^{3}+30\,b{x}^{2}{c}^{2}-24\,{b}^{2}xc+16\,{b}^{3} \right ) }{315\,{c}^{4}}\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 = 1.14971, size = 72, normalized size = 0.67 \begin{align*} \frac{2 \,{\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} \sqrt{c x + b}}{315 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20523, size = 139, normalized size = 1.29 \begin{align*} \frac{2 \,{\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} \sqrt{c x^{2} + b x}}{315 \, c^{4} \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{5}{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 = 1.20601, size = 78, normalized size = 0.72 \begin{align*} \frac{32 \, b^{\frac{9}{2}}}{315 \, c^{4}} + \frac{2 \,{\left (35 \,{\left (c x + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}\right )}}{315 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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