Optimal. Leaf size=136 \[ \frac{256 b^4 \left (b x+c x^2\right )^{3/2}}{3465 c^5 x^{3/2}}-\frac{128 b^3 \left (b x+c x^2\right )^{3/2}}{1155 c^4 \sqrt{x}}+\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac{16 b x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c} \]
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Rubi [A] time = 0.0551495, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {656, 648} \[ \frac{256 b^4 \left (b x+c x^2\right )^{3/2}}{3465 c^5 x^{3/2}}-\frac{128 b^3 \left (b x+c x^2\right )^{3/2}}{1155 c^4 \sqrt{x}}+\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac{16 b x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int x^{7/2} \sqrt{b x+c x^2} \, dx &=\frac{2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}-\frac{(8 b) \int x^{5/2} \sqrt{b x+c x^2} \, dx}{11 c}\\ &=-\frac{16 b x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}+\frac{\left (16 b^2\right ) \int x^{3/2} \sqrt{b x+c x^2} \, dx}{33 c^2}\\ &=\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac{16 b x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}-\frac{\left (64 b^3\right ) \int \sqrt{x} \sqrt{b x+c x^2} \, dx}{231 c^3}\\ &=-\frac{128 b^3 \left (b x+c x^2\right )^{3/2}}{1155 c^4 \sqrt{x}}+\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac{16 b x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}+\frac{\left (128 b^4\right ) \int \frac{\sqrt{b x+c x^2}}{\sqrt{x}} \, dx}{1155 c^4}\\ &=\frac{256 b^4 \left (b x+c x^2\right )^{3/2}}{3465 c^5 x^{3/2}}-\frac{128 b^3 \left (b x+c x^2\right )^{3/2}}{1155 c^4 \sqrt{x}}+\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac{16 b x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}\\ \end{align*}
Mathematica [A] time = 0.0346805, size = 64, normalized size = 0.47 \[ \frac{2 (x (b+c x))^{3/2} \left (240 b^2 c^2 x^2-192 b^3 c x+128 b^4-280 b c^3 x^3+315 c^4 x^4\right )}{3465 c^5 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 66, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 315\,{x}^{4}{c}^{4}-280\,b{x}^{3}{c}^{3}+240\,{b}^{2}{x}^{2}{c}^{2}-192\,{b}^{3}xc+128\,{b}^{4} \right ) }{3465\,{c}^{5}}\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 = 1.16032, size = 86, normalized size = 0.63 \begin{align*} \frac{2 \,{\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} \sqrt{c x + b}}{3465 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88514, size = 170, normalized size = 1.25 \begin{align*} \frac{2 \,{\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} \sqrt{c x^{2} + b x}}{3465 \, c^{5} \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16155, size = 95, normalized size = 0.7 \begin{align*} -\frac{256 \, b^{\frac{11}{2}}}{3465 \, c^{5}} + \frac{2 \,{\left (315 \,{\left (c x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (c x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}\right )}}{3465 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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