Optimal. Leaf size=80 \[ \frac{16 b^2 \left (b x+c x^2\right )^{5/2}}{315 c^3 x^{5/2}}-\frac{8 b \left (b x+c x^2\right )^{5/2}}{63 c^2 x^{3/2}}+\frac{2 \left (b x+c x^2\right )^{5/2}}{9 c \sqrt{x}} \]
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Rubi [A] time = 0.0269342, 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} \[ \frac{16 b^2 \left (b x+c x^2\right )^{5/2}}{315 c^3 x^{5/2}}-\frac{8 b \left (b x+c x^2\right )^{5/2}}{63 c^2 x^{3/2}}+\frac{2 \left (b x+c x^2\right )^{5/2}}{9 c \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int \sqrt{x} \left (b x+c x^2\right )^{3/2} \, dx &=\frac{2 \left (b x+c x^2\right )^{5/2}}{9 c \sqrt{x}}-\frac{(4 b) \int \frac{\left (b x+c x^2\right )^{3/2}}{\sqrt{x}} \, dx}{9 c}\\ &=-\frac{8 b \left (b x+c x^2\right )^{5/2}}{63 c^2 x^{3/2}}+\frac{2 \left (b x+c x^2\right )^{5/2}}{9 c \sqrt{x}}+\frac{\left (8 b^2\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx}{63 c^2}\\ &=\frac{16 b^2 \left (b x+c x^2\right )^{5/2}}{315 c^3 x^{5/2}}-\frac{8 b \left (b x+c x^2\right )^{5/2}}{63 c^2 x^{3/2}}+\frac{2 \left (b x+c x^2\right )^{5/2}}{9 c \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0255045, size = 42, normalized size = 0.52 \[ \frac{2 (x (b+c x))^{5/2} \left (8 b^2-20 b c x+35 c^2 x^2\right )}{315 c^3 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 44, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 35\,{c}^{2}{x}^{2}-20\,bcx+8\,{b}^{2} \right ) }{315\,{c}^{3}} \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 = 1.15089, size = 138, normalized size = 1.72 \begin{align*} \frac{2 \,{\left ({\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 )} x^{3} + 3 \,{\left (15 \, b c^{3} x^{4} + 3 \, b^{2} c^{2} x^{3} - 4 \, b^{3} c x^{2} + 8 \, b^{4} x\right )} x^{2}\right )} \sqrt{c x + b}}{315 \, c^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90306, size = 139, normalized size = 1.74 \begin{align*} \frac{2 \,{\left (35 \, c^{4} x^{4} + 50 \, b c^{3} x^{3} + 3 \, b^{2} c^{2} x^{2} - 4 \, b^{3} c x + 8 \, b^{4}\right )} \sqrt{c x^{2} + b x}}{315 \, 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 \sqrt{x} \left (x \left (b + c x\right )\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29596, size = 149, normalized size = 1.86 \begin{align*} \frac{2}{315} \, c{\left (\frac{16 \, b^{\frac{9}{2}}}{c^{4}} + \frac{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}}{c^{4}}\right )} - \frac{2}{105} \, b{\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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