Optimal. Leaf size=52 \[ \frac{2 \left (b x+c x^2\right )^{3/2}}{5 c \sqrt{x}}-\frac{4 b \left (b x+c x^2\right )^{3/2}}{15 c^2 x^{3/2}} \]
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Rubi [A] time = 0.0152971, 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} \[ \frac{2 \left (b x+c x^2\right )^{3/2}}{5 c \sqrt{x}}-\frac{4 b \left (b x+c x^2\right )^{3/2}}{15 c^2 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int \sqrt{x} \sqrt{b x+c x^2} \, dx &=\frac{2 \left (b x+c x^2\right )^{3/2}}{5 c \sqrt{x}}-\frac{(2 b) \int \frac{\sqrt{b x+c x^2}}{\sqrt{x}} \, dx}{5 c}\\ &=-\frac{4 b \left (b x+c x^2\right )^{3/2}}{15 c^2 x^{3/2}}+\frac{2 \left (b x+c x^2\right )^{3/2}}{5 c \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.017424, size = 31, normalized size = 0.6 \[ \frac{2 (x (b+c x))^{3/2} (3 c x-2 b)}{15 c^2 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 33, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -3\,cx+2\,b \right ) }{15\,{c}^{2}}\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 = 1.02837, size = 41, normalized size = 0.79 \begin{align*} \frac{2 \,{\left (3 \, c^{2} x^{2} + b c x - 2 \, b^{2}\right )} \sqrt{c x + b}}{15 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18565, size = 89, normalized size = 1.71 \begin{align*} \frac{2 \,{\left (3 \, c^{2} x^{2} + b c x - 2 \, b^{2}\right )} \sqrt{c x^{2} + b x}}{15 \, 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 \sqrt{x} \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 = 1.21536, size = 46, normalized size = 0.88 \begin{align*} \frac{4 \, b^{\frac{5}{2}}}{15 \, c^{2}} + \frac{2 \,{\left (3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b\right )}}{15 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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