Optimal. Leaf size=61 \[ \frac{2 \sqrt{f+g x} \left (a g^2+c f^2\right )}{g^3}+\frac{2 c (f+g x)^{5/2}}{5 g^3}-\frac{4 c f (f+g x)^{3/2}}{3 g^3} \]
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Rubi [A] time = 0.0262261, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {697} \[ \frac{2 \sqrt{f+g x} \left (a g^2+c f^2\right )}{g^3}+\frac{2 c (f+g x)^{5/2}}{5 g^3}-\frac{4 c f (f+g x)^{3/2}}{3 g^3} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \frac{a+c x^2}{\sqrt{f+g x}} \, dx &=\int \left (\frac{c f^2+a g^2}{g^2 \sqrt{f+g x}}-\frac{2 c f \sqrt{f+g x}}{g^2}+\frac{c (f+g x)^{3/2}}{g^2}\right ) \, dx\\ &=\frac{2 \left (c f^2+a g^2\right ) \sqrt{f+g x}}{g^3}-\frac{4 c f (f+g x)^{3/2}}{3 g^3}+\frac{2 c (f+g x)^{5/2}}{5 g^3}\\ \end{align*}
Mathematica [A] time = 0.0269882, size = 44, normalized size = 0.72 \[ \frac{2 \sqrt{f+g x} \left (15 a g^2+c \left (8 f^2-4 f g x+3 g^2 x^2\right )\right )}{15 g^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 41, normalized size = 0.7 \begin{align*}{\frac{6\,c{x}^{2}{g}^{2}-8\,cfxg+30\,a{g}^{2}+16\,c{f}^{2}}{15\,{g}^{3}}\sqrt{gx+f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985573, size = 72, normalized size = 1.18 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{g x + f} a + \frac{{\left (3 \,{\left (g x + f\right )}^{\frac{5}{2}} - 10 \,{\left (g x + f\right )}^{\frac{3}{2}} f + 15 \, \sqrt{g x + f} f^{2}\right )} c}{g^{2}}\right )}}{15 \, g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70529, size = 96, normalized size = 1.57 \begin{align*} \frac{2 \,{\left (3 \, c g^{2} x^{2} - 4 \, c f g x + 8 \, c f^{2} + 15 \, a g^{2}\right )} \sqrt{g x + f}}{15 \, g^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.90782, size = 150, normalized size = 2.46 \begin{align*} \begin{cases} - \frac{\frac{2 a f}{\sqrt{f + g x}} + 2 a \left (- \frac{f}{\sqrt{f + g x}} - \sqrt{f + g x}\right ) + \frac{2 c f \left (\frac{f^{2}}{\sqrt{f + g x}} + 2 f \sqrt{f + g x} - \frac{\left (f + g x\right )^{\frac{3}{2}}}{3}\right )}{g^{2}} + \frac{2 c \left (- \frac{f^{3}}{\sqrt{f + g x}} - 3 f^{2} \sqrt{f + g x} + f \left (f + g x\right )^{\frac{3}{2}} - \frac{\left (f + g x\right )^{\frac{5}{2}}}{5}\right )}{g^{2}}}{g} & \text{for}\: g \neq 0 \\\frac{a x + \frac{c x^{3}}{3}}{\sqrt{f}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15214, size = 72, normalized size = 1.18 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{g x + f} a + \frac{{\left (3 \,{\left (g x + f\right )}^{\frac{5}{2}} - 10 \,{\left (g x + f\right )}^{\frac{3}{2}} f + 15 \, \sqrt{g x + f} f^{2}\right )} c}{g^{2}}\right )}}{15 \, g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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