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 = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.59, size = 40, normalized size = 0.66 \[ \frac {2 \, {\left (3 \, c g^{2} x^{2} - 4 \, c f g x + 8 \, c f^{2} + 15 \, a g^{2}\right )} \sqrt {g x + f}}{15 \, g^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 0.87 \[ \frac {2 \, {\left (15 \, \sqrt {g x + f} a + \frac {{\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} c}{g^{2}}\right )}}{15 \, g} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 41, normalized size = 0.67 \[ \frac {2 \sqrt {g x +f}\, \left (3 c \,x^{2} g^{2}-4 c f x g +15 a \,g^{2}+8 c \,f^{2}\right )}{15 g^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 53, normalized size = 0.87 \[ \frac {2 \, {\left (15 \, \sqrt {g x + f} a + \frac {{\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} c}{g^{2}}\right )}}{15 \, g} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.56, size = 44, normalized size = 0.72 \[ \frac {2\,\sqrt {f+g\,x}\,\left (3\,c\,{\left (f+g\,x\right )}^2+15\,a\,g^2+15\,c\,f^2-10\,c\,f\,\left (f+g\,x\right )\right )}{15\,g^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.10, size = 150, normalized size = 2.46 \[ \begin {cases} \frac {- \frac {2 a f}{\sqrt {f + g x}} - 2 a \left (- \frac {f}{\sqrt {f + g x}} - \sqrt {f + g x}\right ) - \frac {2 c f \left (\frac {f^{2}}{\sqrt {f + g x}} + 2 f \sqrt {f + g x} - \frac {\left (f + g x\right )^{\frac {3}{2}}}{3}\right )}{g^{2}} - \frac {2 c \left (- \frac {f^{3}}{\sqrt {f + g x}} - 3 f^{2} \sqrt {f + g x} + f \left (f + g x\right )^{\frac {3}{2}} - \frac {\left (f + g x\right )^{\frac {5}{2}}}{5}\right )}{g^{2}}}{g} & \text {for}\: g \neq 0 \\\frac {a x + \frac {c x^{3}}{3}}{\sqrt {f}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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