Integrand size = 20, antiderivative size = 73 \[ \int \frac {a+b x+c x^2}{\sqrt {f+g x}} \, dx=\frac {2 \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}{g^3}-\frac {2 (2 c f-b g) (f+g x)^{3/2}}{3 g^3}+\frac {2 c (f+g x)^{5/2}}{5 g^3} \]
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Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \[ \int \frac {a+b x+c x^2}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \left (a g^2-b f g+c f^2\right )}{g^3}-\frac {2 (f+g x)^{3/2} (2 c f-b g)}{3 g^3}+\frac {2 c (f+g x)^{5/2}}{5 g^3} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c f^2-b f g+a g^2}{g^2 \sqrt {f+g x}}+\frac {(-2 c f+b g) \sqrt {f+g x}}{g^2}+\frac {c (f+g x)^{3/2}}{g^2}\right ) \, dx \\ & = \frac {2 \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}{g^3}-\frac {2 (2 c f-b g) (f+g x)^{3/2}}{3 g^3}+\frac {2 c (f+g x)^{5/2}}{5 g^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.74 \[ \int \frac {a+b x+c x^2}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \left (5 g (-2 b f+3 a g+b g x)+c \left (8 f^2-4 f g x+3 g^2 x^2\right )\right )}{15 g^3} \]
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Time = 0.48 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.63
method | result | size |
pseudoelliptic | \(\frac {2 \left (\left (\frac {1}{5} c \,x^{2}+\frac {1}{3} b x +a \right ) g^{2}-\frac {2 f \left (\frac {2 c x}{5}+b \right ) g}{3}+\frac {8 c \,f^{2}}{15}\right ) \sqrt {g x +f}}{g^{3}}\) | \(46\) |
gosper | \(\frac {2 \sqrt {g x +f}\, \left (3 c \,x^{2} g^{2}+5 b \,g^{2} x -4 c f x g +15 a \,g^{2}-10 b f g +8 c \,f^{2}\right )}{15 g^{3}}\) | \(53\) |
trager | \(\frac {2 \sqrt {g x +f}\, \left (3 c \,x^{2} g^{2}+5 b \,g^{2} x -4 c f x g +15 a \,g^{2}-10 b f g +8 c \,f^{2}\right )}{15 g^{3}}\) | \(53\) |
risch | \(\frac {2 \sqrt {g x +f}\, \left (3 c \,x^{2} g^{2}+5 b \,g^{2} x -4 c f x g +15 a \,g^{2}-10 b f g +8 c \,f^{2}\right )}{15 g^{3}}\) | \(53\) |
derivativedivides | \(\frac {\frac {2 c \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 b g \left (g x +f \right )^{\frac {3}{2}}}{3}-\frac {4 c f \left (g x +f \right )^{\frac {3}{2}}}{3}+2 a \,g^{2} \sqrt {g x +f}-2 b f g \sqrt {g x +f}+2 c \,f^{2} \sqrt {g x +f}}{g^{3}}\) | \(75\) |
default | \(\frac {\frac {2 c \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 b g \left (g x +f \right )^{\frac {3}{2}}}{3}-\frac {4 c f \left (g x +f \right )^{\frac {3}{2}}}{3}+2 a \,g^{2} \sqrt {g x +f}-2 b f g \sqrt {g x +f}+2 c \,f^{2} \sqrt {g x +f}}{g^{3}}\) | \(75\) |
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Time = 0.47 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.74 \[ \int \frac {a+b x+c x^2}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (3 \, c g^{2} x^{2} + 8 \, c f^{2} - 10 \, b f g + 15 \, a g^{2} - {\left (4 \, c f g - 5 \, b g^{2}\right )} x\right )} \sqrt {g x + f}}{15 \, g^{3}} \]
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Time = 0.45 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.40 \[ \int \frac {a+b x+c x^2}{\sqrt {f+g x}} \, dx=\begin {cases} \frac {2 a \sqrt {f + g x} + \frac {2 b \left (- f \sqrt {f + g x} + \frac {\left (f + g x\right )^{\frac {3}{2}}}{3}\right )}{g} + \frac {2 c \left (f^{2} \sqrt {f + g x} - \frac {2 f \left (f + g x\right )^{\frac {3}{2}}}{3} + \frac {\left (f + g x\right )^{\frac {5}{2}}}{5}\right )}{g^{2}}}{g} & \text {for}\: g \neq 0 \\\frac {a x + \frac {b x^{2}}{2} + \frac {c x^{3}}{3}}{\sqrt {f}} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.05 \[ \int \frac {a+b x+c x^2}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (15 \, \sqrt {g x + f} a + \frac {5 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {g x + f} f\right )} b}{g} + \frac {{\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} c}{g^{2}}\right )}}{15 \, g} \]
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Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.05 \[ \int \frac {a+b x+c x^2}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (15 \, \sqrt {g x + f} a + \frac {5 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {g x + f} f\right )} b}{g} + \frac {{\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} c}{g^{2}}\right )}}{15 \, g} \]
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Time = 11.83 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \frac {a+b x+c x^2}{\sqrt {f+g x}} \, dx=\frac {2\,\sqrt {f+g\,x}\,\left (3\,c\,{\left (f+g\,x\right )}^2+15\,a\,g^2+15\,c\,f^2+5\,b\,g\,\left (f+g\,x\right )-10\,c\,f\,\left (f+g\,x\right )-15\,b\,f\,g\right )}{15\,g^3} \]
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