Integrand size = 20, antiderivative size = 73 \[ \int \frac {a+b x+c x^2}{\sqrt {d+e x}} \, dx=\frac {2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}{e^3}-\frac {2 (2 c d-b e) (d+e x)^{3/2}}{3 e^3}+\frac {2 c (d+e x)^{5/2}}{5 e^3} \]
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Time = 0.02 (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} \[ \int \frac {a+b x+c x^2}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}{e^3}-\frac {2 (d+e x)^{3/2} (2 c d-b e)}{3 e^3}+\frac {2 c (d+e x)^{5/2}}{5 e^3} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c d^2-b d e+a e^2}{e^2 \sqrt {d+e x}}+\frac {(-2 c d+b e) \sqrt {d+e x}}{e^2}+\frac {c (d+e x)^{3/2}}{e^2}\right ) \, dx \\ & = \frac {2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}{e^3}-\frac {2 (2 c d-b e) (d+e x)^{3/2}}{3 e^3}+\frac {2 c (d+e x)^{5/2}}{5 e^3} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.74 \[ \int \frac {a+b x+c x^2}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (5 e (-2 b d+3 a e+b e x)+c \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \]
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Time = 0.17 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.63
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {e x +d}\, \left (\left (\frac {1}{5} c \,x^{2}+\frac {1}{3} b x +a \right ) e^{2}-\frac {2 \left (\frac {2 c x}{5}+b \right ) d e}{3}+\frac {8 c \,d^{2}}{15}\right )}{e^{3}}\) | \(46\) |
gosper | \(\frac {2 \sqrt {e x +d}\, \left (3 c \,x^{2} e^{2}+5 b \,e^{2} x -4 c d e x +15 a \,e^{2}-10 b d e +8 c \,d^{2}\right )}{15 e^{3}}\) | \(53\) |
trager | \(\frac {2 \sqrt {e x +d}\, \left (3 c \,x^{2} e^{2}+5 b \,e^{2} x -4 c d e x +15 a \,e^{2}-10 b d e +8 c \,d^{2}\right )}{15 e^{3}}\) | \(53\) |
risch | \(\frac {2 \sqrt {e x +d}\, \left (3 c \,x^{2} e^{2}+5 b \,e^{2} x -4 c d e x +15 a \,e^{2}-10 b d e +8 c \,d^{2}\right )}{15 e^{3}}\) | \(53\) |
derivativedivides | \(\frac {\frac {2 c \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 b e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {4 c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a \,e^{2} \sqrt {e x +d}-2 b d e \sqrt {e x +d}+2 c \,d^{2} \sqrt {e x +d}}{e^{3}}\) | \(75\) |
default | \(\frac {\frac {2 c \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 b e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {4 c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a \,e^{2} \sqrt {e x +d}-2 b d e \sqrt {e x +d}+2 c \,d^{2} \sqrt {e x +d}}{e^{3}}\) | \(75\) |
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Time = 0.36 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.74 \[ \int \frac {a+b x+c x^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (3 \, c e^{2} x^{2} + 8 \, c d^{2} - 10 \, b d e + 15 \, a e^{2} - {\left (4 \, c d e - 5 \, b e^{2}\right )} x\right )} \sqrt {e x + d}}{15 \, e^{3}} \]
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Time = 0.46 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.40 \[ \int \frac {a+b x+c x^2}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 a \sqrt {d + e x} + \frac {2 b \left (- d \sqrt {d + e x} + \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} + \frac {2 c \left (d^{2} \sqrt {d + e x} - \frac {2 d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}}}{e} & \text {for}\: e \neq 0 \\\frac {a x + \frac {b x^{2}}{2} + \frac {c x^{3}}{3}}{\sqrt {d}} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.05 \[ \int \frac {a+b x+c x^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (15 \, \sqrt {e x + d} a + \frac {5 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} b}{e} + \frac {{\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c}{e^{2}}\right )}}{15 \, e} \]
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Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.05 \[ \int \frac {a+b x+c x^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (15 \, \sqrt {e x + d} a + \frac {5 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} b}{e} + \frac {{\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c}{e^{2}}\right )}}{15 \, e} \]
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Time = 9.79 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \frac {a+b x+c x^2}{\sqrt {d+e x}} \, dx=\frac {2\,\sqrt {d+e\,x}\,\left (3\,c\,{\left (d+e\,x\right )}^2+15\,a\,e^2+15\,c\,d^2+5\,b\,e\,\left (d+e\,x\right )-10\,c\,d\,\left (d+e\,x\right )-15\,b\,d\,e\right )}{15\,e^3} \]
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