Integrand size = 39, antiderivative size = 46 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d \sqrt {d+e x}} \]
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Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {662} \[ \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}} \]
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Rule 662
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d \sqrt {d+e x}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)}}{c d \sqrt {d+e x}} \]
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Time = 0.53 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.70
method | result | size |
default | \(\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}}{\sqrt {e x +d}\, c d}\) | \(32\) |
gosper | \(\frac {2 \left (c d x +a e \right ) \sqrt {e x +d}}{c d \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(50\) |
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none
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{c d e x + c d^{2}} \]
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\[ \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\sqrt {d + e x}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, \sqrt {c d x + a e}}{c d} \]
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none
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, e {\left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{c d e} - \frac {\sqrt {-c d^{2} e + a e^{3}}}{c d e}\right )}}{{\left | e \right |}} \]
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Time = 12.36 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2\,\sqrt {d+e\,x}\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{c\,d\,e\,\left (x+\frac {d}{e}\right )} \]
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