Integrand size = 44, antiderivative size = 125 \[ \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {2 \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 e \sqrt {d+e x}}+\frac {2 g \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d e} \]
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Time = 0.06 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {808, 662} \[ \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 g \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d e}-\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^2 d^2 e \sqrt {d+e x}} \]
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Rule 662
Rule 808
Rubi steps \begin{align*} \text {integral}& = \frac {2 g \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d e}+\frac {1}{3} \left (3 f-\frac {d g}{e}-\frac {2 a e g}{c d}\right ) \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 \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 e \sqrt {d+e x}}+\frac {2 g \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d e} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} (-2 a e g+c d (3 f+g x))}{3 c^2 d^2 \sqrt {d+e x}} \]
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Time = 0.52 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.39
method | result | size |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-c d g x +2 a e g -3 c d f \right )}{3 \sqrt {e x +d}\, c^{2} d^{2}}\) | \(49\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-c d g x +2 a e g -3 c d f \right ) \sqrt {e x +d}}{3 c^{2} d^{2} \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(67\) |
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Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d g x + 3 \, c d f - 2 \, a e g\right )} \sqrt {e x + d}}{3 \, {\left (c^{2} d^{2} e x + c^{2} d^{3}\right )}} \]
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\[ \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\sqrt {d + e x} \left (f + g x\right )}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \]
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Time = 0.24 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, \sqrt {c d x + a e} f}{c d} + \frac {2 \, {\left (c^{2} d^{2} x^{2} - a c d e x - 2 \, a^{2} e^{2}\right )} g}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, e {\left (\frac {3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} {\left (c d f - a e g\right )}}{c^{2} d^{2} e} + \frac {{\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} g}{c^{2} d^{2} e^{3}} - \frac {3 \, \sqrt {-c d^{2} e + a e^{3}} c d e f - \sqrt {-c d^{2} e + a e^{3}} c d^{2} g - 2 \, \sqrt {-c d^{2} e + a e^{3}} a e^{2} g}{c^{2} d^{2} e^{2}}\right )}}{3 \, {\left | e \right |}} \]
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Time = 12.35 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {\left (\frac {\left (4\,a\,e\,g-6\,c\,d\,f\right )\,\sqrt {d+e\,x}}{3\,c^2\,d^2\,e}-\frac {2\,g\,x\,\sqrt {d+e\,x}}{3\,c\,d\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x+\frac {d}{e}} \]
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