Integrand size = 37, antiderivative size = 65 \[ \int \frac {\sqrt {d+e x}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c} \sqrt {d} \sqrt {c d^2-a e^2}} \]
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Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {640, 65, 214} \[ \int \frac {\sqrt {d+e x}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c} \sqrt {d} \sqrt {c d^2-a e^2}} \]
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Rule 65
Rule 214
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{-\frac {c d^2}{e}+a e+\frac {c d x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{e} \\ & = -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c} \sqrt {d} \sqrt {c d^2-a e^2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{\sqrt {c} \sqrt {d} \sqrt {-c d^2+a e^2}} \]
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Time = 3.40 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(\frac {2 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(48\) |
default | \(\frac {2 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(48\) |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(48\) |
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none
Time = 0.39 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.38 \[ \int \frac {\sqrt {d+e x}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\left [\frac {\log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {c^{2} d^{3} - a c d e^{2}} \sqrt {e x + d}}{c d x + a e}\right )}{\sqrt {c^{2} d^{3} - a c d e^{2}}}, \frac {2 \, \sqrt {-c^{2} d^{3} + a c d e^{2}} \arctan \left (\frac {\sqrt {-c^{2} d^{3} + a c d e^{2}} \sqrt {e x + d}}{c d e x + c d^{2}}\right )}{c^{2} d^{3} - a c d e^{2}}\right ] \]
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Time = 1.37 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {d+e x}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\begin {cases} \frac {2 \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e^{2} - c d^{2}}{c d}}} \right )}}{c d \sqrt {\frac {a e^{2} - c d^{2}}{c d}}} & \text {for}\: e \neq 0 \\\frac {\log {\left (x \right )}}{c d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {\sqrt {d+e x}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {d+e x}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}}} \]
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Time = 10.05 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {d+e x}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2\,\mathrm {atan}\left (\frac {c\,d\,\sqrt {d+e\,x}}{\sqrt {a\,c\,d\,e^2-c^2\,d^3}}\right )}{\sqrt {a\,c\,d\,e^2-c^2\,d^3}} \]
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