Integrand size = 37, antiderivative size = 83 \[ \int \frac {(d+e x)^{3/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \sqrt {d+e x}}{c d}-\frac {2 \sqrt {c d^2-a e^2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{3/2} d^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {640, 52, 65, 214} \[ \int \frac {(d+e x)^{3/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \sqrt {d+e x}}{c d}-\frac {2 \sqrt {c d^2-a e^2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{3/2} d^{3/2}} \]
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Rule 52
Rule 65
Rule 214
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {d+e x}}{a e+c d x} \, dx \\ & = \frac {2 \sqrt {d+e x}}{c d}+\frac {\left (c d^2-a e^2\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{c d} \\ & = \frac {2 \sqrt {d+e x}}{c d}+\left (2 \left (\frac {d}{e}-\frac {a e}{c d}\right )\right ) \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 ) \\ & = \frac {2 \sqrt {d+e x}}{c d}-\frac {2 \sqrt {c d^2-a e^2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{3/2} d^{3/2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^{3/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \sqrt {d+e x}}{c d}-\frac {2 \sqrt {-c d^2+a e^2} \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{c^{3/2} d^{3/2}} \]
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Time = 3.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.92
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {e x +d}-\frac {2 \left (e^{2} a -c \,d^{2}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c d}\) | \(76\) |
risch | \(\frac {2 \textit {\_O1} \sqrt {e x +d}}{d}-\frac {2 \left (e^{2} a -c \,d^{2}\right ) \arctan \left (\frac {\sqrt {e x +d}\, d}{\sqrt {\left (\textit {\_O1} a \,e^{2}-d^{2}\right ) d}}\right )}{c^{2} d \sqrt {\left (\textit {\_O1} a \,e^{2}-d^{2}\right ) d}}\) | \(77\) |
derivativedivides | \(\frac {2 \sqrt {e x +d}}{c d}+\frac {2 \left (-e^{2} a +c \,d^{2}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c d \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(82\) |
default | \(\frac {2 \sqrt {e x +d}}{c d}+\frac {2 \left (-e^{2} a +c \,d^{2}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c d \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(82\) |
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Time = 0.38 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.30 \[ \int \frac {(d+e x)^{3/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\left [\frac {\sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {e x + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \, \sqrt {e x + d}}{c d}, -\frac {2 \, {\left (\sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {e x + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - \sqrt {e x + d}\right )}}{c d}\right ] \]
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Time = 1.76 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.08 \[ \int \frac {(d+e x)^{3/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\begin {cases} \frac {2 \left (\frac {e \sqrt {d + e x}}{c d} - \frac {e \left (a e^{2} - c d^{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e^{2} - c d^{2}}{c d}}} \right )}}{c^{2} d^{2} \sqrt {\frac {a e^{2} - c d^{2}}{c d}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\log {\left (x \right )}}{c \sqrt {d}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {(d+e x)^{3/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^{3/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (c d^{2} - a e^{2}\right )} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c d} + \frac {2 \, \sqrt {e x + d}}{c d} \]
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Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.81 \[ \int \frac {(d+e x)^{3/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2\,\sqrt {d+e\,x}}{c\,d}-\frac {2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,\sqrt {d+e\,x}}{\sqrt {a\,e^2-c\,d^2}}\right )\,\sqrt {a\,e^2-c\,d^2}}{c^{3/2}\,d^{3/2}} \]
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