Integrand size = 37, antiderivative size = 131 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e}-\frac {\left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2 \sqrt {c} \sqrt {d} e^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 131, 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 = {678, 635, 212} \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}-\frac {\left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {c} \sqrt {d} e^{3/2}} \]
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Rule 212
Rule 635
Rule 678
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e}-\frac {\left (2 c d^2 e-e \left (c d^2+a e^2\right )\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 e^2} \\ & = \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e}-\frac {\left (2 c d^2 e-e \left (c d^2+a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{e^2} \\ & = \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e}-\frac {\left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2 \sqrt {c} \sqrt {d} e^{3/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {e}+\frac {\left (-c d^2+a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {c} \sqrt {d} \sqrt {a e+c d x} \sqrt {d+e x}}\right )}{e^{3/2}} \]
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Time = 0.54 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (e^{2} a -c \,d^{2}\right ) \ln \left (\frac {\frac {e^{2} a}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{2 \sqrt {c d e}}}{e}\) | \(131\) |
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Time = 0.31 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.57 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\left [\frac {4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} c d e - {\left (c d^{2} - a e^{2}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}{4 \, c d e^{2}}, \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} c d e + {\left (c d^{2} - a e^{2}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right )}{2 \, c d e^{2}}\right ] \]
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\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{d + e x}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.34 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {{\left (c d^{2} - a e^{2}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{2 \, \sqrt {c d e} e} + \frac {\sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}}{e} \]
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Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\int \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x} \,d x \]
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