Integrand size = 29, antiderivative size = 150 \[ \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=-\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}+\frac {3 \sqrt {d+e x}}{4 c d^2 e \sqrt {c d^2-c e^2 x^2}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{4 \sqrt {2} c^{3/2} d^{5/2} e} \]
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Time = 0.05 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {687, 681, 675, 214} \[ \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=-\frac {3 \text {arctanh}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{4 \sqrt {2} c^{3/2} d^{5/2} e}+\frac {3 \sqrt {d+e x}}{4 c d^2 e \sqrt {c d^2-c e^2 x^2}}-\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \]
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Rule 214
Rule 675
Rule 681
Rule 687
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}+\frac {3 \int \frac {\sqrt {d+e x}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx}{4 d} \\ & = -\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}+\frac {3 \sqrt {d+e x}}{4 c d^2 e \sqrt {c d^2-c e^2 x^2}}+\frac {3 \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx}{8 c d^2} \\ & = -\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}+\frac {3 \sqrt {d+e x}}{4 c d^2 e \sqrt {c d^2-c e^2 x^2}}+\frac {(3 e) \text {Subst}\left (\int \frac {1}{-2 c d e^2+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {d+e x}}\right )}{4 c d^2} \\ & = -\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}+\frac {3 \sqrt {d+e x}}{4 c d^2 e \sqrt {c d^2-c e^2 x^2}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{4 \sqrt {2} c^{3/2} d^{5/2} e} \\ \end{align*}
Time = 0.75 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {d} (d+3 e x)-3 \sqrt {2} \sqrt {d+e x} \sqrt {d^2-e^2 x^2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {d+e x}}{\sqrt {d^2-e^2 x^2}}\right )}{8 c d^{5/2} e \sqrt {d+e x} \sqrt {c \left (d^2-e^2 x^2\right )}} \]
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Time = 2.22 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {\sqrt {c \left (-x^{2} e^{2}+d^{2}\right )}\, \left (3 \sqrt {c \left (-e x +d \right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) e x +3 \sqrt {c \left (-e x +d \right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) d -6 \sqrt {c d}\, e x -2 \sqrt {c d}\, d \right )}{8 \left (e x +d \right )^{\frac {3}{2}} c^{2} \left (-e x +d \right ) e \,d^{2} \sqrt {c d}}\) | \(141\) |
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Time = 0.52 (sec) , antiderivative size = 382, normalized size of antiderivative = 2.55 \[ \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\left [\frac {3 \, \sqrt {2} {\left (e^{3} x^{3} + d e^{2} x^{2} - d^{2} e x - d^{3}\right )} \sqrt {c d} \log \left (-\frac {c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 2 \, \sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {c d} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 4 \, \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (3 \, d e x + d^{2}\right )} \sqrt {e x + d}}{16 \, {\left (c^{2} d^{3} e^{4} x^{3} + c^{2} d^{4} e^{3} x^{2} - c^{2} d^{5} e^{2} x - c^{2} d^{6} e\right )}}, -\frac {3 \, \sqrt {2} {\left (e^{3} x^{3} + d e^{2} x^{2} - d^{2} e x - d^{3}\right )} \sqrt {-c d} \arctan \left (\frac {\sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {-c d} \sqrt {e x + d}}{c e^{2} x^{2} - c d^{2}}\right ) + 2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (3 \, d e x + d^{2}\right )} \sqrt {e x + d}}{8 \, {\left (c^{2} d^{3} e^{4} x^{3} + c^{2} d^{4} e^{3} x^{2} - c^{2} d^{5} e^{2} x - c^{2} d^{6} e\right )}}\right ] \]
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\[ \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \sqrt {d + e x}}\, dx \]
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\[ \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac {3}{2}} \sqrt {e x + d}} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right )}{8 \, \sqrt {-c d} c d^{2} e} + \frac {3 \, {\left (e x + d\right )} c - 2 \, c d}{4 \, {\left (2 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c d - {\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}}\right )} c d^{2} e} \]
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Timed out. \[ \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,d^2-c\,e^2\,x^2\right )}^{3/2}\,\sqrt {d+e\,x}} \,d x \]
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