Integrand size = 29, antiderivative size = 98 \[ \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{5/2}} \, dx=-\frac {\sqrt {c d^2-c e^2 x^2}}{e (d+e x)^{3/2}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {2} \sqrt {d} e} \]
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Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {677, 675, 214} \[ \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{5/2}} \, dx=\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {2} \sqrt {d} e}-\frac {\sqrt {c d^2-c e^2 x^2}}{e (d+e x)^{3/2}} \]
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Rule 214
Rule 675
Rule 677
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c d^2-c e^2 x^2}}{e (d+e x)^{3/2}}-\frac {1}{2} c \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx \\ & = -\frac {\sqrt {c d^2-c e^2 x^2}}{e (d+e x)^{3/2}}-(c 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 ) \\ & = -\frac {\sqrt {c d^2-c e^2 x^2}}{e (d+e x)^{3/2}}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {2} \sqrt {d} e} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{5/2}} \, dx=\frac {\sqrt {c \left (d^2-e^2 x^2\right )} \left (-\frac {2}{(d+e x)^{3/2}}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {d+e x}}{\sqrt {d^2-e^2 x^2}}\right )}{\sqrt {d} \sqrt {d^2-e^2 x^2}}\right )}{2 e} \]
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Time = 2.39 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.19
method | result | size |
default | \(\frac {\sqrt {c \left (-x^{2} e^{2}+d^{2}\right )}\, \left (\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) c e x +c d \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right )-2 \sqrt {c \left (-e x +d \right )}\, \sqrt {c d}\right )}{2 \left (e x +d \right )^{\frac {3}{2}} \sqrt {c \left (-e x +d \right )}\, e \sqrt {c d}}\) | \(117\) |
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Time = 0.40 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.98 \[ \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{5/2}} \, dx=\left [\frac {\sqrt {\frac {1}{2}} {\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt {\frac {c}{d}} \log \left (-\frac {c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} - 4 \, \sqrt {\frac {1}{2}} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} d \sqrt {\frac {c}{d}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}}, \frac {\sqrt {\frac {1}{2}} {\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt {-\frac {c}{d}} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} d \sqrt {-\frac {c}{d}}}{c e^{2} x^{2} - c d^{2}}\right ) - \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e}\right ] \]
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\[ \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{5/2}} \, dx=\int \frac {\sqrt {- c \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{5/2}} \, dx=\int { \frac {\sqrt {-c e^{2} x^{2} + c d^{2}}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{5/2}} \, dx=-\frac {\frac {\sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right )}{\sqrt {-c d}} + \frac {2 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{e x + d}}{2 \, e} \]
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Timed out. \[ \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{5/2}} \, dx=\int \frac {\sqrt {c\,d^2-c\,e^2\,x^2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]
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