Integrand size = 24, antiderivative size = 138 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx=-\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-\frac {7 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \]
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Time = 0.04 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {677, 679, 223, 209} \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx=-\frac {7 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}-\frac {7 \sqrt {d^2-e^2 x^2}}{e} \]
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Rule 209
Rule 223
Rule 677
Rule 679
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-\frac {7}{5} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx \\ & = \frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}+\frac {7}{3} \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx \\ & = -\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-7 \int \frac {\sqrt {d^2-e^2 x^2}}{d+e x} \, dx \\ & = -\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-(7 d) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-(7 d) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right ) \\ & = -\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-\frac {7 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.75 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-167 d^3-381 d^2 e x-277 d e^2 x^2-15 e^3 x^3\right )}{15 e (d+e x)^3}+\frac {7 d \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{\sqrt {-e^2}} \]
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Time = 2.61 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.33
method | result | size |
risch | \(-\frac {\sqrt {-x^{2} e^{2}+d^{2}}}{e}-\frac {7 d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}+\frac {128 d^{2} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{15 e^{3} \left (x +\frac {d}{e}\right )^{2}}-\frac {232 d \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{15 e^{2} \left (x +\frac {d}{e}\right )}-\frac {16 d^{3} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 e^{4} \left (x +\frac {d}{e}\right )^{3}}\) | \(184\) |
default | \(\frac {-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{7}}-\frac {2 e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{6}}-\frac {e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{d e \left (x +\frac {d}{e}\right )^{5}}-\frac {4 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{d e \left (x +\frac {d}{e}\right )^{4}}+\frac {5 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{3}}+\frac {2 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {7 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )\right )}{5 d}\right )}{d}\right )}{d}\right )}{d}\right )}{d}\right )}{5 d}}{e^{7}}\) | \(559\) |
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Time = 0.30 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.25 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx=-\frac {167 \, d e^{3} x^{3} + 501 \, d^{2} e^{2} x^{2} + 501 \, d^{3} e x + 167 \, d^{4} - 210 \, {\left (d e^{3} x^{3} + 3 \, d^{2} e^{2} x^{2} + 3 \, d^{3} e x + d^{4}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (15 \, e^{3} x^{3} + 277 \, d e^{2} x^{2} + 381 \, d^{2} e x + 167 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]
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\[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx=\int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}{\left (d + e x\right )^{7}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (122) = 244\).
Time = 0.28 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.91 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx=\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e} - \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{5 \, {\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e} + \frac {42 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{5 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} + \frac {49 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{15 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac {7 \, d \arcsin \left (\frac {e x}{d}\right )}{e} - \frac {266 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{15 \, {\left (e^{2} x + d e\right )}} \]
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Time = 0.39 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.49 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx=-\frac {7 \, d \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{e} + \frac {16 \, {\left (19 \, d + \frac {80 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d}{e^{2} x} + \frac {130 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d}{e^{4} x^{2}} + \frac {60 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d}{e^{6} x^{3}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d}{e^{8} x^{4}}\right )}}{15 \, {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{5} {\left | e \right |}} \]
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Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{7/2}}{{\left (d+e\,x\right )}^7} \,d x \]
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