Integrand size = 24, antiderivative size = 143 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx=\frac {2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\frac {\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \]
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Time = 0.03 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {677, 223, 209} \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx=\frac {\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \sqrt {d^2-e^2 x^2}}{e (d+e x)} \]
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Rule 209
Rule 223
Rule 677
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}-\int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx \\ & = \frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{(d+e x)^4} \, dx \\ & = -\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}-\int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^2} \, dx \\ & = \frac {2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right ) \\ & = \frac {2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx=\frac {8 \sqrt {d^2-e^2 x^2} \left (19 d^3+76 d^2 e x+71 d e^2 x^2+44 e^3 x^3\right )}{105 e (d+e x)^4}-\frac {2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{e} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(610\) vs. \(2(127)=254\).
Time = 2.74 (sec) , antiderivative size = 611, normalized size of antiderivative = 4.27
method | result | size |
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}}}{7 d e \left (x +\frac {d}{e}\right )^{8}}-\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}}}{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}\right )}{7 d}}{e^{8}}\) | \(611\) |
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Time = 0.29 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.40 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx=\frac {2 \, {\left (76 \, e^{4} x^{4} + 304 \, d e^{3} x^{3} + 456 \, d^{2} e^{2} x^{2} + 304 \, d^{3} e x + 76 \, d^{4} - 105 \, {\left (e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 4 \, {\left (44 \, e^{3} x^{3} + 71 \, d e^{2} x^{2} + 76 \, d^{2} e x + 19 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}\right )}}{105 \, {\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} \]
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\[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx=\int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}{\left (d + e x\right )^{8}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 623 vs. \(2 (127) = 254\).
Time = 0.27 (sec) , antiderivative size = 623, normalized size of antiderivative = 4.36 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx=-\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{7 \, {\left (e^{8} x^{7} + 7 \, d e^{7} x^{6} + 21 \, d^{2} e^{6} x^{5} + 35 \, d^{3} e^{5} x^{4} + 35 \, d^{4} e^{4} x^{3} + 21 \, d^{5} e^{3} x^{2} + 7 \, d^{6} e^{2} x + d^{7} e\right )}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{2 \, {\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{7 \, {\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{5 \, {\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e} - \frac {69 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{70 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac {34 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{105 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} + \frac {\arcsin \left (\frac {e x}{d}\right )}{e} + \frac {281 \, \sqrt {-e^{2} x^{2} + d^{2}}}{105 \, {\left (e^{2} x + d e\right )}} \]
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Time = 0.41 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.47 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx=\frac {\arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} - \frac {16 \, {\left (\frac {133 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} + \frac {294 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} + \frac {490 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{6} x^{3}} + \frac {175 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{8} x^{4}} + \frac {105 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e^{10} x^{5}} + 19\right )}}{105 \, {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{7} {\left | e \right |}} \]
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Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{7/2}}{{\left (d+e\,x\right )}^8} \,d x \]
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