Integrand size = 25, antiderivative size = 161 \[ \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {7 d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8} \]
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Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {833, 794, 223, 209} \[ \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {7 d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8}+\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rule 209
Rule 223
Rule 794
Rule 833
Rubi steps \begin{align*} \text {integral}& = \frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {x^5 \left (6 d^3+7 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2} \\ & = \frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {x^3 \left (24 d^5+35 d^4 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4} \\ & = \frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {x \left (48 d^7+105 d^6 e x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^6 e^6} \\ & = \frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {\left (7 d^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^7} \\ & = \frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {\left (7 d^2\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^7} \\ & = \frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {7 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.87 \[ \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (96 d^6+9 d^5 e x-249 d^4 e^2 x^2+4 d^3 e^3 x^3+176 d^2 e^4 x^4-15 d e^5 x^5-15 e^6 x^6\right )}{(d-e x)^3 (d+e x)^2}+210 d^2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{30 e^8} \]
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Time = 0.44 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.56
method | result | size |
default | \(e \left (-\frac {x^{7}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {7 d^{2} \left (\frac {x^{5}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )}{2 e^{2}}\right )+d \left (-\frac {x^{6}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 d^{2} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )}{e^{2}}\right )\) | \(251\) |
risch | \(\frac {\left (e x +2 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{8}}-\frac {7 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{7} \sqrt {e^{2}}}-\frac {7 d^{3} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{15 e^{10} \left (x -\frac {d}{e}\right )^{2}}-\frac {773 d^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{240 e^{9} \left (x -\frac {d}{e}\right )}+\frac {d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{24 e^{10} \left (x +\frac {d}{e}\right )^{2}}-\frac {31 d^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{48 e^{9} \left (x +\frac {d}{e}\right )}-\frac {d^{4} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{20 e^{11} \left (x -\frac {d}{e}\right )^{3}}\) | \(297\) |
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Time = 0.30 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.73 \[ \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {96 \, d^{2} e^{5} x^{5} - 96 \, d^{3} e^{4} x^{4} - 192 \, d^{4} e^{3} x^{3} + 192 \, d^{5} e^{2} x^{2} + 96 \, d^{6} e x - 96 \, d^{7} + 210 \, {\left (d^{2} e^{5} x^{5} - d^{3} e^{4} x^{4} - 2 \, d^{4} e^{3} x^{3} + 2 \, d^{5} e^{2} x^{2} + d^{6} e x - d^{7}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (15 \, e^{6} x^{6} + 15 \, d e^{5} x^{5} - 176 \, d^{2} e^{4} x^{4} - 4 \, d^{3} e^{3} x^{3} + 249 \, d^{4} e^{2} x^{2} - 9 \, d^{5} e x - 96 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (e^{13} x^{5} - d e^{12} x^{4} - 2 \, d^{2} e^{11} x^{3} + 2 \, d^{3} e^{10} x^{2} + d^{4} e^{9} x - d^{5} e^{8}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (144) = 288\).
Time = 13.69 (sec) , antiderivative size = 2004, normalized size of antiderivative = 12.45 \[ \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (141) = 282\).
Time = 0.28 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.01 \[ \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {x^{7}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {7 \, d^{2} x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )}}{30 \, e} - \frac {d x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {7 \, d^{2} x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )}}{6 \, e^{3}} + \frac {6 \, d^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} - \frac {8 \, d^{5} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}} + \frac {16 \, d^{7}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{8}} + \frac {14 \, d^{4} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{7}} - \frac {49 \, d^{2} x}{30 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{7}} - \frac {7 \, d^{2} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}} e^{7}} \]
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\[ \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )} x^{7}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {x^7\,\left (d+e\,x\right )}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
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