Integrand size = 24, antiderivative size = 238 \[ \int \frac {1}{(d+e x)^5 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {128 x}{2145 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {256 x}{2145 d^{11} \sqrt {d^2-e^2 x^2}} \]
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Time = 0.08 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {673, 198, 197} \[ \int \frac {1}{(d+e x)^5 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {256 x}{2145 d^{11} \sqrt {d^2-e^2 x^2}}+\frac {128 x}{2145 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rule 197
Rule 198
Rule 673
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{3 d} \\ & = -\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {6 \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{13 d^2} \\ & = -\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {48 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^3} \\ & = -\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {112 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{429 d^4} \\ & = -\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {32 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^5} \\ & = \frac {32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {128 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{715 d^7} \\ & = \frac {32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {128 x}{2145 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {256 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{2145 d^9} \\ & = \frac {32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {128 x}{2145 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {256 x}{2145 d^{11} \sqrt {d^2-e^2 x^2}} \\ \end{align*}
Time = 0.91 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(d+e x)^5 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-503 d^{10}-370 d^9 e x+1590 d^8 e^2 x^2+3760 d^7 e^3 x^3+1520 d^6 e^4 x^4-3744 d^5 e^5 x^5-4640 d^4 e^6 x^6-640 d^3 e^7 x^7+1920 d^2 e^8 x^8+1280 d e^9 x^9+256 e^{10} x^{10}\right )}{2145 d^{11} e (d-e x)^3 (d+e x)^8} \]
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Time = 2.54 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (-256 e^{10} x^{10}-1280 d \,e^{9} x^{9}-1920 e^{8} d^{2} x^{8}+640 d^{3} e^{7} x^{7}+4640 d^{4} e^{6} x^{6}+3744 d^{5} e^{5} x^{5}-1520 d^{6} e^{4} x^{4}-3760 d^{7} e^{3} x^{3}-1590 d^{8} e^{2} x^{2}+370 d^{9} e x +503 d^{10}\right )}{2145 \left (e x +d \right )^{4} d^{11} e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(143\) |
trager | \(-\frac {\left (-256 e^{10} x^{10}-1280 d \,e^{9} x^{9}-1920 e^{8} d^{2} x^{8}+640 d^{3} e^{7} x^{7}+4640 d^{4} e^{6} x^{6}+3744 d^{5} e^{5} x^{5}-1520 d^{6} e^{4} x^{4}-3760 d^{7} e^{3} x^{3}-1590 d^{8} e^{2} x^{2}+370 d^{9} e x +503 d^{10}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{2145 d^{11} \left (e x +d \right )^{8} \left (-e x +d \right )^{3} e}\) | \(145\) |
default | \(\frac {-\frac {1}{15 d e \left (x +\frac {d}{e}\right )^{5} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {2 e \left (-\frac {1}{13 d e \left (x +\frac {d}{e}\right )^{4} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {9 e \left (-\frac {1}{11 d e \left (x +\frac {d}{e}\right )^{3} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {8 e \left (-\frac {1}{9 d e \left (x +\frac {d}{e}\right )^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {7 e \left (-\frac {1}{7 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {6 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{10 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d^{2}}\right )}{7 d}\right )}{9 d}\right )}{11 d}\right )}{13 d}\right )}{3 d}}{e^{5}}\) | \(431\) |
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Time = 1.03 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.55 \[ \int \frac {1}{(d+e x)^5 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {503 \, e^{11} x^{11} + 2515 \, d e^{10} x^{10} + 3521 \, d^{2} e^{9} x^{9} - 2515 \, d^{3} e^{8} x^{8} - 11066 \, d^{4} e^{7} x^{7} - 7042 \, d^{5} e^{6} x^{6} + 7042 \, d^{6} e^{5} x^{5} + 11066 \, d^{7} e^{4} x^{4} + 2515 \, d^{8} e^{3} x^{3} - 3521 \, d^{9} e^{2} x^{2} - 2515 \, d^{10} e x - 503 \, d^{11} + {\left (256 \, e^{10} x^{10} + 1280 \, d e^{9} x^{9} + 1920 \, d^{2} e^{8} x^{8} - 640 \, d^{3} e^{7} x^{7} - 4640 \, d^{4} e^{6} x^{6} - 3744 \, d^{5} e^{5} x^{5} + 1520 \, d^{6} e^{4} x^{4} + 3760 \, d^{7} e^{3} x^{3} + 1590 \, d^{8} e^{2} x^{2} - 370 \, d^{9} e x - 503 \, d^{10}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{2145 \, {\left (d^{11} e^{12} x^{11} + 5 \, d^{12} e^{11} x^{10} + 7 \, d^{13} e^{10} x^{9} - 5 \, d^{14} e^{9} x^{8} - 22 \, d^{15} e^{8} x^{7} - 14 \, d^{16} e^{7} x^{6} + 14 \, d^{17} e^{6} x^{5} + 22 \, d^{18} e^{5} x^{4} + 5 \, d^{19} e^{4} x^{3} - 7 \, d^{20} e^{3} x^{2} - 5 \, d^{21} e^{2} x - d^{22} e\right )}} \]
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\[ \int \frac {1}{(d+e x)^5 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )^{5}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (206) = 412\).
Time = 0.21 (sec) , antiderivative size = 539, normalized size of antiderivative = 2.26 \[ \int \frac {1}{(d+e x)^5 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {1}{15 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{6} x^{5} + 5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{5} x^{4} + 10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{4} x^{3} + 10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{3} x^{2} + 5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} e\right )}} - \frac {2}{39 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{5} x^{4} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{4} x^{3} + 6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{3} x^{2} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} e\right )}} - \frac {6}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{4} x^{3} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{3} x^{2} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} e\right )}} - \frac {16}{429 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{3} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} e\right )}} - \frac {16}{429 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} e\right )}} + \frac {32 \, x}{715 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{7}} + \frac {128 \, x}{2145 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{9}} + \frac {256 \, x}{2145 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{11}} \]
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Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.69 \[ \int \frac {1}{(d+e x)^5 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {e^{10} {\left (\frac {143 \, {\left (675 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{2} + \frac {100 \, d}{e x + d} - 47\right )}}{d^{11} e^{10} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )} - \frac {143 \, d^{154} e^{140} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {15}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{14} \mathrm {sgn}\left (e\right )^{14} + 1650 \, d^{154} e^{140} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {13}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{14} \mathrm {sgn}\left (e\right )^{14} + 8775 \, d^{154} e^{140} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {11}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{14} \mathrm {sgn}\left (e\right )^{14} + 28600 \, d^{154} e^{140} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {9}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{14} \mathrm {sgn}\left (e\right )^{14} + 64350 \, d^{154} e^{140} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{14} \mathrm {sgn}\left (e\right )^{14} + 108108 \, d^{154} e^{140} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{14} \mathrm {sgn}\left (e\right )^{14} + 150150 \, d^{154} e^{140} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{14} \mathrm {sgn}\left (e\right )^{14} + 257400 \, d^{154} e^{140} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{14} \mathrm {sgn}\left (e\right )^{14}}{d^{165} e^{150} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{15} \mathrm {sgn}\left (e\right )^{15}}\right )} + \frac {262144 i \, \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{11}}}{2196480 \, {\left | e \right |}} \]
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Time = 10.31 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(d+e x)^5 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {128\,x}{2145\,d^9}+\frac {647}{18304\,d^8\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1757\,x}{11440\,d^7}-\frac {3371}{22880\,d^6\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}}{120\,d^4\,e\,{\left (d+e\,x\right )}^8}-\frac {59\,\sqrt {d^2-e^2\,x^2}}{3120\,d^5\,e\,{\left (d+e\,x\right )}^7}-\frac {313\,\sqrt {d^2-e^2\,x^2}}{11440\,d^6\,e\,{\left (d+e\,x\right )}^6}-\frac {149\,\sqrt {d^2-e^2\,x^2}}{4576\,d^7\,e\,{\left (d+e\,x\right )}^5}-\frac {647\,\sqrt {d^2-e^2\,x^2}}{18304\,d^8\,e\,{\left (d+e\,x\right )}^4}+\frac {256\,x\,\sqrt {d^2-e^2\,x^2}}{2145\,d^{11}\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]
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