Optimal. Leaf size=133 \[ -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{715 d^3 e (d+e x)^{10}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{6435 d^4 e (d+e x)^9} \]
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Rubi [A]
time = 0.04, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665}
\begin {gather*} -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{6435 d^4 e (d+e x)^9}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{715 d^3 e (d+e x)^{10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 665
Rule 673
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{12}} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}+\frac {\int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx}{5 d}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}+\frac {2 \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx}{65 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{715 d^3 e (d+e x)^{10}}+\frac {2 \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx}{715 d^3}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{715 d^3 e (d+e x)^{10}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{6435 d^4 e (d+e x)^9}\\ \end {align*}
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Mathematica [A]
time = 0.70, size = 71, normalized size = 0.53 \begin {gather*} -\frac {(d-e x)^4 \sqrt {d^2-e^2 x^2} \left (548 d^3+141 d^2 e x+24 d e^2 x^2+2 e^3 x^3\right )}{6435 d^4 e (d+e x)^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 197, normalized size = 1.48
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (2 e^{3} x^{3}+24 d \,e^{2} x^{2}+141 d^{2} e x +548 d^{3}\right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6435 \left (e x +d \right )^{11} d^{4} e}\) | \(66\) |
trager | \(-\frac {\left (2 e^{7} x^{7}+16 d \,e^{6} x^{6}+57 d^{2} e^{5} x^{5}+120 d^{3} e^{4} x^{4}-1440 d^{4} e^{3} x^{3}+2748 d^{5} e^{2} x^{2}-2051 d^{6} e x +548 d^{7}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{6435 d^{4} \left (e x +d \right )^{8} e}\) | \(104\) |
default | \(\frac {-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{15 d e \left (x +\frac {d}{e}\right )^{12}}+\frac {e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{13 d e \left (x +\frac {d}{e}\right )^{11}}+\frac {2 e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{11 d e \left (x +\frac {d}{e}\right )^{10}}-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{99 d^{2} \left (x +\frac {d}{e}\right )^{9}}\right )}{13 d}\right )}{5 d}}{e^{12}}\) | \(197\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 866 vs.
\(2 (113) = 226\).
time = 0.30, size = 866, normalized size = 6.51 \begin {gather*} -\frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}}}{4 \, {\left (x^{11} e^{12} + 11 \, d x^{10} e^{11} + 55 \, d^{2} x^{9} e^{10} + 165 \, d^{3} x^{8} e^{9} + 330 \, d^{4} x^{7} e^{8} + 462 \, d^{5} x^{6} e^{7} + 462 \, d^{6} x^{5} e^{6} + 330 \, d^{7} x^{4} e^{5} + 165 \, d^{8} x^{3} e^{4} + 55 \, d^{9} x^{2} e^{3} + 11 \, d^{10} x e^{2} + d^{11} e\right )}} + \frac {7 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d}{20 \, {\left (x^{10} e^{11} + 10 \, d x^{9} e^{10} + 45 \, d^{2} x^{8} e^{9} + 120 \, d^{3} x^{7} e^{8} + 210 \, d^{4} x^{6} e^{7} + 252 \, d^{5} x^{5} e^{6} + 210 \, d^{6} x^{4} e^{5} + 120 \, d^{7} x^{3} e^{4} + 45 \, d^{8} x^{2} e^{3} + 10 \, d^{9} x e^{2} + d^{10} e\right )}} - \frac {7 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{24 \, {\left (x^{9} e^{10} + 9 \, d x^{8} e^{9} + 36 \, d^{2} x^{7} e^{8} + 84 \, d^{3} x^{6} e^{7} + 126 \, d^{4} x^{5} e^{6} + 126 \, d^{5} x^{4} e^{5} + 84 \, d^{6} x^{3} e^{4} + 36 \, d^{7} x^{2} e^{3} + 9 \, d^{8} x e^{2} + d^{9} e\right )}} + \frac {7 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3}}{60 \, {\left (x^{8} e^{9} + 8 \, d x^{7} e^{8} + 28 \, d^{2} x^{6} e^{7} + 56 \, d^{3} x^{5} e^{6} + 70 \, d^{4} x^{4} e^{5} + 56 \, d^{5} x^{3} e^{4} + 28 \, d^{6} x^{2} e^{3} + 8 \, d^{7} x e^{2} + d^{8} e\right )}} - \frac {7 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2}}{1560 \, {\left (x^{7} e^{8} + 7 \, d x^{6} e^{7} + 21 \, d^{2} x^{5} e^{6} + 35 \, d^{3} x^{4} e^{5} + 35 \, d^{4} x^{3} e^{4} + 21 \, d^{5} x^{2} e^{3} + 7 \, d^{6} x e^{2} + d^{7} e\right )}} - \frac {7 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{2860 \, {\left (x^{6} e^{7} + 6 \, d x^{5} e^{6} + 15 \, d^{2} x^{4} e^{5} + 20 \, d^{3} x^{3} e^{4} + 15 \, d^{4} x^{2} e^{3} + 6 \, d^{5} x e^{2} + d^{6} e\right )}} - \frac {7 \, \sqrt {-x^{2} e^{2} + d^{2}}}{5148 \, {\left (x^{5} e^{6} + 5 \, d x^{4} e^{5} + 10 \, d^{2} x^{3} e^{4} + 10 \, d^{3} x^{2} e^{3} + 5 \, d^{4} x e^{2} + d^{5} e\right )}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}}}{1287 \, {\left (d x^{4} e^{5} + 4 \, d^{2} x^{3} e^{4} + 6 \, d^{3} x^{2} e^{3} + 4 \, d^{4} x e^{2} + d^{5} e\right )}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}}}{2145 \, {\left (d^{2} x^{3} e^{4} + 3 \, d^{3} x^{2} e^{3} + 3 \, d^{4} x e^{2} + d^{5} e\right )}} - \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}}}{6435 \, {\left (d^{3} x^{2} e^{3} + 2 \, d^{4} x e^{2} + d^{5} e\right )}} - \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}}}{6435 \, {\left (d^{4} x e^{2} + d^{5} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 250 vs.
\(2 (113) = 226\).
time = 3.81, size = 250, normalized size = 1.88 \begin {gather*} -\frac {548 \, x^{8} e^{8} + 4384 \, d x^{7} e^{7} + 15344 \, d^{2} x^{6} e^{6} + 30688 \, d^{3} x^{5} e^{5} + 38360 \, d^{4} x^{4} e^{4} + 30688 \, d^{5} x^{3} e^{3} + 15344 \, d^{6} x^{2} e^{2} + 4384 \, d^{7} x e + 548 \, d^{8} + {\left (2 \, x^{7} e^{7} + 16 \, d x^{6} e^{6} + 57 \, d^{2} x^{5} e^{5} + 120 \, d^{3} x^{4} e^{4} - 1440 \, d^{4} x^{3} e^{3} + 2748 \, d^{5} x^{2} e^{2} - 2051 \, d^{6} x e + 548 \, d^{7}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{6435 \, {\left (d^{4} x^{8} e^{9} + 8 \, d^{5} x^{7} e^{8} + 28 \, d^{6} x^{6} e^{7} + 56 \, d^{7} x^{5} e^{6} + 70 \, d^{8} x^{4} e^{5} + 56 \, d^{9} x^{3} e^{4} + 28 \, d^{10} x^{2} e^{3} + 8 \, d^{11} x e^{2} + d^{12} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 458 vs.
\(2 (113) = 226\).
time = 0.95, size = 458, normalized size = 3.44 \begin {gather*} \frac {2 \, {\left (\frac {1785 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + \frac {38235 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {99190 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-6\right )}}{x^{3}} + \frac {426270 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + \frac {735735 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-10\right )}}{x^{5}} + \frac {1492205 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{\left (-12\right )}}{x^{6}} + \frac {1621620 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{\left (-14\right )}}{x^{7}} + \frac {1904760 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{\left (-16\right )}}{x^{8}} + \frac {1250535 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{9} e^{\left (-18\right )}}{x^{9}} + \frac {909909 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{10} e^{\left (-20\right )}}{x^{10}} + \frac {321750 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{11} e^{\left (-22\right )}}{x^{11}} + \frac {150150 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{12} e^{\left (-24\right )}}{x^{12}} + \frac {19305 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{13} e^{\left (-26\right )}}{x^{13}} + \frac {6435 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{14} e^{\left (-28\right )}}{x^{14}} + 548\right )} e^{\left (-1\right )}}{6435 \, d^{4} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.02, size = 228, normalized size = 1.71 \begin {gather*} \frac {320\,\sqrt {d^2-e^2\,x^2}}{1287\,e\,{\left (d+e\,x\right )}^5}-\frac {824\,d\,\sqrt {d^2-e^2\,x^2}}{715\,e\,{\left (d+e\,x\right )}^6}-\frac {\sqrt {d^2-e^2\,x^2}}{1287\,d\,e\,{\left (d+e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}}{2145\,d^2\,e\,{\left (d+e\,x\right )}^3}-\frac {2\,\sqrt {d^2-e^2\,x^2}}{6435\,d^3\,e\,{\left (d+e\,x\right )}^2}-\frac {2\,\sqrt {d^2-e^2\,x^2}}{6435\,d^4\,e\,\left (d+e\,x\right )}+\frac {368\,d^2\,\sqrt {d^2-e^2\,x^2}}{195\,e\,{\left (d+e\,x\right )}^7}-\frac {16\,d^3\,\sqrt {d^2-e^2\,x^2}}{15\,e\,{\left (d+e\,x\right )}^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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