Optimal. Leaf size=49 \[ \frac {8 x}{15 \sqrt {1-2 x^2}}+\frac {4 x}{15 \left (1-2 x^2\right )^{3/2}}+\frac {x}{5 \left (1-2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {192, 191} \[ \frac {8 x}{15 \sqrt {1-2 x^2}}+\frac {4 x}{15 \left (1-2 x^2\right )^{3/2}}+\frac {x}{5 \left (1-2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rubi steps
\begin {align*} \int \frac {1}{\left (1-2 x^2\right )^{7/2}} \, dx &=\frac {x}{5 \left (1-2 x^2\right )^{5/2}}+\frac {4}{5} \int \frac {1}{\left (1-2 x^2\right )^{5/2}} \, dx\\ &=\frac {x}{5 \left (1-2 x^2\right )^{5/2}}+\frac {4 x}{15 \left (1-2 x^2\right )^{3/2}}+\frac {8}{15} \int \frac {1}{\left (1-2 x^2\right )^{3/2}} \, dx\\ &=\frac {x}{5 \left (1-2 x^2\right )^{5/2}}+\frac {4 x}{15 \left (1-2 x^2\right )^{3/2}}+\frac {8 x}{15 \sqrt {1-2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.57 \[ \frac {x \left (32 x^4-40 x^2+15\right )}{15 \left (1-2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 37, normalized size = 0.76 \[ -\frac {x \sqrt {1-2 x^2} \left (32 x^4-40 x^2+15\right )}{15 \left (2 x^2-1\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 44, normalized size = 0.90 \[ -\frac {{\left (32 \, x^{5} - 40 \, x^{3} + 15 \, x\right )} \sqrt {-2 \, x^{2} + 1}}{15 \, {\left (8 \, x^{6} - 12 \, x^{4} + 6 \, x^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 35, normalized size = 0.71 \[ -\frac {{\left (8 \, {\left (4 \, x^{2} - 5\right )} x^{2} + 15\right )} \sqrt {-2 \, x^{2} + 1} x}{15 \, {\left (2 \, x^{2} - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 25, normalized size = 0.51
method | result | size |
gosper | \(\frac {x \left (32 x^{4}-40 x^{2}+15\right )}{15 \left (-2 x^{2}+1\right )^{\frac {5}{2}}}\) | \(25\) |
meijerg | \(\frac {x \left (32 x^{4}-40 x^{2}+15\right )}{15 \left (-2 x^{2}+1\right )^{\frac {5}{2}}}\) | \(25\) |
trager | \(-\frac {\left (32 x^{4}-40 x^{2}+15\right ) x \sqrt {-2 x^{2}+1}}{15 \left (2 x^{2}-1\right )^{3}}\) | \(34\) |
risch | \(\frac {x \left (32 x^{4}-40 x^{2}+15\right )}{15 \left (2 x^{2}-1\right )^{2} \sqrt {-2 x^{2}+1}}\) | \(34\) |
default | \(\frac {x}{5 \left (-2 x^{2}+1\right )^{\frac {5}{2}}}+\frac {4 x}{15 \left (-2 x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 x}{15 \sqrt {-2 x^{2}+1}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 37, normalized size = 0.76 \[ \frac {8 \, x}{15 \, \sqrt {-2 \, x^{2} + 1}} + \frac {4 \, x}{15 \, {\left (-2 \, x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {x}{5 \, {\left (-2 \, x^{2} + 1\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 179, normalized size = 3.65 \[ \frac {19\,\sqrt {\frac {1}{2}-x^2}}{480\,\left (x^2-\sqrt {2}\,x+\frac {1}{2}\right )}-\frac {19\,\sqrt {\frac {1}{2}-x^2}}{480\,\left (x^2+\sqrt {2}\,x+\frac {1}{2}\right )}-\frac {\sqrt {2}\,\sqrt {\frac {1}{2}-x^2}}{160\,\left (x^3-\frac {3\,\sqrt {2}\,x^2}{2}+\frac {3\,x}{2}-\frac {\sqrt {2}}{4}\right )}-\frac {\sqrt {2}\,\sqrt {\frac {1}{2}-x^2}}{160\,\left (x^3+\frac {3\,\sqrt {2}\,x^2}{2}+\frac {3\,x}{2}+\frac {\sqrt {2}}{4}\right )}-\frac {2\,\sqrt {2}\,\sqrt {\frac {1}{2}-x^2}}{15\,\left (x-\frac {\sqrt {2}}{2}\right )}-\frac {2\,\sqrt {2}\,\sqrt {\frac {1}{2}-x^2}}{15\,\left (x+\frac {\sqrt {2}}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 12.35, size = 291, normalized size = 5.94 \[ \begin {cases} - \frac {32 i x^{5}}{60 x^{4} \sqrt {2 x^{2} - 1} - 60 x^{2} \sqrt {2 x^{2} - 1} + 15 \sqrt {2 x^{2} - 1}} + \frac {40 i x^{3}}{60 x^{4} \sqrt {2 x^{2} - 1} - 60 x^{2} \sqrt {2 x^{2} - 1} + 15 \sqrt {2 x^{2} - 1}} - \frac {15 i x}{60 x^{4} \sqrt {2 x^{2} - 1} - 60 x^{2} \sqrt {2 x^{2} - 1} + 15 \sqrt {2 x^{2} - 1}} & \text {for}\: 2 \left |{x^{2}}\right | > 1 \\\frac {32 x^{5}}{60 x^{4} \sqrt {1 - 2 x^{2}} - 60 x^{2} \sqrt {1 - 2 x^{2}} + 15 \sqrt {1 - 2 x^{2}}} - \frac {40 x^{3}}{60 x^{4} \sqrt {1 - 2 x^{2}} - 60 x^{2} \sqrt {1 - 2 x^{2}} + 15 \sqrt {1 - 2 x^{2}}} + \frac {15 x}{60 x^{4} \sqrt {1 - 2 x^{2}} - 60 x^{2} \sqrt {1 - 2 x^{2}} + 15 \sqrt {1 - 2 x^{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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