Integrand size = 15, antiderivative size = 31 \[ \int x^7 \sqrt {5+3 x^4} \, dx=-\frac {5}{54} \left (5+3 x^4\right )^{3/2}+\frac {1}{90} \left (5+3 x^4\right )^{5/2} \]
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Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int x^7 \sqrt {5+3 x^4} \, dx=\frac {1}{90} \left (3 x^4+5\right )^{5/2}-\frac {5}{54} \left (3 x^4+5\right )^{3/2} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int x \sqrt {5+3 x} \, dx,x,x^4\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \left (-\frac {5}{3} \sqrt {5+3 x}+\frac {1}{3} (5+3 x)^{3/2}\right ) \, dx,x,x^4\right ) \\ & = -\frac {5}{54} \left (5+3 x^4\right )^{3/2}+\frac {1}{90} \left (5+3 x^4\right )^{5/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int x^7 \sqrt {5+3 x^4} \, dx=\frac {1}{270} \left (5+3 x^4\right )^{3/2} \left (-10+9 x^4\right ) \]
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Time = 3.99 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61
method | result | size |
gosper | \(\frac {\left (3 x^{4}+5\right )^{\frac {3}{2}} \left (9 x^{4}-10\right )}{270}\) | \(19\) |
default | \(\frac {\left (3 x^{4}+5\right )^{\frac {3}{2}} \left (9 x^{4}-10\right )}{270}\) | \(19\) |
elliptic | \(\frac {\left (3 x^{4}+5\right )^{\frac {3}{2}} \left (9 x^{4}-10\right )}{270}\) | \(19\) |
pseudoelliptic | \(\frac {\left (3 x^{4}+5\right )^{\frac {3}{2}} \left (9 x^{4}-10\right )}{270}\) | \(19\) |
trager | \(\left (\frac {1}{10} x^{8}+\frac {1}{18} x^{4}-\frac {5}{27}\right ) \sqrt {3 x^{4}+5}\) | \(23\) |
risch | \(\frac {\left (27 x^{8}+15 x^{4}-50\right ) \sqrt {3 x^{4}+5}}{270}\) | \(24\) |
meijerg | \(-\frac {25 \sqrt {5}\, \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1+\frac {3 x^{4}}{5}\right )^{\frac {3}{2}} \left (-\frac {9 x^{4}}{5}+2\right )}{15}\right )}{72 \sqrt {\pi }}\) | \(36\) |
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Time = 0.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int x^7 \sqrt {5+3 x^4} \, dx=\frac {1}{270} \, {\left (27 \, x^{8} + 15 \, x^{4} - 50\right )} \sqrt {3 \, x^{4} + 5} \]
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Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int x^7 \sqrt {5+3 x^4} \, dx=\frac {x^{8} \sqrt {3 x^{4} + 5}}{10} + \frac {x^{4} \sqrt {3 x^{4} + 5}}{18} - \frac {5 \sqrt {3 x^{4} + 5}}{27} \]
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Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int x^7 \sqrt {5+3 x^4} \, dx=\frac {1}{90} \, {\left (3 \, x^{4} + 5\right )}^{\frac {5}{2}} - \frac {5}{54} \, {\left (3 \, x^{4} + 5\right )}^{\frac {3}{2}} \]
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Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int x^7 \sqrt {5+3 x^4} \, dx=\frac {1}{90} \, {\left (3 \, x^{4} + 5\right )}^{\frac {5}{2}} - \frac {5}{54} \, {\left (3 \, x^{4} + 5\right )}^{\frac {3}{2}} \]
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Time = 5.52 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.58 \[ \int x^7 \sqrt {5+3 x^4} \, dx=\frac {{\left (3\,x^4+5\right )}^{3/2}\,\left (9\,x^4-10\right )}{270} \]
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