Integrand size = 15, antiderivative size = 38 \[ \int x^7 \sqrt [4]{a+b x^4} \, dx=-\frac {a \left (a+b x^4\right )^{5/4}}{5 b^2}+\frac {\left (a+b x^4\right )^{9/4}}{9 b^2} \]
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Time = 0.02 (sec) , antiderivative size = 38, 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} \[ \int x^7 \sqrt [4]{a+b x^4} \, dx=\frac {\left (a+b x^4\right )^{9/4}}{9 b^2}-\frac {a \left (a+b x^4\right )^{5/4}}{5 b^2} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int x \sqrt [4]{a+b x} \, dx,x,x^4\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \left (-\frac {a \sqrt [4]{a+b x}}{b}+\frac {(a+b x)^{5/4}}{b}\right ) \, dx,x,x^4\right ) \\ & = -\frac {a \left (a+b x^4\right )^{5/4}}{5 b^2}+\frac {\left (a+b x^4\right )^{9/4}}{9 b^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int x^7 \sqrt [4]{a+b x^4} \, dx=\frac {\sqrt [4]{a+b x^4} \left (-4 a^2+a b x^4+5 b^2 x^8\right )}{45 b^2} \]
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Time = 4.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66
method | result | size |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (-5 b \,x^{4}+4 a \right )}{45 b^{2}}\) | \(25\) |
pseudoelliptic | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (-5 b \,x^{4}+4 a \right )}{45 b^{2}}\) | \(25\) |
trager | \(-\frac {\left (-5 b^{2} x^{8}-a b \,x^{4}+4 a^{2}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{45 b^{2}}\) | \(36\) |
risch | \(-\frac {\left (-5 b^{2} x^{8}-a b \,x^{4}+4 a^{2}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{45 b^{2}}\) | \(36\) |
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none
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int x^7 \sqrt [4]{a+b x^4} \, dx=\frac {{\left (5 \, b^{2} x^{8} + a b x^{4} - 4 \, a^{2}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{45 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (31) = 62\).
Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.66 \[ \int x^7 \sqrt [4]{a+b x^4} \, dx=\begin {cases} - \frac {4 a^{2} \sqrt [4]{a + b x^{4}}}{45 b^{2}} + \frac {a x^{4} \sqrt [4]{a + b x^{4}}}{45 b} + \frac {x^{8} \sqrt [4]{a + b x^{4}}}{9} & \text {for}\: b \neq 0 \\\frac {\sqrt [4]{a} x^{8}}{8} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int x^7 \sqrt [4]{a+b x^4} \, dx=\frac {{\left (b x^{4} + a\right )}^{\frac {9}{4}}}{9 \, b^{2}} - \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}} a}{5 \, b^{2}} \]
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none
Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int x^7 \sqrt [4]{a+b x^4} \, dx=\frac {5 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} - 9 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} a}{45 \, b^{2}} \]
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Time = 5.73 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int x^7 \sqrt [4]{a+b x^4} \, dx={\left (b\,x^4+a\right )}^{1/4}\,\left (\frac {x^8}{9}-\frac {4\,a^2}{45\,b^2}+\frac {a\,x^4}{45\,b}\right ) \]
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