Integrand size = 16, antiderivative size = 106 \[ \int \frac {x^{19}}{\sqrt [4]{a-b x^4}} \, dx=-\frac {a^4 \left (a-b x^4\right )^{3/4}}{3 b^5}+\frac {4 a^3 \left (a-b x^4\right )^{7/4}}{7 b^5}-\frac {6 a^2 \left (a-b x^4\right )^{11/4}}{11 b^5}+\frac {4 a \left (a-b x^4\right )^{15/4}}{15 b^5}-\frac {\left (a-b x^4\right )^{19/4}}{19 b^5} \]
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Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {272, 45} \[ \int \frac {x^{19}}{\sqrt [4]{a-b x^4}} \, dx=-\frac {a^4 \left (a-b x^4\right )^{3/4}}{3 b^5}+\frac {4 a^3 \left (a-b x^4\right )^{7/4}}{7 b^5}-\frac {6 a^2 \left (a-b x^4\right )^{11/4}}{11 b^5}-\frac {\left (a-b x^4\right )^{19/4}}{19 b^5}+\frac {4 a \left (a-b x^4\right )^{15/4}}{15 b^5} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {x^4}{\sqrt [4]{a-b x}} \, dx,x,x^4\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \left (\frac {a^4}{b^4 \sqrt [4]{a-b x}}-\frac {4 a^3 (a-b x)^{3/4}}{b^4}+\frac {6 a^2 (a-b x)^{7/4}}{b^4}-\frac {4 a (a-b x)^{11/4}}{b^4}+\frac {(a-b x)^{15/4}}{b^4}\right ) \, dx,x,x^4\right ) \\ & = -\frac {a^4 \left (a-b x^4\right )^{3/4}}{3 b^5}+\frac {4 a^3 \left (a-b x^4\right )^{7/4}}{7 b^5}-\frac {6 a^2 \left (a-b x^4\right )^{11/4}}{11 b^5}+\frac {4 a \left (a-b x^4\right )^{15/4}}{15 b^5}-\frac {\left (a-b x^4\right )^{19/4}}{19 b^5} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.58 \[ \int \frac {x^{19}}{\sqrt [4]{a-b x^4}} \, dx=\frac {\left (a-b x^4\right )^{3/4} \left (-2048 a^4-1536 a^3 b x^4-1344 a^2 b^2 x^8-1232 a b^3 x^{12}-1155 b^4 x^{16}\right )}{21945 b^5} \]
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Time = 4.60 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.56
method | result | size |
gosper | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (1155 x^{16} b^{4}+1232 a \,b^{3} x^{12}+1344 a^{2} b^{2} x^{8}+1536 a^{3} b \,x^{4}+2048 a^{4}\right )}{21945 b^{5}}\) | \(59\) |
trager | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (1155 x^{16} b^{4}+1232 a \,b^{3} x^{12}+1344 a^{2} b^{2} x^{8}+1536 a^{3} b \,x^{4}+2048 a^{4}\right )}{21945 b^{5}}\) | \(59\) |
risch | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (1155 x^{16} b^{4}+1232 a \,b^{3} x^{12}+1344 a^{2} b^{2} x^{8}+1536 a^{3} b \,x^{4}+2048 a^{4}\right )}{21945 b^{5}}\) | \(59\) |
pseudoelliptic | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (1155 x^{16} b^{4}+1232 a \,b^{3} x^{12}+1344 a^{2} b^{2} x^{8}+1536 a^{3} b \,x^{4}+2048 a^{4}\right )}{21945 b^{5}}\) | \(59\) |
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Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.55 \[ \int \frac {x^{19}}{\sqrt [4]{a-b x^4}} \, dx=-\frac {{\left (1155 \, b^{4} x^{16} + 1232 \, a b^{3} x^{12} + 1344 \, a^{2} b^{2} x^{8} + 1536 \, a^{3} b x^{4} + 2048 \, a^{4}\right )} {\left (-b x^{4} + a\right )}^{\frac {3}{4}}}{21945 \, b^{5}} \]
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Time = 0.88 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.10 \[ \int \frac {x^{19}}{\sqrt [4]{a-b x^4}} \, dx=\begin {cases} - \frac {2048 a^{4} \left (a - b x^{4}\right )^{\frac {3}{4}}}{21945 b^{5}} - \frac {512 a^{3} x^{4} \left (a - b x^{4}\right )^{\frac {3}{4}}}{7315 b^{4}} - \frac {64 a^{2} x^{8} \left (a - b x^{4}\right )^{\frac {3}{4}}}{1045 b^{3}} - \frac {16 a x^{12} \left (a - b x^{4}\right )^{\frac {3}{4}}}{285 b^{2}} - \frac {x^{16} \left (a - b x^{4}\right )^{\frac {3}{4}}}{19 b} & \text {for}\: b \neq 0 \\\frac {x^{20}}{20 \sqrt [4]{a}} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.81 \[ \int \frac {x^{19}}{\sqrt [4]{a-b x^4}} \, dx=-\frac {{\left (-b x^{4} + a\right )}^{\frac {19}{4}}}{19 \, b^{5}} + \frac {4 \, {\left (-b x^{4} + a\right )}^{\frac {15}{4}} a}{15 \, b^{5}} - \frac {6 \, {\left (-b x^{4} + a\right )}^{\frac {11}{4}} a^{2}}{11 \, b^{5}} + \frac {4 \, {\left (-b x^{4} + a\right )}^{\frac {7}{4}} a^{3}}{7 \, b^{5}} - \frac {{\left (-b x^{4} + a\right )}^{\frac {3}{4}} a^{4}}{3 \, b^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.03 \[ \int \frac {x^{19}}{\sqrt [4]{a-b x^4}} \, dx=-\frac {1155 \, {\left (b x^{4} - a\right )}^{4} {\left (-b x^{4} + a\right )}^{\frac {3}{4}} + 5852 \, {\left (b x^{4} - a\right )}^{3} {\left (-b x^{4} + a\right )}^{\frac {3}{4}} a + 11970 \, {\left (b x^{4} - a\right )}^{2} {\left (-b x^{4} + a\right )}^{\frac {3}{4}} a^{2} - 12540 \, {\left (-b x^{4} + a\right )}^{\frac {7}{4}} a^{3} + 7315 \, {\left (-b x^{4} + a\right )}^{\frac {3}{4}} a^{4}}{21945 \, b^{5}} \]
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Time = 5.56 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.57 \[ \int \frac {x^{19}}{\sqrt [4]{a-b x^4}} \, dx=-{\left (a-b\,x^4\right )}^{3/4}\,\left (\frac {2048\,a^4}{21945\,b^5}+\frac {x^{16}}{19\,b}+\frac {16\,a\,x^{12}}{285\,b^2}+\frac {512\,a^3\,x^4}{7315\,b^4}+\frac {64\,a^2\,x^8}{1045\,b^3}\right ) \]
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