Integrand size = 15, antiderivative size = 68 \[ \int \frac {1}{x^{10} \left (a+b x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{a+b x^4}}{9 a x^9}+\frac {8 b \sqrt [4]{a+b x^4}}{45 a^2 x^5}-\frac {32 b^2 \sqrt [4]{a+b x^4}}{45 a^3 x} \]
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Time = 0.01 (sec) , antiderivative size = 68, 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 = {277, 270} \[ \int \frac {1}{x^{10} \left (a+b x^4\right )^{3/4}} \, dx=-\frac {32 b^2 \sqrt [4]{a+b x^4}}{45 a^3 x}+\frac {8 b \sqrt [4]{a+b x^4}}{45 a^2 x^5}-\frac {\sqrt [4]{a+b x^4}}{9 a x^9} \]
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Rule 270
Rule 277
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt [4]{a+b x^4}}{9 a x^9}-\frac {(8 b) \int \frac {1}{x^6 \left (a+b x^4\right )^{3/4}} \, dx}{9 a} \\ & = -\frac {\sqrt [4]{a+b x^4}}{9 a x^9}+\frac {8 b \sqrt [4]{a+b x^4}}{45 a^2 x^5}+\frac {\left (32 b^2\right ) \int \frac {1}{x^2 \left (a+b x^4\right )^{3/4}} \, dx}{45 a^2} \\ & = -\frac {\sqrt [4]{a+b x^4}}{9 a x^9}+\frac {8 b \sqrt [4]{a+b x^4}}{45 a^2 x^5}-\frac {32 b^2 \sqrt [4]{a+b x^4}}{45 a^3 x} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.62 \[ \int \frac {1}{x^{10} \left (a+b x^4\right )^{3/4}} \, dx=\frac {\sqrt [4]{a+b x^4} \left (-5 a^2+8 a b x^4-32 b^2 x^8\right )}{45 a^3 x^9} \]
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Time = 4.38 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.57
method | result | size |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (32 b^{2} x^{8}-8 a b \,x^{4}+5 a^{2}\right )}{45 a^{3} x^{9}}\) | \(39\) |
trager | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (32 b^{2} x^{8}-8 a b \,x^{4}+5 a^{2}\right )}{45 a^{3} x^{9}}\) | \(39\) |
risch | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (32 b^{2} x^{8}-8 a b \,x^{4}+5 a^{2}\right )}{45 a^{3} x^{9}}\) | \(39\) |
pseudoelliptic | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (32 b^{2} x^{8}-8 a b \,x^{4}+5 a^{2}\right )}{45 a^{3} x^{9}}\) | \(39\) |
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Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^{10} \left (a+b x^4\right )^{3/4}} \, dx=-\frac {{\left (32 \, b^{2} x^{8} - 8 \, a b x^{4} + 5 \, a^{2}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{45 \, a^{3} x^{9}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (60) = 120\).
Time = 0.88 (sec) , antiderivative size = 406, normalized size of antiderivative = 5.97 \[ \int \frac {1}{x^{10} \left (a+b x^4\right )^{3/4}} \, dx=\frac {5 a^{4} b^{\frac {17}{4}} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac {3}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac {3}{4}\right )} + \frac {2 a^{3} b^{\frac {21}{4}} x^{4} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac {3}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac {3}{4}\right )} + \frac {21 a^{2} b^{\frac {25}{4}} x^{8} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac {3}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac {3}{4}\right )} + \frac {56 a b^{\frac {29}{4}} x^{12} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac {3}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac {3}{4}\right )} + \frac {32 b^{\frac {33}{4}} x^{16} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac {3}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac {3}{4}\right )} \]
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Time = 0.20 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^{10} \left (a+b x^4\right )^{3/4}} \, dx=-\frac {\frac {45 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2}}{x} - \frac {18 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b}{x^{5}} + \frac {5 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}}}{x^{9}}}{45 \, a^{3}} \]
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\[ \int \frac {1}{x^{10} \left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{10}} \,d x } \]
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Time = 6.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^{10} \left (a+b x^4\right )^{3/4}} \, dx=-\frac {{\left (b\,x^4+a\right )}^{1/4}\,\left (5\,a^2-8\,a\,b\,x^4+32\,b^2\,x^8\right )}{45\,a^3\,x^9} \]
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