Integrand size = 15, antiderivative size = 92 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{18}} \, dx=-\frac {\left (a+b x^4\right )^{5/4}}{17 a x^{17}}+\frac {12 b \left (a+b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac {32 b^2 \left (a+b x^4\right )^{5/4}}{663 a^3 x^9}+\frac {128 b^3 \left (a+b x^4\right )^{5/4}}{3315 a^4 x^5} \]
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Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270} \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{18}} \, dx=\frac {128 b^3 \left (a+b x^4\right )^{5/4}}{3315 a^4 x^5}-\frac {32 b^2 \left (a+b x^4\right )^{5/4}}{663 a^3 x^9}+\frac {12 b \left (a+b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac {\left (a+b x^4\right )^{5/4}}{17 a x^{17}} \]
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Rule 270
Rule 277
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x^4\right )^{5/4}}{17 a x^{17}}-\frac {(12 b) \int \frac {\sqrt [4]{a+b x^4}}{x^{14}} \, dx}{17 a} \\ & = -\frac {\left (a+b x^4\right )^{5/4}}{17 a x^{17}}+\frac {12 b \left (a+b x^4\right )^{5/4}}{221 a^2 x^{13}}+\frac {\left (96 b^2\right ) \int \frac {\sqrt [4]{a+b x^4}}{x^{10}} \, dx}{221 a^2} \\ & = -\frac {\left (a+b x^4\right )^{5/4}}{17 a x^{17}}+\frac {12 b \left (a+b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac {32 b^2 \left (a+b x^4\right )^{5/4}}{663 a^3 x^9}-\frac {\left (128 b^3\right ) \int \frac {\sqrt [4]{a+b x^4}}{x^6} \, dx}{663 a^3} \\ & = -\frac {\left (a+b x^4\right )^{5/4}}{17 a x^{17}}+\frac {12 b \left (a+b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac {32 b^2 \left (a+b x^4\right )^{5/4}}{663 a^3 x^9}+\frac {128 b^3 \left (a+b x^4\right )^{5/4}}{3315 a^4 x^5} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{18}} \, dx=\frac {\left (a+b x^4\right )^{5/4} \left (-195 a^3+180 a^2 b x^4-160 a b^2 x^8+128 b^3 x^{12}\right )}{3315 a^4 x^{17}} \]
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Time = 4.40 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.54
method | result | size |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (-128 b^{3} x^{12}+160 a \,b^{2} x^{8}-180 a^{2} b \,x^{4}+195 a^{3}\right )}{3315 x^{17} a^{4}}\) | \(50\) |
pseudoelliptic | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (-128 b^{3} x^{12}+160 a \,b^{2} x^{8}-180 a^{2} b \,x^{4}+195 a^{3}\right )}{3315 x^{17} a^{4}}\) | \(50\) |
trager | \(-\frac {\left (-128 x^{16} b^{4}+32 a \,b^{3} x^{12}-20 a^{2} b^{2} x^{8}+15 a^{3} b \,x^{4}+195 a^{4}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{3315 x^{17} a^{4}}\) | \(61\) |
risch | \(-\frac {\left (-128 x^{16} b^{4}+32 a \,b^{3} x^{12}-20 a^{2} b^{2} x^{8}+15 a^{3} b \,x^{4}+195 a^{4}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{3315 x^{17} a^{4}}\) | \(61\) |
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Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{18}} \, dx=\frac {{\left (128 \, b^{4} x^{16} - 32 \, a b^{3} x^{12} + 20 \, a^{2} b^{2} x^{8} - 15 \, a^{3} b x^{4} - 195 \, a^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{3315 \, a^{4} x^{17}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 847 vs. \(2 (85) = 170\).
Time = 1.47 (sec) , antiderivative size = 847, normalized size of antiderivative = 9.21 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{18}} \, dx=- \frac {585 a^{7} b^{\frac {37}{4}} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {1}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {1}{4}\right )} - \frac {1800 a^{6} b^{\frac {41}{4}} x^{4} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {1}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {1}{4}\right )} - \frac {1830 a^{5} b^{\frac {45}{4}} x^{8} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {1}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {1}{4}\right )} - \frac {636 a^{4} b^{\frac {49}{4}} x^{12} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {1}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {1}{4}\right )} + \frac {231 a^{3} b^{\frac {53}{4}} x^{16} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {1}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {1}{4}\right )} + \frac {924 a^{2} b^{\frac {57}{4}} x^{20} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {1}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {1}{4}\right )} + \frac {1056 a b^{\frac {61}{4}} x^{24} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {1}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {1}{4}\right )} + \frac {384 b^{\frac {65}{4}} x^{28} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {1}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {1}{4}\right )} \]
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Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{18}} \, dx=\frac {\frac {663 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{3}}{x^{5}} - \frac {1105 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} b^{2}}{x^{9}} + \frac {765 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} b}{x^{13}} - \frac {195 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}}}{x^{17}}}{3315 \, a^{4}} \]
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\[ \int \frac {\sqrt [4]{a+b x^4}}{x^{18}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x^{18}} \,d x } \]
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Time = 6.40 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{18}} \, dx=\frac {128\,b^4\,{\left (b\,x^4+a\right )}^{1/4}}{3315\,a^4\,x}-\frac {b\,{\left (b\,x^4+a\right )}^{1/4}}{221\,a\,x^{13}}-\frac {{\left (b\,x^4+a\right )}^{1/4}}{17\,x^{17}}-\frac {32\,b^3\,{\left (b\,x^4+a\right )}^{1/4}}{3315\,a^3\,x^5}+\frac {4\,b^2\,{\left (b\,x^4+a\right )}^{1/4}}{663\,a^2\,x^9} \]
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