Integrand size = 15, antiderivative size = 92 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx=-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{119 a^2 x^{17}}-\frac {32 b^2 \left (a+b x^4\right )^{9/4}}{1547 a^3 x^{13}}+\frac {128 b^3 \left (a+b x^4\right )^{9/4}}{13923 a^4 x^9} \]
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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} \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx=\frac {128 b^3 \left (a+b x^4\right )^{9/4}}{13923 a^4 x^9}-\frac {32 b^2 \left (a+b x^4\right )^{9/4}}{1547 a^3 x^{13}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{119 a^2 x^{17}}-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}} \]
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Rule 270
Rule 277
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}-\frac {(4 b) \int \frac {\left (a+b x^4\right )^{5/4}}{x^{18}} \, dx}{7 a} \\ & = -\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{119 a^2 x^{17}}+\frac {\left (32 b^2\right ) \int \frac {\left (a+b x^4\right )^{5/4}}{x^{14}} \, dx}{119 a^2} \\ & = -\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{119 a^2 x^{17}}-\frac {32 b^2 \left (a+b x^4\right )^{9/4}}{1547 a^3 x^{13}}-\frac {\left (128 b^3\right ) \int \frac {\left (a+b x^4\right )^{5/4}}{x^{10}} \, dx}{1547 a^3} \\ & = -\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{119 a^2 x^{17}}-\frac {32 b^2 \left (a+b x^4\right )^{9/4}}{1547 a^3 x^{13}}+\frac {128 b^3 \left (a+b x^4\right )^{9/4}}{13923 a^4 x^9} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.58 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx=\frac {\left (a+b x^4\right )^{9/4} \left (-663 a^3+468 a^2 b x^4-288 a b^2 x^8+128 b^3 x^{12}\right )}{13923 a^4 x^{21}} \]
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Time = 4.49 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.54
method | result | size |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (-128 b^{3} x^{12}+288 a \,b^{2} x^{8}-468 a^{2} b \,x^{4}+663 a^{3}\right )}{13923 x^{21} a^{4}}\) | \(50\) |
pseudoelliptic | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (-128 b^{3} x^{12}+288 a \,b^{2} x^{8}-468 a^{2} b \,x^{4}+663 a^{3}\right )}{13923 x^{21} a^{4}}\) | \(50\) |
trager | \(-\frac {\left (-128 b^{5} x^{20}+32 a \,b^{4} x^{16}-20 a^{2} b^{3} x^{12}+15 a^{3} b^{2} x^{8}+858 a^{4} b \,x^{4}+663 a^{5}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{13923 x^{21} a^{4}}\) | \(72\) |
risch | \(-\frac {\left (-128 b^{5} x^{20}+32 a \,b^{4} x^{16}-20 a^{2} b^{3} x^{12}+15 a^{3} b^{2} x^{8}+858 a^{4} b \,x^{4}+663 a^{5}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{13923 x^{21} a^{4}}\) | \(72\) |
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Time = 0.33 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx=\frac {{\left (128 \, b^{5} x^{20} - 32 \, a b^{4} x^{16} + 20 \, a^{2} b^{3} x^{12} - 15 \, a^{3} b^{2} x^{8} - 858 \, a^{4} b x^{4} - 663 \, a^{5}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{13923 \, a^{4} x^{21}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 954 vs. \(2 (85) = 170\).
Time = 2.53 (sec) , antiderivative size = 954, normalized size of antiderivative = 10.37 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx=\text {Too large to display} \]
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Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx=\frac {\frac {1547 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} b^{3}}{x^{9}} - \frac {3213 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} b^{2}}{x^{13}} + \frac {2457 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}} b}{x^{17}} - \frac {663 \, {\left (b x^{4} + a\right )}^{\frac {21}{4}}}{x^{21}}}{13923 \, a^{4}} \]
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\[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{x^{22}} \,d x } \]
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Time = 7.41 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.21 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx=\frac {128\,b^5\,{\left (b\,x^4+a\right )}^{1/4}}{13923\,a^4\,x}-\frac {22\,b\,{\left (b\,x^4+a\right )}^{1/4}}{357\,x^{17}}-\frac {a\,{\left (b\,x^4+a\right )}^{1/4}}{21\,x^{21}}-\frac {32\,b^4\,{\left (b\,x^4+a\right )}^{1/4}}{13923\,a^3\,x^5}+\frac {20\,b^3\,{\left (b\,x^4+a\right )}^{1/4}}{13923\,a^2\,x^9}-\frac {5\,b^2\,{\left (b\,x^4+a\right )}^{1/4}}{4641\,a\,x^{13}} \]
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