Integrand size = 15, antiderivative size = 98 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^9} \, dx=-\frac {5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac {\left (a+b x^4\right )^{5/4}}{8 x^8}-\frac {5 b^2 \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac {5 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}} \]
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Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 43, 65, 218, 212, 209} \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^9} \, dx=-\frac {5 b^2 \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac {5 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac {5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac {\left (a+b x^4\right )^{5/4}}{8 x^8} \]
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Rule 43
Rule 65
Rule 209
Rule 212
Rule 218
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {(a+b x)^{5/4}}{x^3} \, dx,x,x^4\right ) \\ & = -\frac {\left (a+b x^4\right )^{5/4}}{8 x^8}+\frac {1}{32} (5 b) \text {Subst}\left (\int \frac {\sqrt [4]{a+b x}}{x^2} \, dx,x,x^4\right ) \\ & = -\frac {5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac {\left (a+b x^4\right )^{5/4}}{8 x^8}+\frac {1}{128} \left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/4}} \, dx,x,x^4\right ) \\ & = -\frac {5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac {\left (a+b x^4\right )^{5/4}}{8 x^8}+\frac {1}{32} (5 b) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right ) \\ & = -\frac {5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac {\left (a+b x^4\right )^{5/4}}{8 x^8}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{64 \sqrt {a}}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{64 \sqrt {a}} \\ & = -\frac {5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac {\left (a+b x^4\right )^{5/4}}{8 x^8}-\frac {5 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^9} \, dx=\frac {\left (-4 a-9 b x^4\right ) \sqrt [4]{a+b x^4}}{32 x^8}-\frac {5 b^2 \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac {5 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}} \]
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Time = 4.37 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.10
method | result | size |
pseudoelliptic | \(\frac {-10 \arctan \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right ) b^{2} x^{8}-5 \ln \left (\frac {-\left (b \,x^{4}+a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}{-\left (b \,x^{4}+a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}\right ) b^{2} x^{8}-36 b \,x^{4} \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{\frac {3}{4}}-16 a^{\frac {7}{4}} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{128 x^{8} a^{\frac {3}{4}}}\) | \(108\) |
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Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.99 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^9} \, dx=-\frac {5 \, \left (\frac {b^{8}}{a^{3}}\right )^{\frac {1}{4}} x^{8} \log \left (5 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2} + 5 \, \left (\frac {b^{8}}{a^{3}}\right )^{\frac {1}{4}} a\right ) + 5 i \, \left (\frac {b^{8}}{a^{3}}\right )^{\frac {1}{4}} x^{8} \log \left (5 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2} + 5 i \, \left (\frac {b^{8}}{a^{3}}\right )^{\frac {1}{4}} a\right ) - 5 i \, \left (\frac {b^{8}}{a^{3}}\right )^{\frac {1}{4}} x^{8} \log \left (5 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2} - 5 i \, \left (\frac {b^{8}}{a^{3}}\right )^{\frac {1}{4}} a\right ) - 5 \, \left (\frac {b^{8}}{a^{3}}\right )^{\frac {1}{4}} x^{8} \log \left (5 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2} - 5 \, \left (\frac {b^{8}}{a^{3}}\right )^{\frac {1}{4}} a\right ) + 4 \, {\left (9 \, b x^{4} + 4 \, a\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{128 \, x^{8}} \]
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Result contains complex when optimal does not.
Time = 1.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.42 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^9} \, dx=- \frac {b^{\frac {5}{4}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac {7}{4}\right )} \]
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none
Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^9} \, dx=-\frac {5 \, b^{2} \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{64 \, a^{\frac {3}{4}}} + \frac {5 \, b^{2} \log \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{128 \, a^{\frac {3}{4}}} - \frac {9 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{2} - 5 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} a b^{2}}{32 \, {\left ({\left (b x^{4} + a\right )}^{2} - 2 \, {\left (b x^{4} + a\right )} a + a^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (74) = 148\).
Time = 0.30 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.37 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^9} \, dx=\frac {\frac {10 \, \sqrt {2} b^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}}} + \frac {10 \, \sqrt {2} b^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}}} + \frac {5 \, \sqrt {2} b^{3} \log \left (\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right )}{\left (-a\right )^{\frac {3}{4}}} + \frac {5 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{3} \log \left (-\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right )}{a} - \frac {8 \, {\left (9 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{3} - 5 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} a b^{3}\right )}}{b^{2} x^{8}}}{256 \, b} \]
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Time = 6.38 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^9} \, dx=\frac {5\,a\,{\left (b\,x^4+a\right )}^{1/4}}{32\,x^8}-\frac {5\,b^2\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{64\,a^{3/4}}-\frac {9\,{\left (b\,x^4+a\right )}^{5/4}}{32\,x^8}+\frac {b^2\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}\,1{}\mathrm {i}}{a^{1/4}}\right )\,5{}\mathrm {i}}{64\,a^{3/4}} \]
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