Integrand size = 15, antiderivative size = 104 \[ \int \frac {1}{x^9 \left (a+b x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{a+b x^4}}{8 a x^8}+\frac {7 b \sqrt [4]{a+b x^4}}{32 a^2 x^4}-\frac {21 b^2 \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}}-\frac {21 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}} \]
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Time = 0.05 (sec) , antiderivative size = 104, 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, 44, 65, 218, 212, 209} \[ \int \frac {1}{x^9 \left (a+b x^4\right )^{3/4}} \, dx=-\frac {21 b^2 \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}}-\frac {21 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}}+\frac {7 b \sqrt [4]{a+b x^4}}{32 a^2 x^4}-\frac {\sqrt [4]{a+b x^4}}{8 a x^8} \]
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Rule 44
Rule 65
Rule 209
Rule 212
Rule 218
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {1}{x^3 (a+b x)^{3/4}} \, dx,x,x^4\right ) \\ & = -\frac {\sqrt [4]{a+b x^4}}{8 a x^8}-\frac {(7 b) \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/4}} \, dx,x,x^4\right )}{32 a} \\ & = -\frac {\sqrt [4]{a+b x^4}}{8 a x^8}+\frac {7 b \sqrt [4]{a+b x^4}}{32 a^2 x^4}+\frac {\left (21 b^2\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/4}} \, dx,x,x^4\right )}{128 a^2} \\ & = -\frac {\sqrt [4]{a+b x^4}}{8 a x^8}+\frac {7 b \sqrt [4]{a+b x^4}}{32 a^2 x^4}+\frac {(21 b) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )}{32 a^2} \\ & = -\frac {\sqrt [4]{a+b x^4}}{8 a x^8}+\frac {7 b \sqrt [4]{a+b x^4}}{32 a^2 x^4}-\frac {\left (21 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{64 a^{5/2}}-\frac {\left (21 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{64 a^{5/2}} \\ & = -\frac {\sqrt [4]{a+b x^4}}{8 a x^8}+\frac {7 b \sqrt [4]{a+b x^4}}{32 a^2 x^4}-\frac {21 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}}-\frac {21 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^9 \left (a+b x^4\right )^{3/4}} \, dx=\frac {\sqrt [4]{a+b x^4} \left (-4 a+7 b x^4\right )}{32 a^2 x^8}-\frac {21 b^2 \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}}-\frac {21 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}} \]
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Time = 4.40 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.04
method | result | size |
pseudoelliptic | \(\frac {-21 \ln \left (\frac {-\left (b \,x^{4}+a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}{-\left (b \,x^{4}+a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}\right ) b^{2} x^{8}-42 \arctan \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right ) b^{2} x^{8}+28 b \,x^{4} \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{\frac {3}{4}}-16 a^{\frac {7}{4}} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{128 a^{\frac {11}{4}} x^{8}}\) | \(108\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.10 \[ \int \frac {1}{x^9 \left (a+b x^4\right )^{3/4}} \, dx=-\frac {21 \, a^{2} x^{8} \left (\frac {b^{8}}{a^{11}}\right )^{\frac {1}{4}} \log \left (21 \, a^{3} \left (\frac {b^{8}}{a^{11}}\right )^{\frac {1}{4}} + 21 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2}\right ) + 21 i \, a^{2} x^{8} \left (\frac {b^{8}}{a^{11}}\right )^{\frac {1}{4}} \log \left (21 i \, a^{3} \left (\frac {b^{8}}{a^{11}}\right )^{\frac {1}{4}} + 21 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2}\right ) - 21 i \, a^{2} x^{8} \left (\frac {b^{8}}{a^{11}}\right )^{\frac {1}{4}} \log \left (-21 i \, a^{3} \left (\frac {b^{8}}{a^{11}}\right )^{\frac {1}{4}} + 21 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2}\right ) - 21 \, a^{2} x^{8} \left (\frac {b^{8}}{a^{11}}\right )^{\frac {1}{4}} \log \left (-21 \, a^{3} \left (\frac {b^{8}}{a^{11}}\right )^{\frac {1}{4}} + 21 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2}\right ) - 4 \, {\left (7 \, b x^{4} - 4 \, a\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{128 \, a^{2} x^{8}} \]
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Result contains complex when optimal does not.
Time = 1.71 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.38 \[ \int \frac {1}{x^9 \left (a+b x^4\right )^{3/4}} \, dx=- \frac {\Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 b^{\frac {3}{4}} x^{11} \Gamma \left (\frac {15}{4}\right )} \]
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none
Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x^9 \left (a+b x^4\right )^{3/4}} \, dx=\frac {7 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{2} - 11 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} a b^{2}}{32 \, {\left ({\left (b x^{4} + a\right )}^{2} a^{2} - 2 \, {\left (b x^{4} + a\right )} a^{3} + a^{4}\right )}} - \frac {21 \, {\left (\frac {2 \, b^{2} \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}} - \frac {b^{2} \log \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}}\right )}}{128 \, a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (80) = 160\).
Time = 0.30 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.35 \[ \int \frac {1}{x^9 \left (a+b x^4\right )^{3/4}} \, dx=\frac {\frac {42 \, \sqrt {2} b^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{2}} + \frac {42 \, \sqrt {2} b^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{2}} + \frac {21 \, \sqrt {2} b^{3} \log \left (\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right )}{\left (-a\right )^{\frac {3}{4}} a^{2}} + \frac {21 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{3} \log \left (-\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right )}{a^{3}} + \frac {8 \, {\left (7 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{3} - 11 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} a b^{3}\right )}}{a^{2} b^{2} x^{8}}}{256 \, b} \]
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Time = 6.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^9 \left (a+b x^4\right )^{3/4}} \, dx=\frac {7\,{\left (b\,x^4+a\right )}^{5/4}}{32\,a^2\,x^8}-\frac {11\,{\left (b\,x^4+a\right )}^{1/4}}{32\,a\,x^8}-\frac {21\,b^2\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{64\,a^{11/4}}+\frac {b^2\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}\,1{}\mathrm {i}}{a^{1/4}}\right )\,21{}\mathrm {i}}{64\,a^{11/4}} \]
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