Integrand size = 15, antiderivative size = 125 \[ \int \frac {1}{x^9 \left (a+b x^4\right )^{5/4}} \, dx=\frac {45 b^2}{32 a^3 \sqrt [4]{a+b x^4}}-\frac {1}{8 a x^8 \sqrt [4]{a+b x^4}}+\frac {9 b}{32 a^2 x^4 \sqrt [4]{a+b x^4}}+\frac {45 b^2 \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{13/4}}-\frac {45 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{13/4}} \]
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Time = 0.06 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {272, 44, 53, 65, 304, 209, 212} \[ \int \frac {1}{x^9 \left (a+b x^4\right )^{5/4}} \, dx=\frac {45 b^2 \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{13/4}}-\frac {45 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{13/4}}+\frac {45 b^2}{32 a^3 \sqrt [4]{a+b x^4}}+\frac {9 b}{32 a^2 x^4 \sqrt [4]{a+b x^4}}-\frac {1}{8 a x^8 \sqrt [4]{a+b x^4}} \]
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Rule 44
Rule 53
Rule 65
Rule 209
Rule 212
Rule 272
Rule 304
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {1}{x^3 (a+b x)^{5/4}} \, dx,x,x^4\right ) \\ & = -\frac {1}{8 a x^8 \sqrt [4]{a+b x^4}}-\frac {(9 b) \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^{5/4}} \, dx,x,x^4\right )}{32 a} \\ & = -\frac {1}{8 a x^8 \sqrt [4]{a+b x^4}}+\frac {9 b}{32 a^2 x^4 \sqrt [4]{a+b x^4}}+\frac {\left (45 b^2\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{5/4}} \, dx,x,x^4\right )}{128 a^2} \\ & = \frac {45 b^2}{32 a^3 \sqrt [4]{a+b x^4}}-\frac {1}{8 a x^8 \sqrt [4]{a+b x^4}}+\frac {9 b}{32 a^2 x^4 \sqrt [4]{a+b x^4}}+\frac {\left (45 b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt [4]{a+b x}} \, dx,x,x^4\right )}{128 a^3} \\ & = \frac {45 b^2}{32 a^3 \sqrt [4]{a+b x^4}}-\frac {1}{8 a x^8 \sqrt [4]{a+b x^4}}+\frac {9 b}{32 a^2 x^4 \sqrt [4]{a+b x^4}}+\frac {(45 b) \text {Subst}\left (\int \frac {x^2}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )}{32 a^3} \\ & = \frac {45 b^2}{32 a^3 \sqrt [4]{a+b x^4}}-\frac {1}{8 a x^8 \sqrt [4]{a+b x^4}}+\frac {9 b}{32 a^2 x^4 \sqrt [4]{a+b x^4}}-\frac {\left (45 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{64 a^3}+\frac {\left (45 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{64 a^3} \\ & = \frac {45 b^2}{32 a^3 \sqrt [4]{a+b x^4}}-\frac {1}{8 a x^8 \sqrt [4]{a+b x^4}}+\frac {9 b}{32 a^2 x^4 \sqrt [4]{a+b x^4}}+\frac {45 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{13/4}}-\frac {45 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{13/4}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^9 \left (a+b x^4\right )^{5/4}} \, dx=\frac {\frac {2 \sqrt [4]{a} \left (-4 a^2+9 a b x^4+45 b^2 x^8\right )}{x^8 \sqrt [4]{a+b x^4}}+45 b^2 \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-45 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{13/4}} \]
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Time = 4.48 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02
method | result | size |
pseudoelliptic | \(\frac {\frac {45 \arctan \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right ) b^{2} x^{8} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{64}-\frac {45 \ln \left (\frac {-\left (b \,x^{4}+a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}{-\left (b \,x^{4}+a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}\right ) b^{2} x^{8} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{128}+\frac {45 b^{2} x^{8} a^{\frac {1}{4}}}{32}+\frac {9 a^{\frac {5}{4}} b \,x^{4}}{32}-\frac {a^{\frac {9}{4}}}{8}}{a^{\frac {13}{4}} x^{8} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}\) | \(127\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.28 \[ \int \frac {1}{x^9 \left (a+b x^4\right )^{5/4}} \, dx=-\frac {45 \, {\left (a^{3} b x^{12} + a^{4} x^{8}\right )} \left (\frac {b^{8}}{a^{13}}\right )^{\frac {1}{4}} \log \left (91125 \, a^{10} \left (\frac {b^{8}}{a^{13}}\right )^{\frac {3}{4}} + 91125 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{6}\right ) + 45 \, {\left (-i \, a^{3} b x^{12} - i \, a^{4} x^{8}\right )} \left (\frac {b^{8}}{a^{13}}\right )^{\frac {1}{4}} \log \left (91125 i \, a^{10} \left (\frac {b^{8}}{a^{13}}\right )^{\frac {3}{4}} + 91125 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{6}\right ) + 45 \, {\left (i \, a^{3} b x^{12} + i \, a^{4} x^{8}\right )} \left (\frac {b^{8}}{a^{13}}\right )^{\frac {1}{4}} \log \left (-91125 i \, a^{10} \left (\frac {b^{8}}{a^{13}}\right )^{\frac {3}{4}} + 91125 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{6}\right ) - 45 \, {\left (a^{3} b x^{12} + a^{4} x^{8}\right )} \left (\frac {b^{8}}{a^{13}}\right )^{\frac {1}{4}} \log \left (-91125 \, a^{10} \left (\frac {b^{8}}{a^{13}}\right )^{\frac {3}{4}} + 91125 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{6}\right ) - 4 \, {\left (45 \, b^{2} x^{8} + 9 \, a b x^{4} - 4 \, a^{2}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{128 \, {\left (a^{3} b x^{12} + a^{4} x^{8}\right )}} \]
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Result contains complex when optimal does not.
Time = 2.58 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.31 \[ \int \frac {1}{x^9 \left (a+b x^4\right )^{5/4}} \, dx=- \frac {\Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 b^{\frac {5}{4}} x^{13} \Gamma \left (\frac {17}{4}\right )} \]
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none
Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^9 \left (a+b x^4\right )^{5/4}} \, dx=\frac {45 \, {\left (b x^{4} + a\right )}^{2} b^{2} - 81 \, {\left (b x^{4} + a\right )} a b^{2} + 32 \, a^{2} b^{2}}{32 \, {\left ({\left (b x^{4} + a\right )}^{\frac {9}{4}} a^{3} - 2 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} a^{4} + {\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{5}\right )}} + \frac {45 \, b^{2} {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}}\right )}}{128 \, a^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (97) = 194\).
Time = 0.28 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.04 \[ \int \frac {1}{x^9 \left (a+b x^4\right )^{5/4}} \, dx=\frac {45 \, \sqrt {2} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a\right )^{\frac {1}{4}} a^{3}} + \frac {45 \, \sqrt {2} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a\right )^{\frac {1}{4}} a^{3}} + \frac {45 \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} b^{2} \log \left (\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right )}{256 \, a^{4}} + \frac {45 \, \sqrt {2} b^{2} \log \left (-\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right )}{256 \, \left (-a\right )^{\frac {1}{4}} a^{3}} + \frac {b^{2}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{3}} + \frac {13 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{2} - 17 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a b^{2}}{32 \, a^{3} b^{2} x^{8}} \]
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Time = 6.04 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^9 \left (a+b x^4\right )^{5/4}} \, dx=\frac {\frac {b^2}{a}-\frac {81\,b^2\,\left (b\,x^4+a\right )}{32\,a^2}+\frac {45\,b^2\,{\left (b\,x^4+a\right )}^2}{32\,a^3}}{{\left (b\,x^4+a\right )}^{9/4}-2\,a\,{\left (b\,x^4+a\right )}^{5/4}+a^2\,{\left (b\,x^4+a\right )}^{1/4}}+\frac {45\,b^2\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{64\,a^{13/4}}-\frac {45\,b^2\,\mathrm {atanh}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{64\,a^{13/4}} \]
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