Integrand size = 15, antiderivative size = 103 \[ \int x^8 \left (b+a x^4\right )^{3/4} \, dx=\frac {\left (b+a x^4\right )^{3/4} \left (-15 b^2 x+12 a b x^5+32 a^2 x^9\right )}{384 a^2}+\frac {5 b^3 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{256 a^{9/4}}+\frac {5 b^3 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{256 a^{9/4}} \]
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Time = 0.03 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.21, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {285, 327, 246, 218, 212, 209} \[ \int x^8 \left (b+a x^4\right )^{3/4} \, dx=\frac {5 b^3 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{256 a^{9/4}}+\frac {5 b^3 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{256 a^{9/4}}-\frac {5 b^2 x \left (a x^4+b\right )^{3/4}}{128 a^2}+\frac {1}{12} x^9 \left (a x^4+b\right )^{3/4}+\frac {b x^5 \left (a x^4+b\right )^{3/4}}{32 a} \]
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 285
Rule 327
Rubi steps \begin{align*} \text {integral}& = \frac {1}{12} x^9 \left (b+a x^4\right )^{3/4}+\frac {1}{4} b \int \frac {x^8}{\sqrt [4]{b+a x^4}} \, dx \\ & = \frac {b x^5 \left (b+a x^4\right )^{3/4}}{32 a}+\frac {1}{12} x^9 \left (b+a x^4\right )^{3/4}-\frac {\left (5 b^2\right ) \int \frac {x^4}{\sqrt [4]{b+a x^4}} \, dx}{32 a} \\ & = -\frac {5 b^2 x \left (b+a x^4\right )^{3/4}}{128 a^2}+\frac {b x^5 \left (b+a x^4\right )^{3/4}}{32 a}+\frac {1}{12} x^9 \left (b+a x^4\right )^{3/4}+\frac {\left (5 b^3\right ) \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx}{128 a^2} \\ & = -\frac {5 b^2 x \left (b+a x^4\right )^{3/4}}{128 a^2}+\frac {b x^5 \left (b+a x^4\right )^{3/4}}{32 a}+\frac {1}{12} x^9 \left (b+a x^4\right )^{3/4}+\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{128 a^2} \\ & = -\frac {5 b^2 x \left (b+a x^4\right )^{3/4}}{128 a^2}+\frac {b x^5 \left (b+a x^4\right )^{3/4}}{32 a}+\frac {1}{12} x^9 \left (b+a x^4\right )^{3/4}+\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{256 a^2}+\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{256 a^2} \\ & = -\frac {5 b^2 x \left (b+a x^4\right )^{3/4}}{128 a^2}+\frac {b x^5 \left (b+a x^4\right )^{3/4}}{32 a}+\frac {1}{12} x^9 \left (b+a x^4\right )^{3/4}+\frac {5 b^3 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{256 a^{9/4}}+\frac {5 b^3 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{256 a^{9/4}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.95 \[ \int x^8 \left (b+a x^4\right )^{3/4} \, dx=\frac {2 \sqrt [4]{a} x \left (b+a x^4\right )^{3/4} \left (-15 b^2+12 a b x^4+32 a^2 x^8\right )+15 b^3 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+15 b^3 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{768 a^{9/4}} \]
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Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.22
method | result | size |
pseudoelliptic | \(\frac {128 a^{\frac {9}{4}} \left (a \,x^{4}+b \right )^{\frac {3}{4}} x^{9}+48 a^{\frac {5}{4}} b \,x^{5} \left (a \,x^{4}+b \right )^{\frac {3}{4}}-60 b^{2} x \,a^{\frac {1}{4}} \left (a \,x^{4}+b \right )^{\frac {3}{4}}-30 \arctan \left (\frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b^{3}+15 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right ) b^{3}}{1536 a^{\frac {9}{4}}}\) | \(126\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.27 \[ \int x^8 \left (b+a x^4\right )^{3/4} \, dx=\frac {15 \, \left (\frac {b^{12}}{a^{9}}\right )^{\frac {1}{4}} a^{2} \log \left (\frac {125 \, {\left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} b^{9} + \left (\frac {b^{12}}{a^{9}}\right )^{\frac {3}{4}} a^{7} x\right )}}{x}\right ) - 15 i \, \left (\frac {b^{12}}{a^{9}}\right )^{\frac {1}{4}} a^{2} \log \left (\frac {125 \, {\left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} b^{9} + i \, \left (\frac {b^{12}}{a^{9}}\right )^{\frac {3}{4}} a^{7} x\right )}}{x}\right ) + 15 i \, \left (\frac {b^{12}}{a^{9}}\right )^{\frac {1}{4}} a^{2} \log \left (\frac {125 \, {\left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} b^{9} - i \, \left (\frac {b^{12}}{a^{9}}\right )^{\frac {3}{4}} a^{7} x\right )}}{x}\right ) - 15 \, \left (\frac {b^{12}}{a^{9}}\right )^{\frac {1}{4}} a^{2} \log \left (\frac {125 \, {\left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} b^{9} - \left (\frac {b^{12}}{a^{9}}\right )^{\frac {3}{4}} a^{7} x\right )}}{x}\right ) + 4 \, {\left (32 \, a^{2} x^{9} + 12 \, a b x^{5} - 15 \, b^{2} x\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}}{1536 \, a^{2}} \]
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Result contains complex when optimal does not.
Time = 3.90 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.38 \[ \int x^8 \left (b+a x^4\right )^{3/4} \, dx=\frac {b^{\frac {3}{4}} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (83) = 166\).
Time = 0.29 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.83 \[ \int x^8 \left (b+a x^4\right )^{3/4} \, dx=-\frac {5 \, b^{3} {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )}}{512 \, a^{2}} - \frac {\frac {5 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} a^{2} b^{3}}{x^{3}} + \frac {42 \, {\left (a x^{4} + b\right )}^{\frac {7}{4}} a b^{3}}{x^{7}} - \frac {15 \, {\left (a x^{4} + b\right )}^{\frac {11}{4}} b^{3}}{x^{11}}}{384 \, {\left (a^{5} - \frac {3 \, {\left (a x^{4} + b\right )} a^{4}}{x^{4}} + \frac {3 \, {\left (a x^{4} + b\right )}^{2} a^{3}}{x^{8}} - \frac {{\left (a x^{4} + b\right )}^{3} a^{2}}{x^{12}}\right )}} \]
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\[ \int x^8 \left (b+a x^4\right )^{3/4} \, dx=\int { {\left (a x^{4} + b\right )}^{\frac {3}{4}} x^{8} \,d x } \]
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Timed out. \[ \int x^8 \left (b+a x^4\right )^{3/4} \, dx=\int x^8\,{\left (a\,x^4+b\right )}^{3/4} \,d x \]
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