Integrand size = 30, antiderivative size = 107 \[ \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (-b^2+a^2 x^8\right )} \, dx=\frac {x^3}{6 a b \left (b+a x^4\right )^{3/4}}+\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{4\ 2^{3/4} a^{7/4} b}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{4\ 2^{3/4} a^{7/4} b} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.41, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1493, 525, 524} \[ \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (-b^2+a^2 x^8\right )} \, dx=-\frac {x^7 \operatorname {Hypergeometric2F1}\left (1,\frac {7}{4},\frac {11}{4},\frac {2 a x^4}{a x^4+b}\right )}{7 b \left (a x^4+b\right )^{7/4}} \]
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Rule 524
Rule 525
Rule 1493
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^6}{\left (-b+a x^4\right ) \left (b+a x^4\right )^{7/4}} \, dx \\ & = \frac {\left (1+\frac {a x^4}{b}\right )^{3/4} \int \frac {x^6}{\left (-b+a x^4\right ) \left (1+\frac {a x^4}{b}\right )^{7/4}} \, dx}{b \left (b+a x^4\right )^{3/4}} \\ & = -\frac {x^7 \operatorname {Hypergeometric2F1}\left (1,\frac {7}{4},\frac {11}{4},\frac {2 a x^4}{b+a x^4}\right )}{7 b \left (b+a x^4\right )^{7/4}} \\ \end{align*}
Time = 0.71 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.90 \[ \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (-b^2+a^2 x^8\right )} \, dx=\frac {\frac {4 a^{3/4} x^3}{\left (b+a x^4\right )^{3/4}}+3 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )-3 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{24 a^{7/4} b} \]
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Time = 0.40 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.17
method | result | size |
pseudoelliptic | \(\frac {-6 \arctan \left (\frac {2^{\frac {3}{4}} \left (a \,x^{4}+b \right )^{\frac {1}{4}}}{2 a^{\frac {1}{4}} x}\right ) 2^{\frac {1}{4}} a^{\frac {1}{4}} \left (a \,x^{4}+b \right )^{\frac {3}{4}}-3 \ln \left (\frac {-x 2^{\frac {1}{4}} a^{\frac {1}{4}}-\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x 2^{\frac {1}{4}} a^{\frac {1}{4}}-\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}} a^{\frac {1}{4}} \left (a \,x^{4}+b \right )^{\frac {3}{4}}+8 a \,x^{3}}{48 a^{2} \left (a \,x^{4}+b \right )^{\frac {3}{4}} b}\) | \(125\) |
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Timed out. \[ \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (-b^2+a^2 x^8\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (-b^2+a^2 x^8\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (-b^2+a^2 x^8\right )} \, dx=\int { \frac {x^{6}}{{\left (a^{2} x^{8} - b^{2}\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (-b^2+a^2 x^8\right )} \, dx=\int { \frac {x^{6}}{{\left (a^{2} x^{8} - b^{2}\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (-b^2+a^2 x^8\right )} \, dx=-\int \frac {x^6}{\left (b^2-a^2\,x^8\right )\,{\left (a\,x^4+b\right )}^{3/4}} \,d x \]
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