Integrand size = 28, antiderivative size = 110 \[ \int \frac {b-a x^3+x^6}{x^6 \sqrt [4]{b x+a x^4}} \, dx=\frac {4 \left (-3 b+11 a x^3\right ) \left (b x+a x^4\right )^{3/4}}{63 b x^6}+\frac {2 \arctan \left (\frac {\sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 \sqrt [4]{a}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 \sqrt [4]{a}} \]
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Time = 0.16 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.52, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2077, 2036, 335, 281, 246, 218, 212, 209, 2041, 2039} \[ \int \frac {b-a x^3+x^6}{x^6 \sqrt [4]{b x+a x^4}} \, dx=\frac {2 \sqrt [4]{x} \sqrt [4]{a x^3+b} \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a} \sqrt [4]{a x^4+b x}}+\frac {2 \sqrt [4]{x} \sqrt [4]{a x^3+b} \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a} \sqrt [4]{a x^4+b x}}-\frac {4 \left (a x^4+b x\right )^{3/4}}{21 x^6}+\frac {44 a \left (a x^4+b x\right )^{3/4}}{63 b x^3} \]
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 281
Rule 335
Rule 2036
Rule 2039
Rule 2041
Rule 2077
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt [4]{b x+a x^4}}+\frac {b}{x^6 \sqrt [4]{b x+a x^4}}-\frac {a}{x^3 \sqrt [4]{b x+a x^4}}\right ) \, dx \\ & = -\left (a \int \frac {1}{x^3 \sqrt [4]{b x+a x^4}} \, dx\right )+b \int \frac {1}{x^6 \sqrt [4]{b x+a x^4}} \, dx+\int \frac {1}{\sqrt [4]{b x+a x^4}} \, dx \\ & = -\frac {4 \left (b x+a x^4\right )^{3/4}}{21 x^6}+\frac {4 a \left (b x+a x^4\right )^{3/4}}{9 b x^3}-\frac {1}{7} (4 a) \int \frac {1}{x^3 \sqrt [4]{b x+a x^4}} \, dx+\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{b+a x^3}} \, dx}{\sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b x+a x^4\right )^{3/4}}{21 x^6}+\frac {44 a \left (b x+a x^4\right )^{3/4}}{63 b x^3}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b x+a x^4\right )^{3/4}}{21 x^6}+\frac {44 a \left (b x+a x^4\right )^{3/4}}{63 b x^3}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4}} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b x+a x^4\right )^{3/4}}{21 x^6}+\frac {44 a \left (b x+a x^4\right )^{3/4}}{63 b x^3}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b x+a x^4\right )^{3/4}}{21 x^6}+\frac {44 a \left (b x+a x^4\right )^{3/4}}{63 b x^3}+\frac {\left (2 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}}+\frac {\left (2 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b x+a x^4\right )^{3/4}}{21 x^6}+\frac {44 a \left (b x+a x^4\right )^{3/4}}{63 b x^3}+\frac {2 \sqrt [4]{x} \sqrt [4]{b+a x^3} \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{a} \sqrt [4]{b x+a x^4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{b+a x^3} \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{a} \sqrt [4]{b x+a x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.75 \[ \int \frac {b-a x^3+x^6}{x^6 \sqrt [4]{b x+a x^4}} \, dx=\frac {4 \left (-3 b^2+8 a b x^3+11 a^2 x^6+21 b x^6 \sqrt [4]{1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4},\frac {5}{4},-\frac {a x^3}{b}\right )\right )}{63 b x^5 \sqrt [4]{x \left (b+a x^3\right )}} \]
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Time = 0.38 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.08
method | result | size |
pseudoelliptic | \(\frac {44 a^{\frac {5}{4}} {\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {3}{4}} x^{3}-42 \arctan \left (\frac {{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b \,x^{6}+21 \ln \left (\frac {a^{\frac {1}{4}} x +{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}\right ) b \,x^{6}-12 b \,a^{\frac {1}{4}} {\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {3}{4}}}{63 x^{6} a^{\frac {1}{4}} b}\) | \(119\) |
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Timed out. \[ \int \frac {b-a x^3+x^6}{x^6 \sqrt [4]{b x+a x^4}} \, dx=\text {Timed out} \]
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\[ \int \frac {b-a x^3+x^6}{x^6 \sqrt [4]{b x+a x^4}} \, dx=\int \frac {- a x^{3} + b + x^{6}}{x^{6} \sqrt [4]{x \left (a x^{3} + b\right )}}\, dx \]
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\[ \int \frac {b-a x^3+x^6}{x^6 \sqrt [4]{b x+a x^4}} \, dx=\int { \frac {x^{6} - a x^{3} + b}{{\left (a x^{4} + b x\right )}^{\frac {1}{4}} x^{6}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (90) = 180\).
Time = 0.30 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.93 \[ \int \frac {b-a x^3+x^6}{x^6 \sqrt [4]{b x+a x^4}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, \left (-a\right )^{\frac {1}{4}}} - \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, \left (-a\right )^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{6 \, \left (-a\right )^{\frac {1}{4}}} + \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{6 \, a} - \frac {4 \, {\left (3 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{4}} b^{6} - 14 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{4}} a b^{6}\right )}}{63 \, b^{7}} \]
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Time = 5.91 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.70 \[ \int \frac {b-a x^3+x^6}{x^6 \sqrt [4]{b x+a x^4}} \, dx=\frac {4\,x\,{\left (\frac {a\,x^3}{b}+1\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ -\frac {a\,x^3}{b}\right )}{3\,{\left (a\,x^4+b\,x\right )}^{1/4}}-\frac {4\,{\left (a\,x^4+b\,x\right )}^{3/4}}{21\,x^6}+\frac {44\,a\,{\left (a\,x^4+b\,x\right )}^{3/4}}{63\,b\,x^3} \]
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