Integrand size = 35, antiderivative size = 149 \[ \int \frac {b+a x^6}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\frac {4 \left (b+a x^3\right ) \left (b x+a x^4\right )^{3/4}}{21 b^2 x^6}-\frac {2^{3/4} \left (a^{7/4}+a^{3/4} b\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 b^2}-\frac {2^{3/4} \left (a^{7/4}+a^{3/4} b\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 b^2} \]
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Time = 0.74 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.70, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {2081, 6857, 277, 270, 477, 476, 508, 472, 218, 212, 209} \[ \int \frac {b+a x^6}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {2^{3/4} a^{3/4} \sqrt [4]{x} (a+b) \sqrt [4]{a x^3+b} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 b^2 \sqrt [4]{a x^4+b x}}-\frac {2^{3/4} a^{3/4} \sqrt [4]{x} (a+b) \sqrt [4]{a x^3+b} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 b^2 \sqrt [4]{a x^4+b x}}+\frac {4 (a+b) \left (a x^3+b\right )^2}{21 a b^2 x^5 \sqrt [4]{a x^4+b x}}-\frac {4 \left (a x^3+b\right )}{21 a x^5 \sqrt [4]{a x^4+b x}}-\frac {4 \left (a x^3+b\right )}{21 b x^2 \sqrt [4]{a x^4+b x}} \]
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Rule 209
Rule 212
Rule 218
Rule 270
Rule 277
Rule 472
Rule 476
Rule 477
Rule 508
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {b+a x^6}{x^{25/4} \left (-b+a x^3\right ) \sqrt [4]{b+a x^3}} \, dx}{\sqrt [4]{b x+a x^4}} \\ & = \frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \left (\frac {b}{a x^{25/4} \sqrt [4]{b+a x^3}}+\frac {1}{x^{13/4} \sqrt [4]{b+a x^3}}+\frac {a b+b^2}{a x^{25/4} \left (-b+a x^3\right ) \sqrt [4]{b+a x^3}}\right ) \, dx}{\sqrt [4]{b x+a x^4}} \\ & = \frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{x^{13/4} \sqrt [4]{b+a x^3}} \, dx}{\sqrt [4]{b x+a x^4}}+\frac {\left (b \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{x^{25/4} \sqrt [4]{b+a x^3}} \, dx}{a \sqrt [4]{b x+a x^4}}+\frac {\left (b (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{x^{25/4} \left (-b+a x^3\right ) \sqrt [4]{b+a x^3}} \, dx}{a \sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b+a x^3\right )}{21 a x^5 \sqrt [4]{b x+a x^4}}-\frac {4 \left (b+a x^3\right )}{9 b x^2 \sqrt [4]{b x+a x^4}}-\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{x^{13/4} \sqrt [4]{b+a x^3}} \, dx}{7 \sqrt [4]{b x+a x^4}}+\frac {\left (4 b (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{x^{22} \left (-b+a x^{12}\right ) \sqrt [4]{b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{a \sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b+a x^3\right )}{21 a x^5 \sqrt [4]{b x+a x^4}}-\frac {4 \left (b+a x^3\right )}{21 b x^2 \sqrt [4]{b x+a x^4}}+\frac {\left (4 b (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{x^8 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,x^{3/4}\right )}{3 a \sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b+a x^3\right )}{21 a x^5 \sqrt [4]{b x+a x^4}}-\frac {4 \left (b+a x^3\right )}{21 b x^2 \sqrt [4]{b x+a x^4}}+\frac {\left (4 (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {\left (1-a x^4\right )^2}{x^8 \left (-b+2 a b x^4\right )} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 a b \sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b+a x^3\right )}{21 a x^5 \sqrt [4]{b x+a x^4}}-\frac {4 \left (b+a x^3\right )}{21 b x^2 \sqrt [4]{b x+a x^4}}+\frac {\left (4 (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \left (-\frac {1}{b x^8}+\frac {a^2}{b \left (-1+2 a x^4\right )}\right ) \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 a b \sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b+a x^3\right )}{21 a x^5 \sqrt [4]{b x+a x^4}}-\frac {4 \left (b+a x^3\right )}{21 b x^2 \sqrt [4]{b x+a x^4}}+\frac {4 (a+b) \left (b+a x^3\right )^2}{21 a b^2 x^5 \sqrt [4]{b x+a x^4}}+\frac {\left (4 a (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{-1+2 a x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 b^2 \sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b+a x^3\right )}{21 a x^5 \sqrt [4]{b x+a x^4}}-\frac {4 \left (b+a x^3\right )}{21 b x^2 \sqrt [4]{b x+a x^4}}+\frac {4 (a+b) \left (b+a x^3\right )^2}{21 a b^2 x^5 \sqrt [4]{b x+a x^4}}-\frac {\left (2 a (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 b^2 \sqrt [4]{b x+a x^4}}-\frac {\left (2 a (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 b^2 \sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b+a x^3\right )}{21 a x^5 \sqrt [4]{b x+a x^4}}-\frac {4 \left (b+a x^3\right )}{21 b x^2 \sqrt [4]{b x+a x^4}}+\frac {4 (a+b) \left (b+a x^3\right )^2}{21 a b^2 x^5 \sqrt [4]{b x+a x^4}}-\frac {2^{3/4} a^{3/4} (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 b^2 \sqrt [4]{b x+a x^4}}-\frac {2^{3/4} a^{3/4} (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 b^2 \sqrt [4]{b x+a x^4}} \\ \end{align*}
Time = 15.85 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.79 \[ \int \frac {b+a x^6}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\frac {\frac {4 \left (x \left (b+a x^3\right )\right )^{7/4}}{x}-7\ 2^{3/4} a^{3/4} (a+b) x^6 \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{x \left (b+a x^3\right )}}\right )-7\ 2^{3/4} a^{3/4} (a+b) x^6 \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{x \left (b+a x^3\right )}}\right )}{21 b^2 x^6} \]
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Time = 0.70 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(\frac {-7 \,2^{\frac {3}{4}} x^{7} \left (a^{\frac {7}{4}}+a^{\frac {3}{4}} b \right ) \ln \left (\frac {x 2^{\frac {1}{4}} a^{\frac {1}{4}}+{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}{-x 2^{\frac {1}{4}} a^{\frac {1}{4}}+{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}\right )+14 \,2^{\frac {3}{4}} x^{7} \left (a^{\frac {7}{4}}+a^{\frac {3}{4}} b \right ) \arctan \left (\frac {{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,a^{\frac {1}{4}}}\right )+8 {\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {7}{4}}}{42 b^{2} x^{7}}\) | \(125\) |
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Timed out. \[ \int \frac {b+a x^6}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\text {Timed out} \]
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\[ \int \frac {b+a x^6}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\int \frac {a x^{6} + b}{x^{6} \sqrt [4]{x \left (a x^{3} + b\right )} \left (a x^{3} - b\right )}\, dx \]
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\[ \int \frac {b+a x^6}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\int { \frac {a x^{6} + b}{{\left (a x^{4} + b x\right )}^{\frac {1}{4}} {\left (a x^{3} - b\right )} x^{6}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (117) = 234\).
Time = 0.31 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.83 \[ \int \frac {b+a x^6}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{6 \, b^{2}} - \frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{6 \, b^{2}} + \frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{12 \, b^{2}} - \frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{12 \, b^{2}} + \frac {4 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{4}}}{21 \, b^{2}} \]
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Timed out. \[ \int \frac {b+a x^6}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\int \frac {a\,x^6+b}{x^6\,{\left (a\,x^4+b\,x\right )}^{1/4}\,\left (b-a\,x^3\right )} \,d x \]
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