Integrand size = 35, antiderivative size = 136 \[ \int \frac {b+a x^3}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\frac {4 \left (3 b+10 a x^3\right ) \left (b x+a x^4\right )^{3/4}}{63 b^2 x^6}-\frac {2\ 2^{3/4} a^{7/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 b^2}-\frac {2\ 2^{3/4} a^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 b^2} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 3.41 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2081, 477, 476, 525, 524} \[ \int \frac {b+a x^3}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {\operatorname {Gamma}\left (-\frac {3}{4}\right ) \left (a x^3+b\right )^2 \left (\left (4 a^2 x^6-7 a b x^3+3 b^2\right ) \operatorname {Hypergeometric2F1}\left (1,1,\frac {1}{4},-\frac {2 a x^3}{b-a x^3}\right )+32 a x^3 \left (a x^3+b\right ) \operatorname {Hypergeometric2F1}\left (2,2,\frac {5}{4},-\frac {2 a x^3}{b-a x^3}\right )\right )}{21 b^2 x^5 \operatorname {Gamma}\left (\frac {1}{4}\right ) \left (b-a x^3\right )^2 \sqrt [4]{a x^4+b x}} \]
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Rule 476
Rule 477
Rule 524
Rule 525
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {\left (b+a x^3\right )^{3/4}}{x^{25/4} \left (-b+a x^3\right )} \, dx}{\sqrt [4]{b x+a x^4}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {\left (b+a x^{12}\right )^{3/4}}{x^{22} \left (-b+a x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^4}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {\left (b+a x^4\right )^{3/4}}{x^8 \left (-b+a x^4\right )} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{b x+a x^4}} \\ & = \frac {\left (4 \sqrt [4]{x} \left (b+a x^3\right )\right ) \text {Subst}\left (\int \frac {\left (1+\frac {a x^4}{b}\right )^{3/4}}{x^8 \left (-b+a x^4\right )} \, dx,x,x^{3/4}\right )}{3 \left (1+\frac {a x^3}{b}\right )^{3/4} \sqrt [4]{b x+a x^4}} \\ & = -\frac {\left (b+a x^3\right )^2 \operatorname {Gamma}\left (-\frac {3}{4}\right ) \left (\left (3 b^2-7 a b x^3+4 a^2 x^6\right ) \operatorname {Hypergeometric2F1}\left (1,1,\frac {1}{4},-\frac {2 a x^3}{b-a x^3}\right )+32 a x^3 \left (b+a x^3\right ) \operatorname {Hypergeometric2F1}\left (2,2,\frac {5}{4},-\frac {2 a x^3}{b-a x^3}\right )\right )}{21 b^2 x^5 \left (b-a x^3\right )^2 \sqrt [4]{b x+a x^4} \operatorname {Gamma}\left (\frac {1}{4}\right )} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 14.67 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90 \[ \int \frac {b+a x^3}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {\left (x \left (b+a x^3\right )\right )^{7/4} \operatorname {Gamma}\left (-\frac {3}{4}\right ) \left (\left (3 b^2-7 a b x^3+4 a^2 x^6\right ) \operatorname {Hypergeometric2F1}\left (1,1,\frac {1}{4},-\frac {2 a x^3}{b-a x^3}\right )+32 a x^3 \left (b+a x^3\right ) \operatorname {Hypergeometric2F1}\left (2,2,\frac {5}{4},-\frac {2 a x^3}{b-a x^3}\right )\right )}{21 b^2 x^7 \left (b-a x^3\right )^2 \operatorname {Gamma}\left (\frac {1}{4}\right )} \]
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Time = 0.56 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.04
method | result | size |
pseudoelliptic | \(\frac {\left (42 \arctan \left (\frac {{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,a^{\frac {1}{4}}}\right ) a^{2} x^{6}-21 \ln \left (\frac {x 2^{\frac {1}{4}} a^{\frac {1}{4}}+{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}{-x 2^{\frac {1}{4}} a^{\frac {1}{4}}+{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}\right ) a^{2} x^{6}+20 {\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {3}{4}} a^{\frac {5}{4}} x^{3} 2^{\frac {1}{4}}+6 b 2^{\frac {1}{4}} a^{\frac {1}{4}} {\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {3}{4}}\right ) 2^{\frac {3}{4}}}{63 x^{6} a^{\frac {1}{4}} b^{2}}\) | \(142\) |
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Result contains complex when optimal does not.
Time = 82.93 (sec) , antiderivative size = 616, normalized size of antiderivative = 4.53 \[ \int \frac {b+a x^3}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {21 \cdot 8^{\frac {1}{4}} b^{2} x^{6} \left (\frac {a^{7}}{b^{8}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a^{2} b^{4} x^{2} \sqrt {\frac {a^{7}}{b^{8}}} + 8^{\frac {3}{4}} \sqrt {a x^{4} + b x} b^{6} x \left (\frac {a^{7}}{b^{8}}\right )^{\frac {3}{4}} + 4 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a^{5} + 8^{\frac {1}{4}} {\left (3 \, a^{4} b^{2} x^{3} + a^{3} b^{3}\right )} \left (\frac {a^{7}}{b^{8}}\right )^{\frac {1}{4}}}{a x^{3} - b}\right ) + 21 i \cdot 8^{\frac {1}{4}} b^{2} x^{6} \left (\frac {a^{7}}{b^{8}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a^{2} b^{4} x^{2} \sqrt {\frac {a^{7}}{b^{8}}} + i \cdot 8^{\frac {3}{4}} \sqrt {a x^{4} + b x} b^{6} x \left (\frac {a^{7}}{b^{8}}\right )^{\frac {3}{4}} - 4 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a^{5} + 8^{\frac {1}{4}} {\left (-3 i \, a^{4} b^{2} x^{3} - i \, a^{3} b^{3}\right )} \left (\frac {a^{7}}{b^{8}}\right )^{\frac {1}{4}}}{a x^{3} - b}\right ) - 21 i \cdot 8^{\frac {1}{4}} b^{2} x^{6} \left (\frac {a^{7}}{b^{8}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a^{2} b^{4} x^{2} \sqrt {\frac {a^{7}}{b^{8}}} - i \cdot 8^{\frac {3}{4}} \sqrt {a x^{4} + b x} b^{6} x \left (\frac {a^{7}}{b^{8}}\right )^{\frac {3}{4}} - 4 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a^{5} + 8^{\frac {1}{4}} {\left (3 i \, a^{4} b^{2} x^{3} + i \, a^{3} b^{3}\right )} \left (\frac {a^{7}}{b^{8}}\right )^{\frac {1}{4}}}{a x^{3} - b}\right ) - 21 \cdot 8^{\frac {1}{4}} b^{2} x^{6} \left (\frac {a^{7}}{b^{8}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a^{2} b^{4} x^{2} \sqrt {\frac {a^{7}}{b^{8}}} - 8^{\frac {3}{4}} \sqrt {a x^{4} + b x} b^{6} x \left (\frac {a^{7}}{b^{8}}\right )^{\frac {3}{4}} + 4 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a^{5} - 8^{\frac {1}{4}} {\left (3 \, a^{4} b^{2} x^{3} + a^{3} b^{3}\right )} \left (\frac {a^{7}}{b^{8}}\right )^{\frac {1}{4}}}{a x^{3} - b}\right ) - 8 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} {\left (10 \, a x^{3} + 3 \, b\right )}}{126 \, b^{2} x^{6}} \]
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\[ \int \frac {b+a x^3}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\int \frac {a x^{3} + b}{x^{6} \sqrt [4]{x \left (a x^{3} + b\right )} \left (a x^{3} - b\right )}\, dx \]
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\[ \int \frac {b+a x^3}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\int { \frac {a x^{3} + b}{{\left (a x^{4} + b x\right )}^{\frac {1}{4}} {\left (a x^{3} - b\right )} x^{6}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (108) = 216\).
Time = 0.31 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.71 \[ \int \frac {b+a x^3}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {2 \cdot 2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} a \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, b^{2}} - \frac {2 \cdot 2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} a \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, b^{2}} + \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} a \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{3 \, b^{2}} - \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} a \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{3 \, b^{2}} + \frac {4 \, {\left (3 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{4}} b^{12} + 7 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{4}} a b^{12}\right )}}{63 \, b^{14}} \]
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Timed out. \[ \int \frac {b+a x^3}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\int \frac {a\,x^3+b}{x^6\,{\left (a\,x^4+b\,x\right )}^{1/4}\,\left (b-a\,x^3\right )} \,d x \]
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