Integrand size = 24, antiderivative size = 109 \[ \int \frac {1}{\left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\frac {2^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (-b x+a x^4\right )^{3/4}}{-b+a x^3}\right )}{3 \sqrt [4]{a} b}+\frac {2^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (-b x+a x^4\right )^{3/4}}{-b+a x^3}\right )}{3 \sqrt [4]{a} b} \]
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Time = 0.13 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.46, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2081, 477, 476, 385, 218, 212, 209} \[ \int \frac {1}{\left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{a x^3-b} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}\right )}{3 \sqrt [4]{a} b \sqrt [4]{a x^4-b x}}+\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{a x^3-b} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}\right )}{3 \sqrt [4]{a} b \sqrt [4]{a x^4-b x}} \]
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Rule 209
Rule 212
Rule 218
Rule 385
Rule 476
Rule 477
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{-b+a x^3} \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 {x^2}{\sqrt [4]{-b+a x^{12}} \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 {1}{\sqrt [4]{-b+a x^4} \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} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{b-2 a b x^4} \, 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 {2} \sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 b \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 {2} \sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 b \sqrt [4]{-b x+a x^4}} \\ & = \frac {2^{3/4} \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 \sqrt [4]{a} b \sqrt [4]{-b x+a x^4}}+\frac {2^{3/4} \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 \sqrt [4]{a} b \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 = 76, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\frac {4 x \sqrt [4]{1-\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4},\frac {5}{4},\frac {2 a x^3}{b+a x^3}\right )}{3 b \sqrt [4]{1+\frac {a x^3}{b}} \sqrt [4]{-b x+a x^4}} \]
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Time = 0.44 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.85
method | result | size |
pseudoelliptic | \(\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,a^{\frac {1}{4}}}\right )+\ln \left (\frac {-x 2^{\frac {1}{4}} a^{\frac {1}{4}}-{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}}{x 2^{\frac {1}{4}} a^{\frac {1}{4}}-{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}}\right )\right )}{6 a^{\frac {1}{4}} b}\) | \(93\) |
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Result contains complex when optimal does not.
Time = 57.53 (sec) , antiderivative size = 524, normalized size of antiderivative = 4.81 \[ \int \frac {1}{\left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\frac {1}{6} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b x} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {1}{a b^{4}}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{3} - b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + 2 \, {\left (a x^{4} - b x\right )}^{\frac {3}{4}}}{a x^{3} + b}\right ) - \frac {1}{6} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b x} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {1}{a b^{4}}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{3} - b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} - 2 \, {\left (a x^{4} - b x\right )}^{\frac {3}{4}}}{a x^{3} + b}\right ) - \frac {1}{6} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b x} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {1}{a b^{4}}} - \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 i \, a b x^{3} - i \, b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + 2 \, {\left (a x^{4} - b x\right )}^{\frac {3}{4}}}{a x^{3} + b}\right ) + \frac {1}{6} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-4 i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b x} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {1}{a b^{4}}} - \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (-3 i \, a b x^{3} + i \, b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + 2 \, {\left (a x^{4} - b x\right )}^{\frac {3}{4}}}{a x^{3} + b}\right ) \]
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\[ \int \frac {1}{\left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\int \frac {1}{\sqrt [4]{x \left (a x^{3} - b\right )} \left (a x^{3} + b\right )}\, dx \]
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\[ \int \frac {1}{\left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\int { \frac {1}{{\left (a x^{4} - b x\right )}^{\frac {1}{4}} {\left (a x^{3} + b\right )}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (85) = 170\).
Time = 0.30 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.92 \[ \int \frac {1}{\left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, a b} + \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, a b} + \frac {2^{\frac {1}{4}} \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a - \frac {b}{x^{3}}}\right )}{6 \, \left (-a\right )^{\frac {1}{4}} b} + \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a - \frac {b}{x^{3}}}\right )}{6 \, a b} \]
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Timed out. \[ \int \frac {1}{\left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\int \frac {1}{{\left (a\,x^4-b\,x\right )}^{1/4}\,\left (a\,x^3+b\right )} \,d x \]
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