Integrand size = 33, antiderivative size = 123 \[ \int \frac {b+a x^6}{x^3 \left (b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {4 \left (b+4 a x^3+3 b x^3\right ) \left (b x+a x^4\right )^{3/4}}{9 b x^3 \left (b+a x^3\right )}+\frac {2 \arctan \left (\frac {\sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 \sqrt [4]{a}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 \sqrt [4]{a}} \]
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Time = 0.22 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.49, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2081, 1503, 277, 270, 294, 335, 281, 246, 218, 212, 209} \[ \int \frac {b+a x^6}{x^3 \left (b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\frac {2 \sqrt [4]{x} \sqrt [4]{a x^3+b} \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a} \sqrt [4]{a x^4+b x}}+\frac {2 \sqrt [4]{x} \sqrt [4]{a x^3+b} \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a} \sqrt [4]{a x^4+b x}}-\frac {16 a x}{9 b \sqrt [4]{a x^4+b x}}-\frac {4 x}{3 \sqrt [4]{a x^4+b x}}-\frac {4}{9 x^2 \sqrt [4]{a x^4+b x}} \]
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 270
Rule 277
Rule 281
Rule 294
Rule 335
Rule 1503
Rule 2081
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^{13/4} \left (b+a x^3\right )^{5/4}} \, dx}{\sqrt [4]{b x+a x^4}} \\ & = \frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \left (\frac {b}{x^{13/4} \left (b+a x^3\right )^{5/4}}+\frac {a x^{11/4}}{\left (b+a x^3\right )^{5/4}}\right ) \, dx}{\sqrt [4]{b x+a x^4}} \\ & = \frac {\left (a \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {x^{11/4}}{\left (b+a x^3\right )^{5/4}} \, 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^{13/4} \left (b+a x^3\right )^{5/4}} \, dx}{\sqrt [4]{b x+a x^4}} \\ & = -\frac {4}{9 x^2 \sqrt [4]{b x+a x^4}}-\frac {4 x}{3 \sqrt [4]{b x+a x^4}}+\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 {\left (4 a \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{\sqrt [4]{x} \left (b+a x^3\right )^{5/4}} \, dx}{3 \sqrt [4]{b x+a x^4}} \\ & = -\frac {4}{9 x^2 \sqrt [4]{b x+a x^4}}-\frac {4 x}{3 \sqrt [4]{b x+a x^4}}-\frac {16 a x}{9 b \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}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^4}} \\ & = -\frac {4}{9 x^2 \sqrt [4]{b x+a x^4}}-\frac {4 x}{3 \sqrt [4]{b x+a x^4}}-\frac {16 a x}{9 b \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}} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{b x+a x^4}} \\ & = -\frac {4}{9 x^2 \sqrt [4]{b x+a x^4}}-\frac {4 x}{3 \sqrt [4]{b x+a x^4}}-\frac {16 a x}{9 b \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}{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}{9 x^2 \sqrt [4]{b x+a x^4}}-\frac {4 x}{3 \sqrt [4]{b x+a x^4}}-\frac {16 a x}{9 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 {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}{9 x^2 \sqrt [4]{b x+a x^4}}-\frac {4 x}{3 \sqrt [4]{b x+a x^4}}-\frac {16 a x}{9 b \sqrt [4]{b x+a x^4}}+\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 = 10.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.59 \[ \int \frac {b+a x^6}{x^3 \left (b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\frac {4 \left (-5 \left (b+4 a x^3\right )+3 a x^6 \sqrt [4]{1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {5}{4},\frac {9}{4},-\frac {a x^3}{b}\right )\right )}{45 b x^2 \sqrt [4]{x \left (b+a x^3\right )}} \]
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Time = 0.47 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.02
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\frac {8 x^{3} a^{\frac {5}{4}}}{3}+\frac {b \left (\left (12 x^{3}+4\right ) a^{\frac {1}{4}}+3 x^{2} {\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}} \left (2 \arctan \left (\frac {{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )-\ln \left (\frac {a^{\frac {1}{4}} x +{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}\right )\right )\right )}{6}\right )}{3 {\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}} a^{\frac {1}{4}} b \,x^{2}}\) | \(126\) |
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Timed out. \[ \int \frac {b+a x^6}{x^3 \left (b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\text {Timed out} \]
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\[ \int \frac {b+a x^6}{x^3 \left (b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\int \frac {a x^{6} + b}{x^{3} \sqrt [4]{x \left (a x^{3} + b\right )} \left (a x^{3} + b\right )}\, dx \]
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\[ \int \frac {b+a x^6}{x^3 \left (b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\int { \frac {a x^{6} + b}{{\left (a x^{4} + b x\right )}^{\frac {1}{4}} {\left (a x^{3} + b\right )} x^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (103) = 206\).
Time = 0.30 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.76 \[ \int \frac {b+a x^6}{x^3 \left (b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, a} + \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, a} - \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{6 \, a} + \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{6 \, a} - \frac {4 \, {\left (a + b\right )}}{3 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} b} - \frac {4 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{4}}}{9 \, b} \]
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Timed out. \[ \int \frac {b+a x^6}{x^3 \left (b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\int \frac {a\,x^6+b}{x^3\,{\left (a\,x^4+b\,x\right )}^{1/4}\,\left (a\,x^3+b\right )} \,d x \]
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