Integrand size = 30, antiderivative size = 209 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx=-\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+\frac {1}{2} \text {RootSum}\left [3 a^2-2 b-5 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-3 a^2 \log (x)+2 b \log (x)+3 a^2 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-2 b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+2 a \log (x) \text {$\#$1}^4-2 a \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{5 a \text {$\#$1}^3-4 \text {$\#$1}^7}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(750\) vs. \(2(209)=418\).
Time = 1.53 (sec) , antiderivative size = 750, normalized size of antiderivative = 3.59, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2081, 919, 65, 338, 304, 209, 212, 6860, 95, 211, 214} \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx=\frac {\left (-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}+a^2-2 b\right ) \sqrt [4]{a x^4+b x^3} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{-a \sqrt {a^2+16 b}+a^2-4 b}}{\sqrt [4]{a-\sqrt {a^2+16 b}} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a-\sqrt {a^2+16 b}} \left (-a \sqrt {a^2+16 b}+a^2-4 b\right )^{3/4} \sqrt [4]{a x+b}}+\frac {\left (\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}+a^2-2 b\right ) \sqrt [4]{a x^4+b x^3} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {a^2+16 b}+a^2-4 b}}{\sqrt [4]{\sqrt {a^2+16 b}+a} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{\sqrt {a^2+16 b}+a} \left (a \sqrt {a^2+16 b}+a^2-4 b\right )^{3/4} \sqrt [4]{a x+b}}-\frac {\left (-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}+a^2-2 b\right ) \sqrt [4]{a x^4+b x^3} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{-a \sqrt {a^2+16 b}+a^2-4 b}}{\sqrt [4]{a-\sqrt {a^2+16 b}} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a-\sqrt {a^2+16 b}} \left (-a \sqrt {a^2+16 b}+a^2-4 b\right )^{3/4} \sqrt [4]{a x+b}}-\frac {\left (\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}+a^2-2 b\right ) \sqrt [4]{a x^4+b x^3} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {a^2+16 b}+a^2-4 b}}{\sqrt [4]{\sqrt {a^2+16 b}+a} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{\sqrt {a^2+16 b}+a} \left (a \sqrt {a^2+16 b}+a^2-4 b\right )^{3/4} \sqrt [4]{a x+b}}-\frac {\sqrt [4]{a} \sqrt [4]{a x^4+b x^3} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}+\frac {\sqrt [4]{a} \sqrt [4]{a x^4+b x^3} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}} \]
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Rule 65
Rule 95
Rule 209
Rule 211
Rule 212
Rule 214
Rule 304
Rule 338
Rule 919
Rule 2081
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{b+a x}}{-2 b+a x+2 x^2} \, dx}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {\sqrt [4]{b x^3+a x^4} \int \frac {2 a b-\left (a^2-2 b\right ) x}{\sqrt [4]{x} (b+a x)^{3/4} \left (-2 b+a x+2 x^2\right )} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {\sqrt [4]{b x^3+a x^4} \int \left (\frac {-a^2+2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}}{\sqrt [4]{x} \left (a-\sqrt {a^2+16 b}+4 x\right ) (b+a x)^{3/4}}+\frac {-a^2+2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}}{\sqrt [4]{x} \left (a+\sqrt {a^2+16 b}+4 x\right ) (b+a x)^{3/4}}\right ) \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 a \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {\left (2 a \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (-a^2+2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (a+\sqrt {a^2+16 b}+4 x\right ) (b+a x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (-a^2+2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (a-\sqrt {a^2+16 b}+4 x\right ) (b+a x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {\left (\sqrt {a} \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\sqrt {a} \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \left (-a^2+2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{a+\sqrt {a^2+16 b}-\left (-4 b+a \left (a+\sqrt {a^2+16 b}\right )\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \left (-a^2+2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{a-\sqrt {a^2+16 b}-\left (-4 b+a \left (a-\sqrt {a^2+16 b}\right )\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}} \\ & = -\frac {\sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (-a^2+2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+16 b}}-\sqrt {a^2-4 b-a \sqrt {a^2+16 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b-a \sqrt {a^2+16 b}} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (-a^2+2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+16 b}}+\sqrt {a^2-4 b-a \sqrt {a^2+16 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b-a \sqrt {a^2+16 b}} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (-a^2+2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+16 b}}-\sqrt {a^2-4 b+a \sqrt {a^2+16 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b+a \sqrt {a^2+16 b}} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (-a^2+2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+16 b}}+\sqrt {a^2-4 b+a \sqrt {a^2+16 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b+a \sqrt {a^2+16 b}} x^{3/4} \sqrt [4]{b+a x}} \\ & = -\frac {\sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a^2-2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{a^2-4 b-a \sqrt {a^2+16 b}} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+16 b}} \sqrt [4]{b+a x}}\right )}{\sqrt [4]{a-\sqrt {a^2+16 b}} \left (a^2-4 b-a \sqrt {a^2+16 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a^2-2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{a^2-4 b+a \sqrt {a^2+16 b}} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+16 b}} \sqrt [4]{b+a x}}\right )}{\sqrt [4]{a+\sqrt {a^2+16 b}} \left (a^2-4 b+a \sqrt {a^2+16 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a^2-2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a^2-4 b-a \sqrt {a^2+16 b}} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+16 b}} \sqrt [4]{b+a x}}\right )}{\sqrt [4]{a-\sqrt {a^2+16 b}} \left (a^2-4 b-a \sqrt {a^2+16 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a^2-2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a^2-4 b+a \sqrt {a^2+16 b}} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+16 b}} \sqrt [4]{b+a x}}\right )}{\sqrt [4]{a+\sqrt {a^2+16 b}} \left (a^2-4 b+a \sqrt {a^2+16 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx=-\frac {x^{9/4} (b+a x)^{3/4} \left (8 \sqrt [4]{a} \left (\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )+\text {RootSum}\left [3 a^2-2 b-5 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {3 a^2 \log (x)-2 b \log (x)-12 a^2 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+8 b \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )-2 a \log (x) \text {$\#$1}^4+8 a \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{5 a \text {$\#$1}^3-4 \text {$\#$1}^7}\&\right ]\right )}{8 \left (x^3 (b+a x)\right )^{3/4}} \]
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Time = 0.00 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-5 a \,\textit {\_Z}^{4}+3 a^{2}-2 b \right )}{\sum }\frac {\left (2 \textit {\_R}^{4} a -3 a^{2}+2 b \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (4 \textit {\_R}^{4}-5 a \right )}\right )}{2}+\frac {a^{\frac {1}{4}} \ln \left (\frac {x \,a^{\frac {1}{4}}+\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{-x \,a^{\frac {1}{4}}+\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right )}{2}+a^{\frac {1}{4}} \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )\) | \(146\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.46 (sec) , antiderivative size = 3186, normalized size of antiderivative = 15.24 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx=\text {Too large to display} \]
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Not integrable
Time = 1.61 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x + b\right )}}{a x - 2 b + 2 x^{2}}\, dx \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.14 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{a x + 2 \, x^{2} - 2 \, b} \,d x } \]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.57 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx=\frac {1}{2} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{2} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right ) - \frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right ) \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.14 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx=\int \frac {{\left (a\,x^4+b\,x^3\right )}^{1/4}}{2\,x^2+a\,x-2\,b} \,d x \]
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