Integrand size = 20, antiderivative size = 277 \[ \int \frac {c+d x+e x^2}{a+b x^4} \, dx=\frac {d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}-\frac {\left (\sqrt {b} c+\sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {b} c+\sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (\sqrt {b} c-\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {b} c-\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}} \]
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Time = 0.13 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {1890, 281, 211, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {c+d x+e x^2}{a+b x^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} e+\sqrt {b} c\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} e+\sqrt {b} c\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (\sqrt {b} c-\sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {b} c-\sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}} \]
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Rule 210
Rule 211
Rule 281
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 1890
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d x}{a+b x^4}+\frac {c+e x^2}{a+b x^4}\right ) \, dx \\ & = d \int \frac {x}{a+b x^4} \, dx+\int \frac {c+e x^2}{a+b x^4} \, dx \\ & = \frac {1}{2} d \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-e\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{2 b}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+e\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{2 b} \\ & = \frac {d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 b}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 b}-\frac {\left (\sqrt {b} c-\sqrt {a} e\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (\sqrt {b} c-\sqrt {a} e\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} b^{3/4}} \\ & = \frac {d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}-\frac {\left (\sqrt {b} c-\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {b} c-\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {b} c+\sqrt {a} e\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (\sqrt {b} c+\sqrt {a} e\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}} \\ & = \frac {d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}-\frac {\left (\sqrt {b} c+\sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {b} c+\sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (\sqrt {b} c-\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {b} c-\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.83 \[ \int \frac {c+d x+e x^2}{a+b x^4} \, dx=\frac {-2 \left (\sqrt {2} \sqrt {b} c+2 \sqrt [4]{a} \sqrt [4]{b} d+\sqrt {2} \sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \left (\sqrt {2} \sqrt {b} c-2 \sqrt [4]{a} \sqrt [4]{b} d+\sqrt {2} \sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )-\sqrt {2} \left (\sqrt {b} c-\sqrt {a} e\right ) \left (\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )-\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )\right )}{8 a^{3/4} b^{3/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.52 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.13
method | result | size |
risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\textit {\_R}^{2} e +\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b}\) | \(37\) |
default | \(\frac {c \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {d \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{2 \sqrt {a b}}+\frac {e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(226\) |
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Result contains complex when optimal does not.
Time = 2.59 (sec) , antiderivative size = 121386, normalized size of antiderivative = 438.22 \[ \int \frac {c+d x+e x^2}{a+b x^4} \, dx=\text {Too large to display} \]
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Time = 5.31 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.68 \[ \int \frac {c+d x+e x^2}{a+b x^4} \, dx=\operatorname {RootSum} {\left (256 t^{4} a^{3} b^{3} + t^{2} \cdot \left (64 a^{2} b^{2} c e + 32 a^{2} b^{2} d^{2}\right ) + t \left (16 a^{2} b d e^{2} - 16 a b^{2} c^{2} d\right ) + a^{2} e^{4} + 2 a b c^{2} e^{2} - 4 a b c d^{2} e + a b d^{4} + b^{2} c^{4}, \left ( t \mapsto t \log {\left (x + \frac {64 t^{3} a^{4} b^{2} e^{3} - 64 t^{3} a^{3} b^{3} c^{2} e + 128 t^{3} a^{3} b^{3} c d^{2} + 48 t^{2} a^{3} b^{2} c d e^{2} - 32 t^{2} a^{3} b^{2} d^{3} e + 16 t^{2} a^{2} b^{3} c^{3} d + 12 t a^{3} b c e^{4} + 12 t a^{3} b d^{2} e^{3} - 16 t a^{2} b^{2} c^{3} e^{2} + 36 t a^{2} b^{2} c^{2} d^{2} e + 8 t a^{2} b^{2} c d^{4} + 4 t a b^{3} c^{5} + 3 a^{3} d e^{5} + 5 a^{2} b c d^{3} e^{2} - 2 a^{2} b d^{5} e + 5 a b^{2} c^{4} d e - 5 a b^{2} c^{3} d^{3}}{a^{3} e^{6} - a^{2} b c^{2} e^{4} + 8 a^{2} b c d^{2} e^{3} - 4 a^{2} b d^{4} e^{2} - a b^{2} c^{4} e^{2} + 8 a b^{2} c^{3} d^{2} e - 4 a b^{2} c^{2} d^{4} + b^{3} c^{6}} \right )} \right )\right )} \]
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Time = 0.28 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x+e x^2}{a+b x^4} \, dx=\frac {\sqrt {2} {\left (\sqrt {b} c - \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{8 \, a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (\sqrt {b} c - \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{8 \, a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {{\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e - 2 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{4 \, a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {{\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e + 2 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{4 \, a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} \]
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Time = 0.27 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.98 \[ \int \frac {c+d x+e x^2}{a+b x^4} \, dx=-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a b} b^{2} d - \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{3}} - \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a b} b^{2} d - \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{3}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{3}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{3}} \]
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Time = 9.60 (sec) , antiderivative size = 712, normalized size of antiderivative = 2.57 \[ \int \frac {c+d x+e x^2}{a+b x^4} \, dx=\sum _{k=1}^4\ln \left (b^2\,c\,d^2-b^2\,c^2\,e+b^2\,d^3\,x-a\,b\,e^3-{\mathrm {root}\left (256\,a^3\,b^3\,z^4+64\,a^2\,b^2\,c\,e\,z^2+32\,a^2\,b^2\,d^2\,z^2+16\,a^2\,b\,d\,e^2\,z-16\,a\,b^2\,c^2\,d\,z-4\,a\,b\,c\,d^2\,e+2\,a\,b\,c^2\,e^2+a\,b\,d^4+a^2\,e^4+b^2\,c^4,z,k\right )}^2\,a\,b^3\,c\,16-\mathrm {root}\left (256\,a^3\,b^3\,z^4+64\,a^2\,b^2\,c\,e\,z^2+32\,a^2\,b^2\,d^2\,z^2+16\,a^2\,b\,d\,e^2\,z-16\,a\,b^2\,c^2\,d\,z-4\,a\,b\,c\,d^2\,e+2\,a\,b\,c^2\,e^2+a\,b\,d^4+a^2\,e^4+b^2\,c^4,z,k\right )\,b^3\,c^2\,x\,4+{\mathrm {root}\left (256\,a^3\,b^3\,z^4+64\,a^2\,b^2\,c\,e\,z^2+32\,a^2\,b^2\,d^2\,z^2+16\,a^2\,b\,d\,e^2\,z-16\,a\,b^2\,c^2\,d\,z-4\,a\,b\,c\,d^2\,e+2\,a\,b\,c^2\,e^2+a\,b\,d^4+a^2\,e^4+b^2\,c^4,z,k\right )}^2\,a\,b^3\,d\,x\,16+\mathrm {root}\left (256\,a^3\,b^3\,z^4+64\,a^2\,b^2\,c\,e\,z^2+32\,a^2\,b^2\,d^2\,z^2+16\,a^2\,b\,d\,e^2\,z-16\,a\,b^2\,c^2\,d\,z-4\,a\,b\,c\,d^2\,e+2\,a\,b\,c^2\,e^2+a\,b\,d^4+a^2\,e^4+b^2\,c^4,z,k\right )\,a\,b^2\,e^2\,x\,4-\mathrm {root}\left (256\,a^3\,b^3\,z^4+64\,a^2\,b^2\,c\,e\,z^2+32\,a^2\,b^2\,d^2\,z^2+16\,a^2\,b\,d\,e^2\,z-16\,a\,b^2\,c^2\,d\,z-4\,a\,b\,c\,d^2\,e+2\,a\,b\,c^2\,e^2+a\,b\,d^4+a^2\,e^4+b^2\,c^4,z,k\right )\,a\,b^2\,d\,e\,8-2\,b^2\,c\,d\,e\,x\right )\,\mathrm {root}\left (256\,a^3\,b^3\,z^4+64\,a^2\,b^2\,c\,e\,z^2+32\,a^2\,b^2\,d^2\,z^2+16\,a^2\,b\,d\,e^2\,z-16\,a\,b^2\,c^2\,d\,z-4\,a\,b\,c\,d^2\,e+2\,a\,b\,c^2\,e^2+a\,b\,d^4+a^2\,e^4+b^2\,c^4,z,k\right ) \]
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