Integrand size = 40, antiderivative size = 384 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx=\frac {g x}{b}+\frac {h x^2}{2 b}+\frac {i x^3}{3 b}+\frac {(b d-a h) \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}-\frac {\left (\sqrt {b} (b c-a g)+\sqrt {a} (b e-a i)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\left (\sqrt {b} (b c-a g)+\sqrt {a} (b e-a i)\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{7/4}}-\frac {\left (\sqrt {b} (b c-a g)-\sqrt {a} (b e-a i)\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\left (\sqrt {b} (b c-a g)-\sqrt {a} (b e-a i)\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}+\frac {f \log \left (a+b x^4\right )}{4 b} \]
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Time = 0.37 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {1899, 1833, 1824, 649, 211, 266, 1901, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {b} (b c-a g)+\sqrt {a} (b e-a i)\right )}{2 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {b} (b c-a g)+\sqrt {a} (b e-a i)\right )}{2 \sqrt {2} a^{3/4} b^{7/4}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (b c-a g)-\sqrt {a} (b e-a i)\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (b c-a g)-\sqrt {a} (b e-a i)\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ) (b d-a h)}{2 \sqrt {a} b^{3/2}}+\frac {f \log \left (a+b x^4\right )}{4 b}+\frac {g x}{b}+\frac {h x^2}{2 b}+\frac {i x^3}{3 b} \]
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Rule 210
Rule 211
Rule 266
Rule 631
Rule 642
Rule 649
Rule 1176
Rule 1179
Rule 1182
Rule 1824
Rule 1833
Rule 1899
Rule 1901
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x \left (d+f x^2+h x^4\right )}{a+b x^4}+\frac {c+e x^2+g x^4+i x^6}{a+b x^4}\right ) \, dx \\ & = \int \frac {x \left (d+f x^2+h x^4\right )}{a+b x^4} \, dx+\int \frac {c+e x^2+g x^4+i x^6}{a+b x^4} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {d+f x+h x^2}{a+b x^2} \, dx,x,x^2\right )+\int \left (\frac {g}{b}+\frac {i x^2}{b}+\frac {b c-a g+(b e-a i) x^2}{b \left (a+b x^4\right )}\right ) \, dx \\ & = \frac {g x}{b}+\frac {i x^3}{3 b}+\frac {1}{2} \text {Subst}\left (\int \left (\frac {h}{b}+\frac {b d-a h+b f x}{b \left (a+b x^2\right )}\right ) \, dx,x,x^2\right )+\frac {\int \frac {b c-a g+(b e-a i) x^2}{a+b x^4} \, dx}{b} \\ & = \frac {g x}{b}+\frac {h x^2}{2 b}+\frac {i x^3}{3 b}+\frac {\text {Subst}\left (\int \frac {b d-a h+b f x}{a+b x^2} \, dx,x,x^2\right )}{2 b}-\frac {\left (b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{2 b^2}+\frac {\left (b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{2 b^2} \\ & = \frac {g x}{b}+\frac {h x^2}{2 b}+\frac {i x^3}{3 b}+\frac {1}{2} f \text {Subst}\left (\int \frac {x}{a+b x^2} \, dx,x,x^2\right )+\frac {(b d-a h) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{2 b}+\frac {\left (b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\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} \sqrt [4]{a} b^{7/4}}+\frac {\left (b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\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} \sqrt [4]{a} b^{7/4}}+\frac {\left (b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 b^2}+\frac {\left (b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 b^2} \\ & = \frac {g x}{b}+\frac {h x^2}{2 b}+\frac {i x^3}{3 b}+\frac {(b d-a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}+\frac {\left (b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}-\frac {\left (b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {f \log \left (a+b x^4\right )}{4 b}+\frac {\left (b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\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} \sqrt [4]{a} b^{7/4}}-\frac {\left (b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\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} \sqrt [4]{a} b^{7/4}} \\ & = \frac {g x}{b}+\frac {h x^2}{2 b}+\frac {i x^3}{3 b}+\frac {(b d-a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}-\frac {\left (b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {\left (b e+\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {\left (b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}-\frac {\left (b e-\frac {\sqrt {b} (b c-a g)}{\sqrt {a}}-a i\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} b^{7/4}}+\frac {f \log \left (a+b x^4\right )}{4 b} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.11 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx=\frac {24 b^{3/4} g x+12 b^{3/4} h x^2+8 b^{3/4} i x^3+\frac {6 \left (-\sqrt {2} b^{3/2} c-2 \sqrt [4]{a} b^{5/4} d-\sqrt {2} \sqrt {a} b e+\sqrt {2} a \sqrt {b} g+2 a^{5/4} \sqrt [4]{b} h+\sqrt {2} a^{3/2} i\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac {6 \left (\sqrt {2} b^{3/2} c-2 \sqrt [4]{a} b^{5/4} d+\sqrt {2} \sqrt {a} b e-\sqrt {2} a \sqrt {b} g+2 a^{5/4} \sqrt [4]{b} h-\sqrt {2} a^{3/2} i\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{3/4}}-\frac {3 \sqrt {2} \left (b^{3/2} c-\sqrt {a} b e-a \sqrt {b} g+a^{3/2} i\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{3/4}}+\frac {3 \sqrt {2} \left (b^{3/2} c-\sqrt {a} b e-a \sqrt {b} g+a^{3/2} i\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{3/4}}+6 b^{3/4} f \log \left (a+b x^4\right )}{24 b^{7/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.60 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.23
method | result | size |
risch | \(\frac {i \,x^{3}}{3 b}+\frac {h \,x^{2}}{2 b}+\frac {g x}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (b c -a g +\left (-a h +b d \right ) \textit {\_R} +\left (-a i +b e \right ) \textit {\_R}^{2}+\textit {\_R}^{3} b f \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b^{2}}\) | \(88\) |
default | \(\frac {\frac {1}{3} i \,x^{3}+\frac {1}{2} h \,x^{2}+g x}{b}+\frac {\frac {\left (-a g +b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (-a h +b d \right ) \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{2 \sqrt {a b}}+\frac {\left (-a i +b e \right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {f \ln \left (b \,x^{4}+a \right )}{4}}{b}\) | \(283\) |
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Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.04 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx=\frac {2 \, i x^{3} + 3 \, h x^{2} + 6 \, g x}{6 \, b} + \frac {\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} f + b^{2} c - \sqrt {a} b^{\frac {3}{2}} e - a b g + a^{\frac {3}{2}} \sqrt {b} i\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} f - b^{2} c + \sqrt {a} b^{\frac {3}{2}} e + a b g - a^{\frac {3}{2}} \sqrt {b} i\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {9}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} e - \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} g - \sqrt {2} a^{\frac {7}{4}} b^{\frac {3}{4}} i - 2 \, \sqrt {a} b^{2} d + 2 \, a^{\frac {3}{2}} b h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {9}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} e - \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} g - \sqrt {2} a^{\frac {7}{4}} b^{\frac {3}{4}} i + 2 \, \sqrt {a} b^{2} d - 2 \, a^{\frac {3}{2}} b h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}}}{8 \, b} \]
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Time = 0.27 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.15 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx=\frac {f \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, b} + \frac {2 \, b^{2} i x^{3} + 3 \, b^{2} h x^{2} + 6 \, b^{2} g x}{6 \, b^{3}} - \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a b} b^{3} d + \sqrt {2} \sqrt {a b} a b^{2} h - \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g - \left (a b^{3}\right )^{\frac {3}{4}} b e + \left (a b^{3}\right )^{\frac {3}{4}} a i\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{4}} - \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a b} b^{3} d + \sqrt {2} \sqrt {a b} a b^{2} h - \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g - \left (a b^{3}\right )^{\frac {3}{4}} b e + \left (a b^{3}\right )^{\frac {3}{4}} a i\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c - \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g - \left (a b^{3}\right )^{\frac {3}{4}} b e + \left (a b^{3}\right )^{\frac {3}{4}} a i\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{4}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c - \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g - \left (a b^{3}\right )^{\frac {3}{4}} b e + \left (a b^{3}\right )^{\frac {3}{4}} a i\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{4}} \]
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Time = 9.42 (sec) , antiderivative size = 3798, normalized size of antiderivative = 9.89 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx=\text {Too large to display} \]
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