Integrand size = 35, antiderivative size = 462 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^4\right )^4} \, dx=\frac {x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac {x \left (7 (11 b c+a g)+12 (5 b d+a h) x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac {(5 b d+a h) \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}}-\frac {\left (77 b c+15 \sqrt {a} \sqrt {b} e+7 a g\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{5/4}}+\frac {\left (77 b c+15 \sqrt {a} \sqrt {b} e+7 a g\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{5/4}}-\frac {\left (77 b c-15 \sqrt {a} \sqrt {b} e+7 a g\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{5/4}}+\frac {\left (77 b c-15 \sqrt {a} \sqrt {b} e+7 a g\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{5/4}} \]
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Time = 0.41 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.343, Rules used = {1872, 1868, 1869, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^4\right )^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt {a} \sqrt {b} e+7 a g+77 b c\right )}{256 \sqrt {2} a^{15/4} b^{5/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (15 \sqrt {a} \sqrt {b} e+7 a g+77 b c\right )}{256 \sqrt {2} a^{15/4} b^{5/4}}+\frac {\arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ) (a h+5 b d)}{32 a^{7/2} b^{3/2}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (-15 \sqrt {a} \sqrt {b} e+7 a g+77 b c\right )}{512 \sqrt {2} a^{15/4} b^{5/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (-15 \sqrt {a} \sqrt {b} e+7 a g+77 b c\right )}{512 \sqrt {2} a^{15/4} b^{5/4}}+\frac {x \left (7 (a g+11 b c)+12 x (a h+5 b d)+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (2 x (a h+5 b d)+a g+11 b c+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac {x \left (x (b d-a h)-a g+b c+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3} \]
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Rule 210
Rule 211
Rule 281
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 1868
Rule 1869
Rule 1872
Rule 1890
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}-\frac {\int \frac {-b (11 b c+a g)-2 b (5 b d+a h) x-9 b^2 e x^2-8 b^2 f x^3}{\left (a+b x^4\right )^3} \, dx}{12 a b^2} \\ & = \frac {x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac {\int \frac {7 b (11 b c+a g)+12 b (5 b d+a h) x+45 b^2 e x^2}{\left (a+b x^4\right )^2} \, dx}{96 a^2 b^2} \\ & = \frac {x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac {x \left (7 (11 b c+a g)+12 (5 b d+a h) x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}-\frac {\int \frac {-21 b (11 b c+a g)-24 b (5 b d+a h) x-45 b^2 e x^2}{a+b x^4} \, dx}{384 a^3 b^2} \\ & = \frac {x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac {x \left (7 (11 b c+a g)+12 (5 b d+a h) x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}-\frac {\int \left (-\frac {24 b (5 b d+a h) x}{a+b x^4}+\frac {-21 b (11 b c+a g)-45 b^2 e x^2}{a+b x^4}\right ) \, dx}{384 a^3 b^2} \\ & = \frac {x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac {x \left (7 (11 b c+a g)+12 (5 b d+a h) x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}-\frac {\int \frac {-21 b (11 b c+a g)-45 b^2 e x^2}{a+b x^4} \, dx}{384 a^3 b^2}+\frac {(5 b d+a h) \int \frac {x}{a+b x^4} \, dx}{16 a^3 b} \\ & = \frac {x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac {x \left (7 (11 b c+a g)+12 (5 b d+a h) x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac {\left (77 b c+15 \sqrt {a} \sqrt {b} e+7 a g\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{256 a^{7/2} b^{3/2}}-\frac {\left (15 \sqrt {b} e-\frac {7 (11 b c+a g)}{\sqrt {a}}\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{256 a^3 b^{3/2}}+\frac {(5 b d+a h) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{32 a^3 b} \\ & = \frac {x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac {x \left (7 (11 b c+a g)+12 (5 b d+a h) x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac {(5 b d+a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}}-\frac {\left (77 b c-15 \sqrt {a} \sqrt {b} e+7 a g\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}{512 \sqrt {2} a^{15/4} b^{5/4}}-\frac {\left (77 b c-15 \sqrt {a} \sqrt {b} e+7 a g\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}{512 \sqrt {2} a^{15/4} b^{5/4}}+\frac {\left (77 b c+15 \sqrt {a} \sqrt {b} e+7 a g\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{512 a^{7/2} b^{3/2}}+\frac {\left (77 b c+15 \sqrt {a} \sqrt {b} e+7 a g\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{512 a^{7/2} b^{3/2}} \\ & = \frac {x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac {x \left (7 (11 b c+a g)+12 (5 b d+a h) x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac {(5 b d+a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}}-\frac {\left (77 b c-15 \sqrt {a} \sqrt {b} e+7 a g\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{5/4}}+\frac {\left (77 b c-15 \sqrt {a} \sqrt {b} e+7 a g\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{5/4}}+\frac {\left (77 b c+15 \sqrt {a} \sqrt {b} e+7 a g\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{5/4}}-\frac {\left (77 b c+15 \sqrt {a} \sqrt {b} e+7 a g\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{5/4}} \\ & = \frac {x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac {x \left (7 (11 b c+a g)+12 (5 b d+a h) x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac {(5 b d+a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}}-\frac {\left (77 b c+15 \sqrt {a} \sqrt {b} e+7 a g\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{5/4}}+\frac {\left (77 b c+15 \sqrt {a} \sqrt {b} e+7 a g\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{5/4}}-\frac {\left (77 b c-15 \sqrt {a} \sqrt {b} e+7 a g\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{5/4}}+\frac {\left (77 b c-15 \sqrt {a} \sqrt {b} e+7 a g\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{5/4}} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^4\right )^4} \, dx=\frac {\frac {8 a^{3/4} \sqrt {b} x \left (77 b c+7 a g+60 b d x+12 a h x+45 b e x^2\right )}{a+b x^4}+\frac {32 a^{7/4} \sqrt {b} x (11 b c+b x (10 d+9 e x)+a (g+2 h x))}{\left (a+b x^4\right )^2}-\frac {256 a^{11/4} \sqrt {b} (-b x (c+x (d+e x))+a (f+x (g+h x)))}{\left (a+b x^4\right )^3}-6 \left (77 \sqrt {2} b^{5/4} c+80 \sqrt [4]{a} b d+15 \sqrt {2} \sqrt {a} b^{3/4} e+7 \sqrt {2} a \sqrt [4]{b} g+16 a^{5/4} h\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+6 \left (77 \sqrt {2} b^{5/4} c-80 \sqrt [4]{a} b d+15 \sqrt {2} \sqrt {a} b^{3/4} e+7 \sqrt {2} a \sqrt [4]{b} g-16 a^{5/4} h\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )-3 \sqrt {2} \sqrt [4]{b} \left (77 b c-15 \sqrt {a} \sqrt {b} e+7 a g\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+3 \sqrt {2} \sqrt [4]{b} \left (77 b c-15 \sqrt {a} \sqrt {b} e+7 a g\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{3072 a^{15/4} b^{3/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.54 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.47
method | result | size |
risch | \(\frac {\frac {15 e \,b^{2} x^{11}}{128 a^{3}}+\frac {\left (a h +5 b d \right ) b \,x^{10}}{32 a^{3}}+\frac {7 \left (a g +11 b c \right ) b \,x^{9}}{384 a^{3}}+\frac {21 b e \,x^{7}}{64 a^{2}}+\frac {\left (a h +5 b d \right ) x^{6}}{12 a^{2}}+\frac {3 \left (a g +11 b c \right ) x^{5}}{64 a^{2}}+\frac {113 e \,x^{3}}{384 a}-\frac {\left (a h -11 b d \right ) x^{2}}{32 a b}-\frac {\left (7 a g -51 b c \right ) x}{128 a b}-\frac {f}{12 b}}{\left (b \,x^{4}+a \right )^{3}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (15 \textit {\_R}^{2} e +\frac {8 \left (a h +5 b d \right ) \textit {\_R}}{b}+\frac {7 a g +77 b c}{b}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{512 a^{3} b}\) | \(216\) |
default | \(\frac {\frac {15 e \,b^{2} x^{11}}{128 a^{3}}+\frac {\left (a h +5 b d \right ) b \,x^{10}}{32 a^{3}}+\frac {7 \left (a g +11 b c \right ) b \,x^{9}}{384 a^{3}}+\frac {21 b e \,x^{7}}{64 a^{2}}+\frac {\left (a h +5 b d \right ) x^{6}}{12 a^{2}}+\frac {3 \left (a g +11 b c \right ) x^{5}}{64 a^{2}}+\frac {113 e \,x^{3}}{384 a}-\frac {\left (a h -11 b d \right ) x^{2}}{32 a b}-\frac {\left (7 a g -51 b c \right ) x}{128 a b}-\frac {f}{12 b}}{\left (b \,x^{4}+a \right )^{3}}+\frac {\frac {\left (7 a g +77 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (8 a h +40 b d \right ) \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{2 \sqrt {a b}}+\frac {15 e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{128 a^{3} b}\) | \(399\) |
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Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^4\right )^4} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^4\right )^4} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.12 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^4\right )^4} \, dx=\frac {45 \, b^{3} e x^{11} + 126 \, a b^{2} e x^{7} + 12 \, {\left (5 \, b^{3} d + a b^{2} h\right )} x^{10} + 7 \, {\left (11 \, b^{3} c + a b^{2} g\right )} x^{9} + 113 \, a^{2} b e x^{3} + 32 \, {\left (5 \, a b^{2} d + a^{2} b h\right )} x^{6} + 18 \, {\left (11 \, a b^{2} c + a^{2} b g\right )} x^{5} - 32 \, a^{3} f + 12 \, {\left (11 \, a^{2} b d - a^{3} h\right )} x^{2} + 3 \, {\left (51 \, a^{2} b c - 7 \, a^{3} g\right )} x}{384 \, {\left (a^{3} b^{4} x^{12} + 3 \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{4} + a^{6} b\right )}} + \frac {\frac {\sqrt {2} {\left (77 \, b^{\frac {3}{2}} c - 15 \, \sqrt {a} b e + 7 \, a \sqrt {b} g\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 {3}{4}}} - \frac {\sqrt {2} {\left (77 \, b^{\frac {3}{2}} c - 15 \, \sqrt {a} b e + 7 \, a \sqrt {b} g\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 {3}{4}}} + \frac {2 \, {\left (77 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {7}{4}} c + 15 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} e + 7 \, \sqrt {2} a^{\frac {5}{4}} b^{\frac {3}{4}} g - 80 \, \sqrt {a} b^{\frac {3}{2}} d - 16 \, a^{\frac {3}{2}} \sqrt {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 {3}{4}}} + \frac {2 \, {\left (77 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {7}{4}} c + 15 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} e + 7 \, \sqrt {2} a^{\frac {5}{4}} b^{\frac {3}{4}} g + 80 \, \sqrt {a} b^{\frac {3}{2}} d + 16 \, a^{\frac {3}{2}} \sqrt {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 {3}{4}}}}{1024 \, a^{3} b} \]
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Time = 0.27 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.11 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^4\right )^4} \, dx=\frac {\sqrt {2} {\left (40 \, \sqrt {2} \sqrt {a b} b^{2} d + 8 \, \sqrt {2} \sqrt {a b} a b h + 77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g + 15 \, \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 )}{512 \, a^{4} b^{3}} + \frac {\sqrt {2} {\left (40 \, \sqrt {2} \sqrt {a b} b^{2} d + 8 \, \sqrt {2} \sqrt {a b} a b h + 77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g + 15 \, \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 )}{512 \, a^{4} b^{3}} + \frac {\sqrt {2} {\left (77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g - 15 \, \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 )}{1024 \, a^{4} b^{3}} - \frac {\sqrt {2} {\left (77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g - 15 \, \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 )}{1024 \, a^{4} b^{3}} + \frac {45 \, b^{3} e x^{11} + 60 \, b^{3} d x^{10} + 12 \, a b^{2} h x^{10} + 77 \, b^{3} c x^{9} + 7 \, a b^{2} g x^{9} + 126 \, a b^{2} e x^{7} + 160 \, a b^{2} d x^{6} + 32 \, a^{2} b h x^{6} + 198 \, a b^{2} c x^{5} + 18 \, a^{2} b g x^{5} + 113 \, a^{2} b e x^{3} + 132 \, a^{2} b d x^{2} - 12 \, a^{3} h x^{2} + 153 \, a^{2} b c x - 21 \, a^{3} g x - 32 \, a^{3} f}{384 \, {\left (b x^{4} + a\right )}^{3} a^{3} b} \]
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Time = 10.10 (sec) , antiderivative size = 1743, normalized size of antiderivative = 3.77 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^4\right )^4} \, dx=\text {Too large to display} \]
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