Integrand size = 45, antiderivative size = 417 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a+b x^4\right )^2} \, dx=\frac {x \left (b c-a g+(b d-a h) x+(b e-a i) x^2+(b f-a j) x^3\right )}{4 a b \left (a+b x^4\right )}+\frac {(b d+a h) \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}-\frac {\left (\sqrt {b} (3 b c+a g)+\sqrt {a} (b e+3 a i)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{7/4}}+\frac {\left (\sqrt {b} (3 b c+a g)+\sqrt {a} (b e+3 a i)\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{7/4}}-\frac {\left (\sqrt {b} (3 b c+a g)-\sqrt {a} (b e+3 a i)\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{7/4}}+\frac {\left (\sqrt {b} (3 b c+a g)-\sqrt {a} (b e+3 a i)\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{7/4}}+\frac {j \log \left (a+b x^4\right )}{4 b^2} \]
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Time = 0.35 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1872, 1890, 1182, 1176, 631, 210, 1179, 642, 1262, 649, 211, 266} \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a+b x^4\right )^2} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {b} (a g+3 b c)+\sqrt {a} (3 a i+b e)\right )}{8 \sqrt {2} a^{7/4} b^{7/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {b} (a g+3 b c)+\sqrt {a} (3 a i+b e)\right )}{8 \sqrt {2} a^{7/4} b^{7/4}}+\frac {\arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ) (a h+b d)}{4 a^{3/2} b^{3/2}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (a g+3 b c)-\sqrt {a} (3 a i+b e)\right )}{16 \sqrt {2} a^{7/4} b^{7/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (a g+3 b c)-\sqrt {a} (3 a i+b e)\right )}{16 \sqrt {2} a^{7/4} b^{7/4}}+\frac {j \log \left (a+b x^4\right )}{4 b^2}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)+x^3 (b f-a j)-a g+b c\right )}{4 a b \left (a+b x^4\right )} \]
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Rule 210
Rule 211
Rule 266
Rule 631
Rule 642
Rule 649
Rule 1176
Rule 1179
Rule 1182
Rule 1262
Rule 1872
Rule 1890
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (b c-a g+(b d-a h) x+(b e-a i) x^2+(b f-a j) x^3\right )}{4 a b \left (a+b x^4\right )}-\frac {\int \frac {-b (3 b c+a g)-2 b (b d+a h) x-b (b e+3 a i) x^2-4 a b j x^3}{a+b x^4} \, dx}{4 a b^2} \\ & = \frac {x \left (b c-a g+(b d-a h) x+(b e-a i) x^2+(b f-a j) x^3\right )}{4 a b \left (a+b x^4\right )}-\frac {\int \left (\frac {-b (3 b c+a g)-b (b e+3 a i) x^2}{a+b x^4}+\frac {x \left (-2 b (b d+a h)-4 a b j x^2\right )}{a+b x^4}\right ) \, dx}{4 a b^2} \\ & = \frac {x \left (b c-a g+(b d-a h) x+(b e-a i) x^2+(b f-a j) x^3\right )}{4 a b \left (a+b x^4\right )}-\frac {\int \frac {-b (3 b c+a g)-b (b e+3 a i) x^2}{a+b x^4} \, dx}{4 a b^2}-\frac {\int \frac {x \left (-2 b (b d+a h)-4 a b j x^2\right )}{a+b x^4} \, dx}{4 a b^2} \\ & = \frac {x \left (b c-a g+(b d-a h) x+(b e-a i) x^2+(b f-a j) x^3\right )}{4 a b \left (a+b x^4\right )}-\frac {\text {Subst}\left (\int \frac {-2 b (b d+a h)-4 a b j x}{a+b x^2} \, dx,x,x^2\right )}{8 a b^2}-\frac {\left (b e-\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}+3 a i\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{8 a b^2}+\frac {\left (b e+\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}+3 a i\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{8 a b^2} \\ & = \frac {x \left (b c-a g+(b d-a h) x+(b e-a i) x^2+(b f-a j) x^3\right )}{4 a b \left (a+b x^4\right )}+\frac {(b d+a h) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{4 a b}+\frac {\left (b e-\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}+3 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}{16 \sqrt {2} a^{5/4} b^{7/4}}+\frac {\left (b e-\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}+3 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}{16 \sqrt {2} a^{5/4} b^{7/4}}+\frac {\left (b e+\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}+3 a i\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a b^2}+\frac {\left (b e+\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}+3 a i\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a b^2}+\frac {j \text {Subst}\left (\int \frac {x}{a+b x^2} \, dx,x,x^2\right )}{2 b} \\ & = \frac {x \left (b c-a g+(b d-a h) x+(b e-a i) x^2+(b f-a j) x^3\right )}{4 a b \left (a+b x^4\right )}+\frac {(b d+a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}+\frac {\left (b e-\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}+3 a i\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{5/4} b^{7/4}}-\frac {\left (b e-\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}+3 a i\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{5/4} b^{7/4}}+\frac {j \log \left (a+b x^4\right )}{4 b^2}+\frac {\left (b e+\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}+3 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 )}{8 \sqrt {2} a^{5/4} b^{7/4}}-\frac {\left (b e+\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}+3 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 )}{8 \sqrt {2} a^{5/4} b^{7/4}} \\ & = \frac {x \left (b c-a g+(b d-a h) x+(b e-a i) x^2+(b f-a j) x^3\right )}{4 a b \left (a+b x^4\right )}+\frac {(b d+a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}-\frac {\left (b e+\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}+3 a i\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{5/4} b^{7/4}}+\frac {\left (b e+\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}+3 a i\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{5/4} b^{7/4}}+\frac {\left (b e-\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}+3 a i\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{5/4} b^{7/4}}-\frac {\left (b e-\frac {\sqrt {b} (3 b c+a g)}{\sqrt {a}}+3 a i\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{5/4} b^{7/4}}+\frac {j \log \left (a+b x^4\right )}{4 b^2} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.10 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a+b x^4\right )^2} \, dx=\frac {\frac {8 \left (a^2 j+b^2 x (c+x (d+e x))-a b (f+x (g+x (h+i x)))\right )}{a \left (a+b x^4\right )}-\frac {2 \sqrt [4]{b} \left (3 \sqrt {2} b^{3/2} c+4 \sqrt [4]{a} b^{5/4} d+\sqrt {2} \sqrt {a} b e+\sqrt {2} a \sqrt {b} g+4 a^{5/4} \sqrt [4]{b} h+3 \sqrt {2} a^{3/2} i\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {2 \sqrt [4]{b} \left (3 \sqrt {2} b^{3/2} c-4 \sqrt [4]{a} b^{5/4} d+\sqrt {2} \sqrt {a} b e+\sqrt {2} a \sqrt {b} g-4 a^{5/4} \sqrt [4]{b} h+3 \sqrt {2} a^{3/2} i\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {\sqrt {2} \sqrt [4]{b} \left (-3 b^{3/2} c+\sqrt {a} b e-a \sqrt {b} g+3 a^{3/2} i\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{7/4}}+\frac {\sqrt {2} \sqrt [4]{b} \left (3 b^{3/2} c-\sqrt {a} b e+a \sqrt {b} g-3 a^{3/2} i\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{7/4}}+8 j \log \left (a+b x^4\right )}{32 b^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.55 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.37
method | result | size |
risch | \(\frac {-\frac {\left (a i -b e \right ) x^{3}}{4 a b}-\frac {\left (a h -b d \right ) x^{2}}{4 a b}-\frac {\left (a g -b c \right ) x}{4 a b}+\frac {a j -b f}{4 b^{2}}}{b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (4 j \,\textit {\_R}^{3}+\frac {\left (3 a i +b e \right ) \textit {\_R}^{2}}{a}+\frac {2 \left (a h +b d \right ) \textit {\_R}}{a}+\frac {a g +3 b c}{a}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{16 b^{2}}\) | \(154\) |
default | \(\frac {-\frac {\left (a i -b e \right ) x^{3}}{4 a b}-\frac {\left (a h -b d \right ) x^{2}}{4 a b}-\frac {\left (a g -b c \right ) x}{4 a b}+\frac {a j -b f}{4 b^{2}}}{b \,x^{4}+a}+\frac {\frac {\left (a g +3 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (2 a h +2 b d \right ) \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{2 \sqrt {a b}}+\frac {\left (3 a i +b e \right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {a j \ln \left (b \,x^{4}+a \right )}{b}}{4 b a}\) | \(350\) |
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Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a+b x^4\right )^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a+b x^4\right )^2} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.10 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a+b x^4\right )^2} \, dx=\frac {{\left (b^{2} e - a b i\right )} x^{3} - a b f + a^{2} j + {\left (b^{2} d - a b h\right )} x^{2} + {\left (b^{2} c - a b g\right )} x}{4 \, {\left (a b^{3} x^{4} + a^{2} b^{2}\right )}} + \frac {\frac {\sqrt {2} {\left (4 \, \sqrt {2} a^{\frac {7}{4}} b^{\frac {1}{4}} j + 3 \, b^{2} c - \sqrt {a} b^{\frac {3}{2}} e + a b g - 3 \, a^{\frac {3}{2}} \sqrt {b} i\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {\sqrt {2} {\left (4 \, \sqrt {2} a^{\frac {7}{4}} b^{\frac {1}{4}} j - 3 \, b^{2} c + \sqrt {a} b^{\frac {3}{2}} e - a b g + 3 \, a^{\frac {3}{2}} \sqrt {b} i\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {9}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} e + \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} g + 3 \, \sqrt {2} a^{\frac {7}{4}} b^{\frac {3}{4}} i - 4 \, \sqrt {a} b^{2} d - 4 \, a^{\frac {3}{2}} b h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {9}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} e + \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} g + 3 \, \sqrt {2} a^{\frac {7}{4}} b^{\frac {3}{4}} i + 4 \, \sqrt {a} b^{2} d + 4 \, a^{\frac {3}{2}} b h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}}}{32 \, a b} \]
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Time = 0.28 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.17 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a+b x^4\right )^2} \, dx=\frac {j \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, b^{2}} + \frac {{\left (b e - a i\right )} x^{3} + {\left (b d - a h\right )} x^{2} + {\left (b c - a g\right )} x - \frac {a b f - a^{2} j}{b}}{4 \, {\left (b x^{4} + a\right )} a b} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a b} b^{3} d + 2 \, \sqrt {2} \sqrt {a b} a b^{2} h + 3 \, \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 + 3 \, \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 )}{16 \, a^{2} b^{4}} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a b} b^{3} d + 2 \, \sqrt {2} \sqrt {a b} a b^{2} h + 3 \, \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 + 3 \, \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 )}{16 \, a^{2} b^{4}} + \frac {\sqrt {2} {\left (3 \, \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 - 3 \, \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 )}{32 \, a^{2} b^{4}} - \frac {\sqrt {2} {\left (3 \, \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 - 3 \, \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 )}{32 \, a^{2} b^{4}} \]
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Time = 10.39 (sec) , antiderivative size = 3939, normalized size of antiderivative = 9.45 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a+b x^4\right )^2} \, dx=\text {Too large to display} \]
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