Integrand size = 19, antiderivative size = 288 \[ \int \frac {\left (c+d x^4\right )^3}{a+b x^4} \, dx=\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{b^3}+\frac {d^2 (3 b c-a d) x^5}{5 b^2}+\frac {d^3 x^9}{9 b}-\frac {(b c-a d)^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{13/4}}+\frac {(b c-a d)^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{13/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{13/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{13/4}} \]
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Time = 0.16 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {398, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\left (c+d x^4\right )^3}{a+b x^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) (b c-a d)^3}{2 \sqrt {2} a^{3/4} b^{13/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{2 \sqrt {2} a^{3/4} b^{13/4}}-\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{13/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{13/4}}+\frac {d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac {d^2 x^5 (3 b c-a d)}{5 b^2}+\frac {d^3 x^9}{9 b} \]
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Rule 210
Rule 217
Rule 398
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right )}{b^3}+\frac {d^2 (3 b c-a d) x^4}{b^2}+\frac {d^3 x^8}{b}+\frac {b^3 c^3-3 a b^2 c^2 d+3 a^2 b c d^2-a^3 d^3}{b^3 \left (a+b x^4\right )}\right ) \, dx \\ & = \frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{b^3}+\frac {d^2 (3 b c-a d) x^5}{5 b^2}+\frac {d^3 x^9}{9 b}+\frac {(b c-a d)^3 \int \frac {1}{a+b x^4} \, dx}{b^3} \\ & = \frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{b^3}+\frac {d^2 (3 b c-a d) x^5}{5 b^2}+\frac {d^3 x^9}{9 b}+\frac {(b c-a d)^3 \int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx}{2 \sqrt {a} b^3}+\frac {(b c-a d)^3 \int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx}{2 \sqrt {a} b^3} \\ & = \frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{b^3}+\frac {d^2 (3 b c-a d) x^5}{5 b^2}+\frac {d^3 x^9}{9 b}+\frac {(b c-a d)^3 \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 \sqrt {a} b^{7/2}}+\frac {(b c-a d)^3 \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 \sqrt {a} b^{7/2}}-\frac {(b c-a d)^3 \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^{13/4}}-\frac {(b c-a d)^3 \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^{13/4}} \\ & = \frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{b^3}+\frac {d^2 (3 b c-a d) x^5}{5 b^2}+\frac {d^3 x^9}{9 b}-\frac {(b c-a d)^3 \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^{13/4}}+\frac {(b c-a d)^3 \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^{13/4}}+\frac {(b c-a d)^3 \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^{13/4}}-\frac {(b c-a d)^3 \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^{13/4}} \\ & = \frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{b^3}+\frac {d^2 (3 b c-a d) x^5}{5 b^2}+\frac {d^3 x^9}{9 b}-\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{13/4}}+\frac {(b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{13/4}}-\frac {(b c-a d)^3 \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^{13/4}}+\frac {(b c-a d)^3 \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^{13/4}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.94 \[ \int \frac {\left (c+d x^4\right )^3}{a+b x^4} \, dx=\frac {360 a^{3/4} \sqrt [4]{b} d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x-72 a^{3/4} b^{5/4} d^2 (-3 b c+a d) x^5+40 a^{3/4} b^{9/4} d^3 x^9-90 \sqrt {2} (b c-a d)^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+90 \sqrt {2} (b c-a d)^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )-45 \sqrt {2} (b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+45 \sqrt {2} (b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{360 a^{3/4} b^{13/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.98 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.45
method | result | size |
risch | \(\frac {d^{3} x^{9}}{9 b}-\frac {d^{3} a \,x^{5}}{5 b^{2}}+\frac {3 d^{2} c \,x^{5}}{5 b}+\frac {d^{3} a^{2} x}{b^{3}}-\frac {3 d^{2} a c x}{b^{2}}+\frac {3 d \,c^{2} x}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b^{4}}\) | \(131\) |
default | \(\frac {d \left (\frac {1}{9} b^{2} d^{2} x^{9}-\frac {1}{5} a b \,d^{2} x^{5}+\frac {3}{5} b^{2} c d \,x^{5}+a^{2} d^{2} x -3 a b c d x +3 b^{2} c^{2} x \right )}{b^{3}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b^{3} a}\) | \(203\) |
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 1642, normalized size of antiderivative = 5.70 \[ \int \frac {\left (c+d x^4\right )^3}{a+b x^4} \, dx=\text {Too large to display} \]
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Time = 1.30 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.05 \[ \int \frac {\left (c+d x^4\right )^3}{a+b x^4} \, dx=x^{5} \left (- \frac {a d^{3}}{5 b^{2}} + \frac {3 c d^{2}}{5 b}\right ) + x \left (\frac {a^{2} d^{3}}{b^{3}} - \frac {3 a c d^{2}}{b^{2}} + \frac {3 c^{2} d}{b}\right ) + \operatorname {RootSum} {\left (256 t^{4} a^{3} b^{13} + a^{12} d^{12} - 12 a^{11} b c d^{11} + 66 a^{10} b^{2} c^{2} d^{10} - 220 a^{9} b^{3} c^{3} d^{9} + 495 a^{8} b^{4} c^{4} d^{8} - 792 a^{7} b^{5} c^{5} d^{7} + 924 a^{6} b^{6} c^{6} d^{6} - 792 a^{5} b^{7} c^{7} d^{5} + 495 a^{4} b^{8} c^{8} d^{4} - 220 a^{3} b^{9} c^{9} d^{3} + 66 a^{2} b^{10} c^{10} d^{2} - 12 a b^{11} c^{11} d + b^{12} c^{12}, \left ( t \mapsto t \log {\left (- \frac {4 t a b^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )} \right )\right )} + \frac {d^{3} x^{9}}{9 b} \]
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none
Time = 0.29 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.34 \[ \int \frac {\left (c+d x^4\right )^3}{a+b x^4} \, dx=\frac {5 \, b^{2} d^{3} x^{9} + 9 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{5} + 45 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x}{45 \, b^{3}} + \frac {\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{8 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (219) = 438\).
Time = 0.29 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.67 \[ \int \frac {\left (c+d x^4\right )^3}{a+b x^4} \, dx=\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{4}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{4}} + \frac {5 \, b^{8} d^{3} x^{9} + 27 \, b^{8} c d^{2} x^{5} - 9 \, a b^{7} d^{3} x^{5} + 135 \, b^{8} c^{2} d x - 135 \, a b^{7} c d^{2} x + 45 \, a^{2} b^{6} d^{3} x}{45 \, b^{9}} \]
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Time = 0.24 (sec) , antiderivative size = 1433, normalized size of antiderivative = 4.98 \[ \int \frac {\left (c+d x^4\right )^3}{a+b x^4} \, dx=\text {Too large to display} \]
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