Integrand size = 17, antiderivative size = 394 \[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^3} \, dx=\frac {x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac {9 d^2 e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}-\frac {3 d \left (7 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}+\frac {3 d \left (7 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}-\frac {3 d \left (7 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} c^{3/4}}+\frac {3 d \left (7 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} c^{3/4}} \]
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Time = 0.24 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {1868, 1869, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^3} \, dx=-\frac {3 d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt {a} e^2+7 \sqrt {c} d^2\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}+\frac {3 d \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (5 \sqrt {a} e^2+7 \sqrt {c} d^2\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}+\frac {9 d^2 e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}-\frac {3 d \left (7 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} c^{3/4}}+\frac {3 d \left (7 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} c^{3/4}}+\frac {x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2} \]
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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 1890
Rubi steps \begin{align*} \text {integral}& = -\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}-\frac {\int \frac {-7 d^3-18 d^2 e x-15 d e^2 x^2}{\left (a+c x^4\right )^2} \, dx}{8 a} \\ & = \frac {x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac {\int \frac {21 d^3+36 d^2 e x+15 d e^2 x^2}{a+c x^4} \, dx}{32 a^2} \\ & = \frac {x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac {\int \left (\frac {36 d^2 e x}{a+c x^4}+\frac {21 d^3+15 d e^2 x^2}{a+c x^4}\right ) \, dx}{32 a^2} \\ & = \frac {x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac {\int \frac {21 d^3+15 d e^2 x^2}{a+c x^4} \, dx}{32 a^2}+\frac {\left (9 d^2 e\right ) \int \frac {x}{a+c x^4} \, dx}{8 a^2} \\ & = \frac {x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac {\left (9 d^2 e\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{16 a^2}+\frac {\left (3 d \left (\frac {7 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{64 a^2 c}+\frac {\left (3 d \left (\frac {7 \sqrt {c} d^2}{\sqrt {a}}+5 e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{64 a^2 c} \\ & = \frac {x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac {9 d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}-\frac {\left (3 d \left (\frac {7 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt {2} a^{9/4} c^{3/4}}-\frac {\left (3 d \left (\frac {7 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt {2} a^{9/4} c^{3/4}}+\frac {\left (3 d \left (\frac {7 \sqrt {c} d^2}{\sqrt {a}}+5 e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^2 c}+\frac {\left (3 d \left (\frac {7 \sqrt {c} d^2}{\sqrt {a}}+5 e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^2 c} \\ & = \frac {x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac {9 d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}-\frac {3 d \left (\frac {7 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{9/4} c^{3/4}}+\frac {3 d \left (\frac {7 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{9/4} c^{3/4}}+\frac {\left (3 d \left (7 \sqrt {c} d^2+5 \sqrt {a} e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}-\frac {\left (3 d \left (7 \sqrt {c} d^2+5 \sqrt {a} e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}} \\ & = \frac {x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac {9 d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}-\frac {3 d \left (7 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}+\frac {3 d \left (7 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}-\frac {3 d \left (\frac {7 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{9/4} c^{3/4}}+\frac {3 d \left (\frac {7 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{9/4} c^{3/4}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {8 a d x \left (7 d^2+18 d e x+15 e^2 x^2\right )}{a+c x^4}-\frac {32 a^2 \left (a e^3-c d x \left (d^2+3 d e x+3 e^2 x^2\right )\right )}{c \left (a+c x^4\right )^2}-\frac {6 \sqrt [4]{a} d \left (7 \sqrt {2} \sqrt {c} d^2+24 \sqrt [4]{a} \sqrt [4]{c} d e+5 \sqrt {2} \sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac {6 \sqrt [4]{a} d \left (7 \sqrt {2} \sqrt {c} d^2-24 \sqrt [4]{a} \sqrt [4]{c} d e+5 \sqrt {2} \sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac {3 \sqrt {2} \left (-7 \sqrt [4]{a} \sqrt {c} d^3+5 a^{3/4} d e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{c^{3/4}}+\frac {3 \sqrt {2} \left (7 \sqrt [4]{a} \sqrt {c} d^3-5 a^{3/4} d e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{c^{3/4}}}{256 a^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.81 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.36
method | result | size |
risch | \(\frac {\frac {15 c d \,e^{2} x^{7}}{32 a^{2}}+\frac {9 c \,d^{2} e \,x^{6}}{16 a^{2}}+\frac {7 d^{3} c \,x^{5}}{32 a^{2}}+\frac {27 d \,e^{2} x^{3}}{32 a}+\frac {15 e \,x^{2} d^{2}}{16 a}+\frac {11 d^{3} x}{32 a}-\frac {e^{3}}{8 c}}{\left (c \,x^{4}+a \right )^{2}}+\frac {3 d \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (5 e^{2} \textit {\_R}^{2}+12 e d \textit {\_R} +7 d^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{128 a^{2} c}\) | \(141\) |
default | \(d^{3} \left (\frac {x}{8 a \left (c \,x^{4}+a \right )^{2}}+\frac {\frac {7 x}{32 a \left (c \,x^{4}+a \right )}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a^{2}}}{a}\right )+3 d^{2} e \left (\frac {x^{2}}{8 a \left (c \,x^{4}+a \right )^{2}}+\frac {\frac {3 x^{2}}{16 a \left (c \,x^{4}+a \right )}+\frac {3 \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{16 a \sqrt {a c}}}{a}\right )+3 d \,e^{2} \left (\frac {x^{3}}{8 a \left (c \,x^{4}+a \right )^{2}}+\frac {\frac {5 x^{3}}{32 a \left (c \,x^{4}+a \right )}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{a}\right )+e^{3} \left (\frac {x^{4}}{8 a \left (c \,x^{4}+a \right )^{2}}+\frac {x^{4}}{8 a^{2} \left (c \,x^{4}+a \right )}\right )\) | \(403\) |
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Result contains complex when optimal does not.
Time = 13.23 (sec) , antiderivative size = 95566, normalized size of antiderivative = 242.55 \[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^3} \, dx=\text {Too large to display} \]
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Time = 131.88 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.05 \[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^3} \, dx=\operatorname {RootSum} {\left (268435456 t^{4} a^{11} c^{3} + 63111168 t^{2} a^{6} c^{2} d^{4} e^{2} + t \left (4147200 a^{4} c d^{4} e^{5} - 8128512 a^{3} c^{2} d^{8} e\right ) + 50625 a^{2} d^{4} e^{8} + 245106 a c d^{8} e^{4} + 194481 c^{2} d^{12}, \left ( t \mapsto t \log {\left (x + \frac {262144000 t^{3} a^{10} c^{2} e^{6} + 3714056192 t^{3} a^{9} c^{3} d^{4} e^{2} - 539688960 t^{2} a^{7} c^{2} d^{4} e^{5} + 202309632 t^{2} a^{6} c^{3} d^{8} e + 77328000 t a^{5} c d^{4} e^{8} + 660699648 t a^{4} c^{2} d^{8} e^{4} + 19361664 t a^{3} c^{3} d^{12} + 3037500 a^{3} d^{4} e^{11} - 26360640 a^{2} c d^{8} e^{7} - 60566940 a c^{2} d^{12} e^{3}}{421875 a^{3} d^{3} e^{12} - 29598075 a^{2} c d^{7} e^{8} - 58012227 a c^{2} d^{11} e^{4} + 3176523 c^{3} d^{15}} \right )} \right )\right )} + \frac {- 4 a^{2} e^{3} + 11 a c d^{3} x + 30 a c d^{2} e x^{2} + 27 a c d e^{2} x^{3} + 7 c^{2} d^{3} x^{5} + 18 c^{2} d^{2} e x^{6} + 15 c^{2} d e^{2} x^{7}}{32 a^{4} c + 64 a^{3} c^{2} x^{4} + 32 a^{2} c^{3} x^{8}} \]
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Time = 0.28 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^3} \, dx=\frac {15 \, c^{2} d e^{2} x^{7} + 18 \, c^{2} d^{2} e x^{6} + 7 \, c^{2} d^{3} x^{5} + 27 \, a c d e^{2} x^{3} + 30 \, a c d^{2} e x^{2} + 11 \, a c d^{3} x - 4 \, a^{2} e^{3}}{32 \, {\left (a^{2} c^{3} x^{8} + 2 \, a^{3} c^{2} x^{4} + a^{4} c\right )}} + \frac {3 \, d {\left (\frac {\sqrt {2} {\left (7 \, \sqrt {c} d^{2} - 5 \, \sqrt {a} e^{2}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (7 \, \sqrt {c} d^{2} - 5 \, \sqrt {a} e^{2}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} + \frac {2 \, {\left (7 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {3}{4}} d^{2} + 5 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{2} - 24 \, \sqrt {a} \sqrt {c} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {3}{4}}} + \frac {2 \, {\left (7 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {3}{4}} d^{2} + 5 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{2} + 24 \, \sqrt {a} \sqrt {c} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {3}{4}}}\right )}}{256 \, a^{2}} \]
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Time = 0.32 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^3} \, dx=\frac {3 \, \sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a c} c^{2} d^{2} e + 7 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} c^{3}} + \frac {3 \, \sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a c} c^{2} d^{2} e + 7 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} c^{3}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{3} c^{3}} - \frac {3 \, \sqrt {2} {\left (7 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{3} c^{3}} + \frac {15 \, c^{2} d e^{2} x^{7} + 18 \, c^{2} d^{2} e x^{6} + 7 \, c^{2} d^{3} x^{5} + 27 \, a c d e^{2} x^{3} + 30 \, a c d^{2} e x^{2} + 11 \, a c d^{3} x - 4 \, a^{2} e^{3}}{32 \, {\left (c x^{4} + a\right )}^{2} a^{2} c} \]
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Time = 0.27 (sec) , antiderivative size = 721, normalized size of antiderivative = 1.83 \[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {11\,d^3\,x}{32\,a}-\frac {e^3}{8\,c}+\frac {7\,c\,d^3\,x^5}{32\,a^2}+\frac {15\,d^2\,e\,x^2}{16\,a}+\frac {27\,d\,e^2\,x^3}{32\,a}+\frac {9\,c\,d^2\,e\,x^6}{16\,a^2}+\frac {15\,c\,d\,e^2\,x^7}{32\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}+\left (\sum _{k=1}^4\ln \left (\frac {c\,d^2\,\left (6867\,c\,d^5\,e^2-1125\,a\,d\,e^6+7992\,c\,d^4\,e^3\,x-{\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+63111168\,a^6\,c^2\,d^4\,e^2\,z^2-8128512\,a^3\,c^2\,d^8\,e\,z+4147200\,a^4\,c\,d^4\,e^5\,z+245106\,a\,c\,d^8\,e^4+50625\,a^2\,d^4\,e^8+194481\,c^2\,d^{12},z,k\right )}^2\,a^5\,c^2\,d\,114688+\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+63111168\,a^6\,c^2\,d^4\,e^2\,z^2-8128512\,a^3\,c^2\,d^8\,e\,z+4147200\,a^4\,c\,d^4\,e^5\,z+245106\,a\,c\,d^8\,e^4+50625\,a^2\,d^4\,e^8+194481\,c^2\,d^{12},z,k\right )\,a^3\,c\,e^4\,x\,9600-\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+63111168\,a^6\,c^2\,d^4\,e^2\,z^2-8128512\,a^3\,c^2\,d^8\,e\,z+4147200\,a^4\,c\,d^4\,e^5\,z+245106\,a\,c\,d^8\,e^4+50625\,a^2\,d^4\,e^8+194481\,c^2\,d^{12},z,k\right )\,a^2\,c^2\,d^4\,x\,18816+{\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+63111168\,a^6\,c^2\,d^4\,e^2\,z^2-8128512\,a^3\,c^2\,d^8\,e\,z+4147200\,a^4\,c\,d^4\,e^5\,z+245106\,a\,c\,d^8\,e^4+50625\,a^2\,d^4\,e^8+194481\,c^2\,d^{12},z,k\right )}^2\,a^5\,c^2\,e\,x\,196608-\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+63111168\,a^6\,c^2\,d^4\,e^2\,z^2-8128512\,a^3\,c^2\,d^8\,e\,z+4147200\,a^4\,c\,d^4\,e^5\,z+245106\,a\,c\,d^8\,e^4+50625\,a^2\,d^4\,e^8+194481\,c^2\,d^{12},z,k\right )\,a^3\,c\,d\,e^3\,46080\right )\,3}{a^6\,32768}\right )\,\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+63111168\,a^6\,c^2\,d^4\,e^2\,z^2-8128512\,a^3\,c^2\,d^8\,e\,z+4147200\,a^4\,c\,d^4\,e^5\,z+245106\,a\,c\,d^8\,e^4+50625\,a^2\,d^4\,e^8+194481\,c^2\,d^{12},z,k\right )\right ) \]
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