Integrand size = 17, antiderivative size = 322 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^2} \, dx=\frac {x (d+e x)^2}{4 a \left (a+c x^4\right )}+\frac {d e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c}}-\frac {\left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{3/4}}+\frac {\left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{3/4}}-\frac {\left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{3/4}}+\frac {\left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{3/4}} \]
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Time = 0.18 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {1869, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^2} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} e^2+3 \sqrt {c} d^2\right )}{8 \sqrt {2} a^{7/4} c^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} e^2+3 \sqrt {c} d^2\right )}{8 \sqrt {2} a^{7/4} c^{3/4}}+\frac {d e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c}}-\frac {\left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{3/4}}+\frac {\left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{3/4}}+\frac {x (d+e x)^2}{4 a \left (a+c x^4\right )} \]
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Rule 210
Rule 211
Rule 281
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 1869
Rule 1890
Rubi steps \begin{align*} \text {integral}& = \frac {x (d+e x)^2}{4 a \left (a+c x^4\right )}-\frac {\int \frac {-3 d^2-4 d e x-e^2 x^2}{a+c x^4} \, dx}{4 a} \\ & = \frac {x (d+e x)^2}{4 a \left (a+c x^4\right )}-\frac {\int \left (-\frac {4 d e x}{a+c x^4}+\frac {-3 d^2-e^2 x^2}{a+c x^4}\right ) \, dx}{4 a} \\ & = \frac {x (d+e x)^2}{4 a \left (a+c x^4\right )}-\frac {\int \frac {-3 d^2-e^2 x^2}{a+c x^4} \, dx}{4 a}+\frac {(d e) \int \frac {x}{a+c x^4} \, dx}{a} \\ & = \frac {x (d+e x)^2}{4 a \left (a+c x^4\right )}+\frac {(d e) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 a}+\frac {\left (\frac {3 \sqrt {c} d^2}{\sqrt {a}}-e^2\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 a c}+\frac {\left (\frac {3 \sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 a c} \\ & = \frac {x (d+e x)^2}{4 a \left (a+c x^4\right )}+\frac {d e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c}}+\frac {\left (\frac {3 \sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a c}+\frac {\left (\frac {3 \sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a c}-\frac {\left (3 \sqrt {c} d^2-\sqrt {a} e^2\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}{16 \sqrt {2} a^{7/4} c^{3/4}}-\frac {\left (3 \sqrt {c} d^2-\sqrt {a} e^2\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}{16 \sqrt {2} a^{7/4} c^{3/4}} \\ & = \frac {x (d+e x)^2}{4 a \left (a+c x^4\right )}+\frac {d e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c}}-\frac {\left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{3/4}}+\frac {\left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{3/4}}+\frac {\left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{3/4}}-\frac {\left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{3/4}} \\ & = \frac {x (d+e x)^2}{4 a \left (a+c x^4\right )}+\frac {d e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c}}-\frac {\left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{3/4}}+\frac {\left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{3/4}}-\frac {\left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{3/4}}+\frac {\left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{3/4}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^2} \, dx=\frac {\frac {8 a x (d+e x)^2}{a+c x^4}-\frac {2 \sqrt [4]{a} \left (3 \sqrt {2} \sqrt {c} d^2+8 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt {2} \sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac {2 \sqrt [4]{a} \left (3 \sqrt {2} \sqrt {c} d^2-8 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt {2} \sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac {\sqrt {2} \left (-3 \sqrt [4]{a} \sqrt {c} d^2+a^{3/4} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{c^{3/4}}+\frac {\sqrt {2} \left (3 \sqrt [4]{a} \sqrt {c} d^2-a^{3/4} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{c^{3/4}}}{32 a^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.82 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.28
method | result | size |
risch | \(\frac {\frac {e^{2} x^{3}}{4 a}+\frac {e d \,x^{2}}{2 a}+\frac {d^{2} x}{4 a}}{c \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (e^{2} \textit {\_R}^{2}+4 e d \textit {\_R} +3 d^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{16 a c}\) | \(90\) |
default | \(d^{2} \left (\frac {x}{4 a \left (c \,x^{4}+a \right )}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}\right )+2 e d \left (\frac {x^{2}}{4 a \left (c \,x^{4}+a \right )}+\frac {\arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{4 a \sqrt {a c}}\right )+e^{2} \left (\frac {x^{3}}{4 a \left (c \,x^{4}+a \right )}+\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a c \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )\) | \(293\) |
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Result contains complex when optimal does not.
Time = 2.74 (sec) , antiderivative size = 90963, normalized size of antiderivative = 282.49 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]
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Time = 3.73 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^2} \, dx=\operatorname {RootSum} {\left (65536 t^{4} a^{7} c^{3} + 11264 t^{2} a^{4} c^{2} d^{2} e^{2} + t \left (256 a^{3} c d e^{5} - 2304 a^{2} c^{2} d^{5} e\right ) + a^{2} e^{8} + 82 a c d^{4} e^{4} + 81 c^{2} d^{8}, \left ( t \mapsto t \log {\left (x + \frac {4096 t^{3} a^{7} c^{2} e^{6} + 356352 t^{3} a^{6} c^{3} d^{4} e^{2} - 23552 t^{2} a^{5} c^{2} d^{3} e^{5} + 27648 t^{2} a^{4} c^{3} d^{7} e + 912 t a^{4} c d^{2} e^{8} + 43584 t a^{3} c^{2} d^{6} e^{4} + 3888 t a^{2} c^{3} d^{10} + 12 a^{3} d e^{11} - 1088 a^{2} c d^{5} e^{7} - 7020 a c^{2} d^{9} e^{3}}{a^{3} e^{12} - 649 a^{2} c d^{4} e^{8} - 5841 a c^{2} d^{8} e^{4} + 729 c^{3} d^{12}} \right )} \right )\right )} + \frac {d^{2} x + 2 d e x^{2} + e^{2} x^{3}}{4 a^{2} + 4 a c x^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^2} \, dx=\frac {e^{2} x^{3} + 2 \, d e x^{2} + d^{2} x}{4 \, {\left (a c x^{4} + a^{2}\right )}} + \frac {\frac {\sqrt {2} {\left (3 \, \sqrt {c} d^{2} - \sqrt {a} e^{2}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (3 \, \sqrt {c} d^{2} - \sqrt {a} e^{2}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {3}{4}} d^{2} + \sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{2} - 8 \, \sqrt {a} \sqrt {c} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {3}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {3}{4}} d^{2} + \sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{2} + 8 \, \sqrt {a} \sqrt {c} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {3}{4}}}}{32 \, a} \]
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Time = 0.30 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^2} \, dx=\frac {e^{2} x^{3} + 2 \, d e x^{2} + d^{2} x}{4 \, {\left (c x^{4} + a\right )} a} + \frac {\sqrt {2} {\left (4 \, \sqrt {2} \sqrt {a c} c^{2} d e + 3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {3}{4}} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (4 \, \sqrt {2} \sqrt {a c} c^{2} d e + 3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {3}{4}} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {3}{4}} e^{2}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{2} c^{3}} - \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {3}{4}} e^{2}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{2} c^{3}} \]
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Time = 9.25 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.21 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^2} \, dx=\frac {\frac {d^2\,x}{4\,a}+\frac {e^2\,x^3}{4\,a}+\frac {d\,e\,x^2}{2\,a}}{c\,x^4+a}+\left (\sum _{k=1}^4\ln \left (\frac {39\,c^2\,d^4\,e^2-a\,c\,e^6}{64\,a^3}-\mathrm {root}\left (65536\,a^7\,c^3\,z^4+11264\,a^4\,c^2\,d^2\,e^2\,z^2-2304\,a^2\,c^2\,d^5\,e\,z+256\,a^3\,c\,d\,e^5\,z+82\,a\,c\,d^4\,e^4+81\,c^2\,d^8+a^2\,e^8,z,k\right )\,\left (\mathrm {root}\left (65536\,a^7\,c^3\,z^4+11264\,a^4\,c^2\,d^2\,e^2\,z^2-2304\,a^2\,c^2\,d^5\,e\,z+256\,a^3\,c\,d\,e^5\,z+82\,a\,c\,d^4\,e^4+81\,c^2\,d^8+a^2\,e^8,z,k\right )\,\left (12\,c^3\,d^2-16\,c^3\,d\,e\,x\right )+\frac {x\,\left (18\,a\,c^3\,d^4-2\,a^2\,c^2\,e^4\right )}{8\,a^3}+\frac {2\,c^2\,d\,e^3}{a}\right )+\frac {5\,c^2\,d^3\,e^3\,x}{8\,a^3}\right )\,\mathrm {root}\left (65536\,a^7\,c^3\,z^4+11264\,a^4\,c^2\,d^2\,e^2\,z^2-2304\,a^2\,c^2\,d^5\,e\,z+256\,a^3\,c\,d\,e^5\,z+82\,a\,c\,d^4\,e^4+81\,c^2\,d^8+a^2\,e^8,z,k\right )\right ) \]
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