Integrand size = 17, antiderivative size = 291 \[ \int \frac {(d+e x)^2}{a+c x^4} \, dx=\frac {d e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}-\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}-\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}} \]
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Time = 0.14 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {1890, 281, 211, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {(d+e x)^2}{a+c x^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}-\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}}+\frac {d e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}} \]
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Rule 210
Rule 211
Rule 281
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 1890
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 d e x}{a+c x^4}+\frac {d^2+e^2 x^2}{a+c x^4}\right ) \, dx \\ & = (2 d e) \int \frac {x}{a+c x^4} \, dx+\int \frac {d^2+e^2 x^2}{a+c x^4} \, dx \\ & = (d e) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )+\frac {\left (\frac {\sqrt {c} d^2}{\sqrt {a}}-e^2\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 c}+\frac {\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 c} \\ & = \frac {d e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\frac {\left (\frac {\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}{4 c}+\frac {\left (\frac {\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}{4 c}-\frac {\left (\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}{4 \sqrt {2} a^{3/4} c^{3/4}}-\frac {\left (\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}{4 \sqrt {2} a^{3/4} c^{3/4}} \\ & = \frac {d e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}-\frac {\left (\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 )}{4 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\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 )}{4 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\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 )}{2 \sqrt {2} a^{3/4} c^{3/4}}-\frac {\left (\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 )}{2 \sqrt {2} a^{3/4} c^{3/4}} \\ & = \frac {d e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}-\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}-\frac {\left (\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 )}{4 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\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 )}{4 \sqrt {2} a^{3/4} c^{3/4}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^2}{a+c x^4} \, dx=\frac {-2 \left (\sqrt {2} \sqrt {c} d^2+4 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt {2} \sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \left (\sqrt {2} \sqrt {c} d^2-4 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt {2} \sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-\sqrt {2} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )-\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )\right )}{8 a^{3/4} c^{3/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.81 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.15
method | result | size |
risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (\textit {\_R}^{2} e^{2}+2 \textit {\_R} d e +d^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 c}\) | \(43\) |
default | \(\frac {d^{2} \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {e d \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{\sqrt {a c}}+\frac {e^{2} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}\) | \(230\) |
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Result contains complex when optimal does not.
Time = 2.48 (sec) , antiderivative size = 86139, normalized size of antiderivative = 296.01 \[ \int \frac {(d+e x)^2}{a+c x^4} \, dx=\text {Too large to display} \]
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Time = 1.69 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x)^2}{a+c x^4} \, dx=\operatorname {RootSum} {\left (256 t^{4} a^{3} c^{3} + 192 t^{2} a^{2} c^{2} d^{2} e^{2} + t \left (32 a^{2} c d e^{5} - 32 a c^{2} d^{5} e\right ) + a^{2} e^{8} + 2 a c d^{4} e^{4} + c^{2} d^{8}, \left ( t \mapsto t \log {\left (x + \frac {64 t^{3} a^{4} c^{2} e^{6} + 448 t^{3} a^{3} c^{3} d^{4} e^{2} - 160 t^{2} a^{3} c^{2} d^{3} e^{5} + 32 t^{2} a^{2} c^{3} d^{7} e + 60 t a^{3} c d^{2} e^{8} + 256 t a^{2} c^{2} d^{6} e^{4} + 4 t a c^{3} d^{10} + 6 a^{3} d e^{11} - 24 a^{2} c d^{5} e^{7} - 30 a c^{2} d^{9} e^{3}}{a^{3} e^{12} - 33 a^{2} c d^{4} e^{8} - 33 a c^{2} d^{8} e^{4} + c^{3} d^{12}} \right )} \right )\right )} \]
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Time = 0.27 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x)^2}{a+c x^4} \, dx=\frac {\sqrt {2} {\left (\sqrt {c} d^{2} - \sqrt {a} e^{2}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{8 \, a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (\sqrt {c} d^{2} - \sqrt {a} e^{2}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{8 \, a^{\frac {3}{4}} c^{\frac {3}{4}}} + \frac {{\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {3}{4}} d^{2} + \sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{2} - 4 \, \sqrt {a} \sqrt {c} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{4 \, a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {3}{4}}} + \frac {{\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {3}{4}} d^{2} + \sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{2} + 4 \, \sqrt {a} \sqrt {c} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{4 \, a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {3}{4}}} \]
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Time = 0.29 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^2}{a+c x^4} \, dx=\frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a c} c^{2} d e + \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {3}{4}} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{3}} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a c} c^{2} d e + \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {3}{4}} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {3}{4}} e^{2}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{3}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {3}{4}} e^{2}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{3}} \]
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Time = 9.20 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.91 \[ \int \frac {(d+e x)^2}{a+c x^4} \, dx=\sum _{k=1}^4\ln \left (3\,c^2\,d^4\,e^2-a\,c\,e^6+4\,c^2\,d^3\,e^3\,x-\mathrm {root}\left (256\,a^3\,c^3\,z^4+192\,a^2\,c^2\,d^2\,e^2\,z^2+32\,a^2\,c\,d\,e^5\,z-32\,a\,c^2\,d^5\,e\,z+2\,a\,c\,d^4\,e^4+c^2\,d^8+a^2\,e^8,z,k\right )\,c^3\,d^4\,x\,4-{\mathrm {root}\left (256\,a^3\,c^3\,z^4+192\,a^2\,c^2\,d^2\,e^2\,z^2+32\,a^2\,c\,d\,e^5\,z-32\,a\,c^2\,d^5\,e\,z+2\,a\,c\,d^4\,e^4+c^2\,d^8+a^2\,e^8,z,k\right )}^2\,a\,c^3\,d^2\,16+\mathrm {root}\left (256\,a^3\,c^3\,z^4+192\,a^2\,c^2\,d^2\,e^2\,z^2+32\,a^2\,c\,d\,e^5\,z-32\,a\,c^2\,d^5\,e\,z+2\,a\,c\,d^4\,e^4+c^2\,d^8+a^2\,e^8,z,k\right )\,a\,c^2\,e^4\,x\,4-\mathrm {root}\left (256\,a^3\,c^3\,z^4+192\,a^2\,c^2\,d^2\,e^2\,z^2+32\,a^2\,c\,d\,e^5\,z-32\,a\,c^2\,d^5\,e\,z+2\,a\,c\,d^4\,e^4+c^2\,d^8+a^2\,e^8,z,k\right )\,a\,c^2\,d\,e^3\,16+{\mathrm {root}\left (256\,a^3\,c^3\,z^4+192\,a^2\,c^2\,d^2\,e^2\,z^2+32\,a^2\,c\,d\,e^5\,z-32\,a\,c^2\,d^5\,e\,z+2\,a\,c\,d^4\,e^4+c^2\,d^8+a^2\,e^8,z,k\right )}^2\,a\,c^3\,d\,e\,x\,32\right )\,\mathrm {root}\left (256\,a^3\,c^3\,z^4+192\,a^2\,c^2\,d^2\,e^2\,z^2+32\,a^2\,c\,d\,e^5\,z-32\,a\,c^2\,d^5\,e\,z+2\,a\,c\,d^4\,e^4+c^2\,d^8+a^2\,e^8,z,k\right ) \]
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