Integrand size = 17, antiderivative size = 855 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}-\frac {\sqrt {c} d^2 e^5 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt {c} d^2 e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )}-\frac {\sqrt [4]{c} d e^4 \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} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d \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} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d e^4 \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} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \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} \left (c d^4+a e^4\right )}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d e^4 \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} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d \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} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d e^4 \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} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \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} \left (c d^4+a e^4\right )}-\frac {e^7 \log \left (a+c x^4\right )}{4 \left (c d^4+a e^4\right )^2} \]
[Out]
Time = 0.59 (sec) , antiderivative size = 855, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.824, Rules used = {6874, 1868, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642, 1262, 649, 266} \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\frac {\log (d+e x) e^7}{\left (c d^4+a e^4\right )^2}-\frac {\log \left (c x^4+a\right ) e^7}{4 \left (c d^4+a e^4\right )^2}-\frac {\sqrt {c} d^2 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e^5}{2 \sqrt {a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^4}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^4}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac {\sqrt {c} d^2 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e}{4 a^{3/2} \left (c d^4+a e^4\right )}+\frac {a e^3+c x \left (d^3-e x d^2+e^2 x^2 d\right )}{4 a \left (c d^4+a e^4\right ) \left (c x^4+a\right )}-\frac {\sqrt [4]{c} d \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} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}-\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )} \]
[In]
[Out]
Rule 210
Rule 211
Rule 266
Rule 281
Rule 631
Rule 642
Rule 649
Rule 1176
Rule 1179
Rule 1182
Rule 1262
Rule 1868
Rule 1890
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^8}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {c \left (d^3-d^2 e x+d e^2 x^2-e^3 x^3\right )}{\left (c d^4+a e^4\right ) \left (a+c x^4\right )^2}-\frac {c e^4 \left (-d^3+d^2 e x-d e^2 x^2+e^3 x^3\right )}{\left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}\right ) \, dx \\ & = \frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\left (c e^4\right ) \int \frac {-d^3+d^2 e x-d e^2 x^2+e^3 x^3}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^2}+\frac {c \int \frac {d^3-d^2 e x+d e^2 x^2-e^3 x^3}{\left (a+c x^4\right )^2} \, dx}{c d^4+a e^4} \\ & = \frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\left (c e^4\right ) \int \left (\frac {-d^3-d e^2 x^2}{a+c x^4}+\frac {x \left (d^2 e+e^3 x^2\right )}{a+c x^4}\right ) \, dx}{\left (c d^4+a e^4\right )^2}-\frac {c \int \frac {-3 d^3+2 d^2 e x-d e^2 x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )} \\ & = \frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\left (c e^4\right ) \int \frac {-d^3-d e^2 x^2}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^2}-\frac {\left (c e^4\right ) \int \frac {x \left (d^2 e+e^3 x^2\right )}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^2}-\frac {c \int \left (\frac {2 d^2 e x}{a+c x^4}+\frac {-3 d^3-d e^2 x^2}{a+c x^4}\right ) \, dx}{4 a \left (c d^4+a e^4\right )} \\ & = \frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\left (c e^4\right ) \text {Subst}\left (\int \frac {d^2 e+e^3 x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^2}+\frac {\left (d e^4 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}-e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^2}+\frac {\left (d e^4 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^2}-\frac {c \int \frac {-3 d^3-d e^2 x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )}-\frac {\left (c d^2 e\right ) \int \frac {x}{a+c x^4} \, dx}{2 a \left (c d^4+a e^4\right )} \\ & = \frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\left (c d^2 e^5\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^2}-\frac {\left (c e^7\right ) \text {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^2}+\frac {\left (d e^4 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+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}{4 \left (c d^4+a e^4\right )^2}+\frac {\left (d e^4 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+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}{4 \left (c d^4+a e^4\right )^2}-\frac {\left (\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2-\sqrt {a} 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}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac {\left (\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2-\sqrt {a} 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}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac {\left (c d^2 e\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{4 a \left (c d^4+a e^4\right )}+\frac {\left (d \left (\frac {3 \sqrt {c} d^2}{\sqrt {a}}-e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 a \left (c d^4+a e^4\right )}+\frac {\left (d \left (\frac {3 \sqrt {c} d^2}{\sqrt {a}}+e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 a \left (c d^4+a e^4\right )} \\ & = \frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}-\frac {\sqrt {c} d^2 e^5 \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt {c} d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d e^4 \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} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d e^4 \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} \left (c d^4+a e^4\right )^2}-\frac {e^7 \log \left (a+c x^4\right )}{4 \left (c d^4+a e^4\right )^2}+\frac {\left (\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2+\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 )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac {\left (\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2+\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 )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac {\left (d \left (\frac {3 \sqrt {c} d^2}{\sqrt {a}}+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}{16 a \left (c d^4+a e^4\right )}+\frac {\left (d \left (\frac {3 \sqrt {c} d^2}{\sqrt {a}}+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}{16 a \left (c d^4+a e^4\right )}-\frac {\left (\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} 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}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}-\frac {\left (\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} 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}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )} \\ & = \frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}-\frac {\sqrt {c} d^2 e^5 \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt {c} d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )}-\frac {\sqrt [4]{c} d e^4 \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} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d e^4 \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} \left (c d^4+a e^4\right )^2}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d e^4 \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} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d \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} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d e^4 \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} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \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} \left (c d^4+a e^4\right )}-\frac {e^7 \log \left (a+c x^4\right )}{4 \left (c d^4+a e^4\right )^2}+\frac {\left (\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\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 )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}-\frac {\left (\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\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 )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )} \\ & = \frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}-\frac {\sqrt {c} d^2 e^5 \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt {c} d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )}-\frac {\sqrt [4]{c} d e^4 \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} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d \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} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d e^4 \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} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \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} \left (c d^4+a e^4\right )}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d e^4 \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} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d \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} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d e^4 \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} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \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} \left (c d^4+a e^4\right )}-\frac {e^7 \log \left (a+c x^4\right )}{4 \left (c d^4+a e^4\right )^2} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 558, normalized size of antiderivative = 0.65 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\frac {\frac {8 \left (c d^4+a e^4\right ) \left (a e^3+c d x \left (d^2-d e x+e^2 x^2\right )\right )}{a \left (a+c x^4\right )}-\frac {2 \sqrt [4]{c} d \left (3 \sqrt {2} c^{3/2} d^6-4 \sqrt [4]{a} c^{5/4} d^5 e+\sqrt {2} \sqrt {a} c d^4 e^2+7 \sqrt {2} a \sqrt {c} d^2 e^4-12 a^{5/4} \sqrt [4]{c} d e^5+5 \sqrt {2} a^{3/2} e^6\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {2 \sqrt [4]{c} d \left (3 \sqrt {2} c^{3/2} d^6+4 \sqrt [4]{a} c^{5/4} d^5 e+\sqrt {2} \sqrt {a} c d^4 e^2+7 \sqrt {2} a \sqrt {c} d^2 e^4+12 a^{5/4} \sqrt [4]{c} d e^5+5 \sqrt {2} a^{3/2} e^6\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+32 e^7 \log (d+e x)+\frac {\sqrt {2} \sqrt [4]{c} \left (-3 c^{3/2} d^7+\sqrt {a} c d^5 e^2-7 a \sqrt {c} d^3 e^4+5 a^{3/2} d e^6\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}+\frac {\sqrt {2} \sqrt [4]{c} \left (3 c^{3/2} d^7-\sqrt {a} c d^5 e^2+7 a \sqrt {c} d^3 e^4-5 a^{3/2} d e^6\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}-8 e^7 \log \left (a+c x^4\right )}{32 \left (c d^4+a e^4\right )^2} \]
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Time = 0.94 (sec) , antiderivative size = 430, normalized size of antiderivative = 0.50
method | result | size |
default | \(\frac {c \left (\frac {\frac {d \,e^{2} \left (e^{4} a +d^{4} c \right ) x^{3}}{4 a}-\frac {d^{2} e \left (e^{4} a +d^{4} c \right ) x^{2}}{4 a}+\frac {d^{3} \left (e^{4} a +d^{4} c \right ) x}{4 a}+\frac {e^{3} \left (e^{4} a +d^{4} c \right )}{4 c}}{c \,x^{4}+a}+\frac {\frac {\left (7 a \,d^{3} e^{4}+3 c \,d^{7}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (-6 a \,d^{2} e^{5}-2 c \,d^{6} e \right ) \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{2 \sqrt {a c}}+\frac {\left (5 a d \,e^{6}+c \,d^{5} e^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}-\frac {a \,e^{7} \ln \left (c \,x^{4}+a \right )}{c}}{4 a}\right )}{\left (e^{4} a +d^{4} c \right )^{2}}+\frac {e^{7} \ln \left (e x +d \right )}{\left (e^{4} a +d^{4} c \right )^{2}}\) | \(430\) |
risch | \(\frac {\frac {c d \,e^{2} x^{3}}{4 a \left (e^{4} a +d^{4} c \right )}-\frac {d^{2} c e \,x^{2}}{4 a \left (e^{4} a +d^{4} c \right )}+\frac {d^{3} c x}{4 a \left (e^{4} a +d^{4} c \right )}+\frac {e^{3}}{4 e^{4} a +4 d^{4} c}}{c \,x^{4}+a}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{9} e^{8}+2 a^{8} c \,d^{4} e^{4}+a^{7} c^{2} d^{8}\right ) \textit {\_Z}^{4}+16 a^{7} e^{7} \textit {\_Z}^{3}+\left (96 a^{5} e^{6}+20 a^{4} c \,d^{4} e^{2}\right ) \textit {\_Z}^{2}+\left (256 a^{3} e^{5}+72 a^{2} c \,d^{4} e \right ) \textit {\_Z} +256 e^{4} a +81 d^{4} c \right )}{\sum }\textit {\_R} \ln \left (\left (\left (5 a^{9} e^{14}+7 a^{8} c \,d^{4} e^{10}-a^{7} c^{2} d^{8} e^{6}-3 a^{6} c^{3} d^{12} e^{2}\right ) \textit {\_R}^{4}+\left (60 a^{7} e^{13}+64 a^{6} c \,d^{4} e^{9}-4 a^{5} c^{2} d^{8} e^{5}-8 a^{4} c^{3} d^{12} e \right ) \textit {\_R}^{3}+\left (240 a^{5} e^{12}+295 a^{4} c \,d^{4} e^{8}+30 a^{3} c^{2} d^{8} e^{4}-9 a^{2} c^{3} d^{12}\right ) \textit {\_R}^{2}+\left (320 a^{3} e^{11}+560 a^{2} c \,d^{4} e^{7}+64 a \,c^{2} d^{8} e^{3}\right ) \textit {\_R} +324 c \,d^{4} e^{6}\right ) x +\left (6 a^{9} d \,e^{13}+10 a^{8} c \,d^{5} e^{9}+2 a^{7} c^{2} d^{9} e^{5}-2 a^{6} c^{3} d^{13} e \right ) \textit {\_R}^{4}+\left (51 a^{7} d \,e^{12}+67 a^{6} c \,d^{5} e^{8}+13 a^{5} c^{2} d^{9} e^{4}-3 a^{4} c^{3} d^{13}\right ) \textit {\_R}^{3}+\left (136 a^{5} d \,e^{11}+240 a^{4} c \,d^{5} e^{7}+56 a^{3} c^{2} d^{9} e^{3}\right ) \textit {\_R}^{2}+\left (176 a^{3} d \,e^{10}+384 a^{2} c \,d^{5} e^{6}+48 a \,c^{2} d^{9} e^{2}\right ) \textit {\_R} +256 a d \,e^{9}+324 c \,d^{5} e^{5}\right )\right )}{16}+\frac {e^{7} \ln \left (e x +d \right )}{a^{2} e^{8}+2 a c \,d^{4} e^{4}+c^{2} d^{8}}\) | \(653\) |
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Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 601, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\frac {e^{7} \log \left (e x + d\right )}{c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}} - \frac {c {\left (\frac {\sqrt {2} {\left (4 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {1}{4}} e^{7} - 3 \, c^{2} d^{7} + \sqrt {a} c^{\frac {3}{2}} d^{5} e^{2} - 7 \, a c d^{3} e^{4} + 5 \, a^{\frac {3}{2}} \sqrt {c} d e^{6}\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 {5}{4}}} + \frac {\sqrt {2} {\left (4 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {1}{4}} e^{7} + 3 \, c^{2} d^{7} - \sqrt {a} c^{\frac {3}{2}} d^{5} e^{2} + 7 \, a c d^{3} e^{4} - 5 \, a^{\frac {3}{2}} \sqrt {c} d e^{6}\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 {5}{4}}} - \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {9}{4}} d^{7} + \sqrt {2} a^{\frac {3}{4}} c^{\frac {7}{4}} d^{5} e^{2} + 7 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {5}{4}} d^{3} e^{4} + 5 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {3}{4}} d e^{6} + 4 \, \sqrt {a} c^{2} d^{6} e + 12 \, a^{\frac {3}{2}} c d^{2} e^{5}\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 {5}{4}}} - \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {9}{4}} d^{7} + \sqrt {2} a^{\frac {3}{4}} c^{\frac {7}{4}} d^{5} e^{2} + 7 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {5}{4}} d^{3} e^{4} + 5 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {3}{4}} d e^{6} - 4 \, \sqrt {a} c^{2} d^{6} e - 12 \, a^{\frac {3}{2}} c d^{2} e^{5}\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 {5}{4}}}\right )}}{32 \, {\left (a c^{2} d^{8} + 2 \, a^{2} c d^{4} e^{4} + a^{3} e^{8}\right )}} + \frac {c d e^{2} x^{3} - c d^{2} e x^{2} + c d^{3} x + a e^{3}}{4 \, {\left (a^{2} c d^{4} + a^{3} e^{4} + {\left (a c^{2} d^{4} + a^{2} c e^{4}\right )} x^{4}\right )}} \]
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Time = 0.46 (sec) , antiderivative size = 795, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\frac {e^{8} \log \left ({\left | e x + d \right |}\right )}{c^{2} d^{8} e + 2 \, a c d^{4} e^{5} + a^{2} e^{9}} - \frac {e^{7} \log \left ({\left | c x^{4} + a \right |}\right )}{4 \, {\left (c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}\right )}} + \frac {{\left (4 \, \sqrt {2} \sqrt {a c} c^{2} d^{2} e + 3 \, \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 )}{8 \, {\left (\sqrt {2} a^{2} c^{3} d^{4} + \sqrt {2} a^{3} c^{2} e^{4} + 4 \, \sqrt {2} \sqrt {a c} a^{2} c^{2} d^{2} e^{2} - 4 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c^{2} d^{3} e - 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a^{2} d e^{3}\right )}} + \frac {{\left (4 \, \sqrt {2} \sqrt {a c} c^{2} d^{2} e + 3 \, \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 )}{8 \, {\left (\sqrt {2} a^{2} c^{3} d^{4} + \sqrt {2} a^{3} c^{2} e^{4} + 4 \, \sqrt {2} \sqrt {a c} a^{2} c^{2} d^{2} e^{2} + 4 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c^{2} d^{3} e + 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a^{2} d e^{3}\right )}} + \frac {{\left (3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{7} + 7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{3} e^{4} - \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{5} e^{2} - 5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d e^{6}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, {\left (a^{2} c^{4} d^{8} + 2 \, a^{3} c^{3} d^{4} e^{4} + a^{4} c^{2} e^{8}\right )}} - \frac {{\left (3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{7} + 7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{3} e^{4} - \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{5} e^{2} - 5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d e^{6}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, {\left (a^{2} c^{4} d^{8} + 2 \, a^{3} c^{3} d^{4} e^{4} + a^{4} c^{2} e^{8}\right )}} + \frac {a c d^{4} e^{3} + a^{2} e^{7} + {\left (c^{2} d^{5} e^{2} + a c d e^{6}\right )} x^{3} - {\left (c^{2} d^{6} e + a c d^{2} e^{5}\right )} x^{2} + {\left (c^{2} d^{7} + a c d^{3} e^{4}\right )} x}{4 \, {\left (c d^{4} + a e^{4}\right )}^{2} {\left (c x^{4} + a\right )} a} \]
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Time = 9.66 (sec) , antiderivative size = 1591, normalized size of antiderivative = 1.86 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]
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