Optimal. Leaf size=855 \[ \frac{\log (d+e x) e^7}{\left (c d^4+a e^4\right )^2}-\frac{\log \left (c x^4+a\right ) e^7}{4 \left (c d^4+a e^4\right )^2}-\frac{\sqrt{c} d^2 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right ) e^5}{2 \sqrt{a} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^4}{2 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^4}{2 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^4}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^4}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac{\sqrt{c} d^2 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right ) e}{4 a^{3/2} \left (c d^4+a e^4\right )}+\frac{a e^3+c x \left (d^3-e x d^2+e^2 x^2 d\right )}{4 a \left (c d^4+a e^4\right ) \left (c x^4+a\right )}-\frac{\sqrt [4]{c} d \left (3 \sqrt{c} d^2+\sqrt{a} e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )}+\frac{\sqrt [4]{c} d \left (3 \sqrt{c} d^2+\sqrt{a} e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )}-\frac{\sqrt [4]{c} d \left (3 \sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )}+\frac{\sqrt [4]{c} d \left (3 \sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.849351, antiderivative size = 855, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 14, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.824, Rules used = {6742, 1854, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628, 1248, 635, 260} \[ \frac{\log (d+e x) e^7}{\left (c d^4+a e^4\right )^2}-\frac{\log \left (c x^4+a\right ) e^7}{4 \left (c d^4+a e^4\right )^2}-\frac{\sqrt{c} d^2 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right ) e^5}{2 \sqrt{a} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^4}{2 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^4}{2 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^4}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^4}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac{\sqrt{c} d^2 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right ) e}{4 a^{3/2} \left (c d^4+a e^4\right )}+\frac{a e^3+c x \left (d^3-e x d^2+e^2 x^2 d\right )}{4 a \left (c d^4+a e^4\right ) \left (c x^4+a\right )}-\frac{\sqrt [4]{c} d \left (3 \sqrt{c} d^2+\sqrt{a} e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )}+\frac{\sqrt [4]{c} d \left (3 \sqrt{c} d^2+\sqrt{a} e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )}-\frac{\sqrt [4]{c} d \left (3 \sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )}+\frac{\sqrt [4]{c} d \left (3 \sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6742
Rule 1854
Rule 1876
Rule 275
Rule 205
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 1248
Rule 635
Rule 260
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \left (a+c x^4\right )^2} \, dx &=\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.421641, size = 558, normalized size = 0.65 \[ \frac{\frac{\sqrt{2} \sqrt [4]{c} \left (5 a^{3/2} d e^6+\sqrt{a} c d^5 e^2-7 a \sqrt{c} d^3 e^4-3 c^{3/2} d^7\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{7/4}}+\frac{\sqrt{2} \sqrt [4]{c} \left (-5 a^{3/2} d e^6-\sqrt{a} c d^5 e^2+7 a \sqrt{c} d^3 e^4+3 c^{3/2} d^7\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{7/4}}-\frac{2 \sqrt [4]{c} d \left (-12 a^{5/4} \sqrt [4]{c} d e^5+5 \sqrt{2} a^{3/2} e^6-4 \sqrt [4]{a} c^{5/4} d^5 e+\sqrt{2} \sqrt{a} c d^4 e^2+7 \sqrt{2} a \sqrt{c} d^2 e^4+3 \sqrt{2} c^{3/2} d^6\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{2 \sqrt [4]{c} d \left (12 a^{5/4} \sqrt [4]{c} d e^5+5 \sqrt{2} a^{3/2} e^6+4 \sqrt [4]{a} c^{5/4} d^5 e+\sqrt{2} \sqrt{a} c d^4 e^2+7 \sqrt{2} a \sqrt{c} d^2 e^4+3 \sqrt{2} c^{3/2} d^6\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}+\frac{8 \left (a e^4+c d^4\right ) \left (a e^3+c d x \left (d^2-d e x+e^2 x^2\right )\right )}{a \left (a+c x^4\right )}-8 e^7 \log \left (a+c x^4\right )+32 e^7 \log (d+e x)}{32 \left (a e^4+c d^4\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 1122, normalized size = 1.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29737, size = 1038, normalized size = 1.21 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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