Optimal. Leaf size=864 \[ \frac{x e^4}{2 d \left (c d^2+a e^2\right )^2 \left (e x^2+d\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{2 d^{3/2} \left (c d^2+a e^2\right )^2}+\frac{4 c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{\left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2-4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^2}{2 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2-4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^2}{2 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2+4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^2}{4 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2+4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^2}{4 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}-\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c x \left (c d^2-2 c e x^2 d-a e^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (c x^4+a\right )} \]
[Out]
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Rubi [A] time = 0.906108, antiderivative size = 864, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {1239, 199, 205, 1179, 1168, 1162, 617, 204, 1165, 628} \[ \frac{x e^4}{2 d \left (c d^2+a e^2\right )^2 \left (e x^2+d\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{2 d^{3/2} \left (c d^2+a e^2\right )^2}+\frac{4 c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{\left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2-4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^2}{2 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2-4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^2}{2 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2+4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^2}{4 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2+4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^2}{4 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}-\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c x \left (c d^2-2 c e x^2 d-a e^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (c x^4+a\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1239
Rule 199
Rule 205
Rule 1179
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^2} \, dx &=\int \left (\frac{e^4}{\left (c d^2+a e^2\right )^2 \left (d+e x^2\right )^2}+\frac{4 c d e^4}{\left (c d^2+a e^2\right )^3 \left (d+e x^2\right )}+\frac{c \left (c d^2-a e^2-2 c d e x^2\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^4\right )^2}-\frac{c e^2 \left (-3 c d^2+a e^2+4 c d e x^2\right )}{\left (c d^2+a e^2\right )^3 \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac{\left (c e^2\right ) \int \frac{-3 c d^2+a e^2+4 c d e x^2}{a+c x^4} \, dx}{\left (c d^2+a e^2\right )^3}+\frac{\left (4 c d e^4\right ) \int \frac{1}{d+e x^2} \, dx}{\left (c d^2+a e^2\right )^3}+\frac{c \int \frac{c d^2-a e^2-2 c d e x^2}{\left (a+c x^4\right )^2} \, dx}{\left (c d^2+a e^2\right )^2}+\frac{e^4 \int \frac{1}{\left (d+e x^2\right )^2} \, dx}{\left (c d^2+a e^2\right )^2}\\ &=\frac{e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac{c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac{4 c \sqrt{d} e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac{\left (\sqrt{c} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-a e^2\right )\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{2 \sqrt{a} \left (c d^2+a e^2\right )^3}+\frac{\left (\sqrt{c} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-a e^2\right )\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{2 \sqrt{a} \left (c d^2+a e^2\right )^3}-\frac{c \int \frac{-3 \left (c d^2-a e^2\right )+2 c d e x^2}{a+c x^4} \, dx}{4 a \left (c d^2+a e^2\right )^2}+\frac{e^4 \int \frac{1}{d+e x^2} \, dx}{2 d \left (c d^2+a e^2\right )^2}\\ &=\frac{e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac{c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac{4 c \sqrt{d} e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}+\frac{\left (\sqrt{c} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-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 \sqrt{a} \left (c d^2+a e^2\right )^3}+\frac{\left (\sqrt{c} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-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 \sqrt{a} \left (c d^2+a e^2\right )^3}-\frac{\left (c^{3/4} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}-\frac{\left (c^{3/4} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}+\frac{\left (\sqrt{c} \left (3 c d^2-2 \sqrt{a} \sqrt{c} d e-3 a e^2\right )\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{8 a^{3/2} \left (c d^2+a e^2\right )^2}+\frac{\left (\sqrt{c} \left (3 c d^2+2 \sqrt{a} \sqrt{c} d e-3 a e^2\right )\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{8 a^{3/2} \left (c d^2+a e^2\right )^2}\\ &=\frac{e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac{c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac{4 c \sqrt{d} e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}-\frac{c^{3/4} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}+\frac{c^{3/4} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}+\frac{\left (c^{3/4} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}-\frac{\left (c^{3/4} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}+\frac{\left (\sqrt{c} \left (3 c d^2-2 \sqrt{a} \sqrt{c} d e-3 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^{3/2} \left (c d^2+a e^2\right )^2}+\frac{\left (\sqrt{c} \left (3 c d^2-2 \sqrt{a} \sqrt{c} d e-3 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^{3/2} \left (c d^2+a e^2\right )^2}-\frac{\left (c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}-\frac{\left (c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}\\ &=\frac{e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac{c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac{4 c \sqrt{d} e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}-\frac{c^{3/4} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}+\frac{c^{3/4} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}-\frac{c^{3/4} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}+\frac{c^{3/4} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}+\frac{\left (c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}-\frac{\left (c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}\\ &=\frac{e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac{c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac{4 c \sqrt{d} e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}-\frac{c^{3/4} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}+\frac{c^{3/4} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}-\frac{c^{3/4} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}+\frac{c^{3/4} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.623756, size = 540, normalized size = 0.62 \[ \frac{-\frac{\sqrt{2} c^{3/4} \left (18 a^{3/2} \sqrt{c} d e^3-7 a^2 e^4+2 \sqrt{a} c^{3/2} d^3 e+12 a c d^2 e^2+3 c^2 d^4\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{7/4}}+\frac{\sqrt{2} c^{3/4} \left (18 a^{3/2} \sqrt{c} d e^3-7 a^2 e^4+2 \sqrt{a} c^{3/2} d^3 e+12 a c d^2 e^2+3 c^2 d^4\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{7/4}}+\frac{2 \sqrt{2} c^{3/4} \left (18 a^{3/2} \sqrt{c} d e^3+7 a^2 e^4+2 \sqrt{a} c^{3/2} d^3 e-12 a c d^2 e^2-3 c^2 d^4\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}-\frac{2 \sqrt{2} c^{3/4} \left (18 a^{3/2} \sqrt{c} d e^3+7 a^2 e^4+2 \sqrt{a} c^{3/2} d^3 e-12 a c d^2 e^2-3 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}+\frac{16 e^4 x \left (a e^2+c d^2\right )}{d \left (d+e x^2\right )}+\frac{8 c x \left (a e^2+c d^2\right ) \left (c d \left (d-2 e x^2\right )-a e^2\right )}{a \left (a+c x^4\right )}+\frac{16 e^{7/2} \left (a e^2+9 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{3/2}}}{32 \left (a e^2+c d^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 1169, normalized size = 1.4 \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 = 1.15424, size = 1146, normalized size = 1.33 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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