Optimal. Leaf size=337 \[ \frac{\left (\sqrt{c} d-\sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )}-\frac{\left (\sqrt{c} d-\sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )}-\frac{\sqrt{d} \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{a e^2+c d^2}-\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.267139, antiderivative size = 337, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {1288, 205, 1168, 1162, 617, 204, 1165, 628} \[ \frac{\left (\sqrt{c} d-\sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )}-\frac{\left (\sqrt{c} d-\sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )}-\frac{\sqrt{d} \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{a e^2+c d^2}-\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 1288
Rule 205
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{x^2}{\left (d+e x^2\right ) \left (a+c x^4\right )} \, dx &=\int \left (-\frac{d e}{\left (c d^2+a e^2\right ) \left (d+e x^2\right )}+\frac{a e+c d x^2}{\left (c d^2+a e^2\right ) \left (a+c x^4\right )}\right ) \, dx\\ &=\frac{\int \frac{a e+c d x^2}{a+c x^4} \, dx}{c d^2+a e^2}-\frac{(d e) \int \frac{1}{d+e x^2} \, dx}{c d^2+a e^2}\\ &=-\frac{\sqrt{d} \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{c d^2+a e^2}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{2 \left (c d^2+a e^2\right )}+\frac{\left (d+\frac{\sqrt{a} e}{\sqrt{c}}\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{2 \left (c d^2+a e^2\right )}\\ &=-\frac{\sqrt{d} \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{c d^2+a e^2}+\frac{\left (\sqrt [4]{c} \left (d-\frac{\sqrt{a} e}{\sqrt{c}}\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} \sqrt [4]{a} \left (c d^2+a e^2\right )}+\frac{\left (\sqrt [4]{c} \left (d-\frac{\sqrt{a} e}{\sqrt{c}}\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} \sqrt [4]{a} \left (c d^2+a e^2\right )}+\frac{\left (d+\frac{\sqrt{a} e}{\sqrt{c}}\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^2+a e^2\right )}+\frac{\left (d+\frac{\sqrt{a} e}{\sqrt{c}}\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^2+a e^2\right )}\\ &=-\frac{\sqrt{d} \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{c d^2+a e^2}+\frac{\sqrt [4]{c} \left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^2+a e^2\right )}-\frac{\sqrt [4]{c} \left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^2+a e^2\right )}+\frac{\left (\sqrt [4]{c} \left (d+\frac{\sqrt{a} e}{\sqrt{c}}\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} \sqrt [4]{a} \left (c d^2+a e^2\right )}-\frac{\left (\sqrt [4]{c} \left (d+\frac{\sqrt{a} e}{\sqrt{c}}\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} \sqrt [4]{a} \left (c d^2+a e^2\right )}\\ &=-\frac{\sqrt{d} \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{c d^2+a e^2}-\frac{\sqrt [4]{c} \left (d+\frac{\sqrt{a} e}{\sqrt{c}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \left (c d^2+a e^2\right )}+\frac{\sqrt [4]{c} \left (d+\frac{\sqrt{a} e}{\sqrt{c}}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \left (c d^2+a e^2\right )}+\frac{\sqrt [4]{c} \left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^2+a e^2\right )}-\frac{\sqrt [4]{c} \left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^2+a e^2\right )}\\ \end{align*}
Mathematica [A] time = 0.13345, size = 232, normalized size = 0.69 \[ \frac{\sqrt{2} \left (\left (\sqrt{c} d-\sqrt{a} e\right ) \left (\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )\right )-2 \left (\sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \left (\sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )\right )-8 \sqrt [4]{a} \sqrt [4]{c} \sqrt{d} \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 351, normalized size = 1. \begin{align*}{\frac{e\sqrt{2}}{4\,a{e}^{2}+4\,c{d}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{e\sqrt{2}}{8\,a{e}^{2}+8\,c{d}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{e\sqrt{2}}{4\,a{e}^{2}+4\,c{d}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{d\sqrt{2}}{8\,a{e}^{2}+8\,c{d}^{2}}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{d\sqrt{2}}{4\,a{e}^{2}+4\,c{d}^{2}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{d\sqrt{2}}{4\,a{e}^{2}+4\,c{d}^{2}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{de}{a{e}^{2}+c{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \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 [B] time = 1.91362, size = 7464, normalized size = 22.15 \begin{align*} \text{result too large to display} \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 = 1.13798, size = 454, normalized size = 1.35 \begin{align*} -\frac{\sqrt{d} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\frac{1}{2}}}{c d^{2} + a e^{2}} + \frac{{\left (\left (a c^{3}\right )^{\frac{1}{4}} a c e + \left (a c^{3}\right )^{\frac{3}{4}} d\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 )}{2 \,{\left (\sqrt{2} a c^{3} d^{2} + \sqrt{2} a^{2} c^{2} e^{2}\right )}} + \frac{{\left (\left (a c^{3}\right )^{\frac{1}{4}} a c e + \left (a c^{3}\right )^{\frac{3}{4}} d\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 )}{2 \,{\left (\sqrt{2} a c^{3} d^{2} + \sqrt{2} a^{2} c^{2} e^{2}\right )}} + \frac{{\left (\left (a c^{3}\right )^{\frac{1}{4}} a c e - \left (a c^{3}\right )^{\frac{3}{4}} d\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{4 \,{\left (\sqrt{2} a c^{3} d^{2} + \sqrt{2} a^{2} c^{2} e^{2}\right )}} - \frac{{\left (\left (a c^{3}\right )^{\frac{1}{4}} a c e - \left (a c^{3}\right )^{\frac{3}{4}} d\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{4 \,{\left (\sqrt{2} a c^{3} d^{2} + \sqrt{2} a^{2} c^{2} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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