Optimal. Leaf size=284 \[ -\frac{\left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ),\frac{1}{8}\right )}{12 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{\sqrt{x^4+3 x^2+4} x}{77 \left (x^2+2\right )}-\frac{\left (4 x^2+13\right ) x}{308 \sqrt{x^4+3 x^2+4}}+\frac{25}{176} \sqrt{\frac{5}{77}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right )-\frac{\sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{77 \sqrt{x^4+3 x^2+4}}+\frac{425 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{3696 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]
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Rubi [A] time = 0.16267, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {1221, 1178, 1197, 1103, 1195, 1216, 1706} \[ \frac{\sqrt{x^4+3 x^2+4} x}{77 \left (x^2+2\right )}-\frac{\left (4 x^2+13\right ) x}{308 \sqrt{x^4+3 x^2+4}}+\frac{25}{176} \sqrt{\frac{5}{77}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right )-\frac{\left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{12 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{\sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{77 \sqrt{x^4+3 x^2+4}}+\frac{425 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{3696 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1221
Rule 1178
Rule 1197
Rule 1103
Rule 1195
Rule 1216
Rule 1706
Rubi steps
\begin{align*} \int \frac{1}{\left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}} \, dx &=\frac{1}{44} \int \frac{-8-5 x^2}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx+\frac{25}{44} \int \frac{1}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx\\ &=-\frac{x \left (13+4 x^2\right )}{308 \sqrt{4+3 x^2+x^4}}+\frac{\int \frac{-4+16 x^2}{\sqrt{4+3 x^2+x^4}} \, dx}{1232}-\frac{25}{132} \int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx+\frac{125}{66} \int \frac{1+\frac{x^2}{2}}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx\\ &=-\frac{x \left (13+4 x^2\right )}{308 \sqrt{4+3 x^2+x^4}}+\frac{25}{176} \sqrt{\frac{5}{77}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{4+3 x^2+x^4}}\right )-\frac{25 \left (2+x^2\right ) \sqrt{\frac{4+3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{264 \sqrt{2} \sqrt{4+3 x^2+x^4}}+\frac{425 \left (2+x^2\right ) \sqrt{\frac{4+3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{3696 \sqrt{2} \sqrt{4+3 x^2+x^4}}+\frac{1}{44} \int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx-\frac{2}{77} \int \frac{1-\frac{x^2}{2}}{\sqrt{4+3 x^2+x^4}} \, dx\\ &=-\frac{x \left (13+4 x^2\right )}{308 \sqrt{4+3 x^2+x^4}}+\frac{x \sqrt{4+3 x^2+x^4}}{77 \left (2+x^2\right )}+\frac{25}{176} \sqrt{\frac{5}{77}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{4+3 x^2+x^4}}\right )-\frac{\sqrt{2} \left (2+x^2\right ) \sqrt{\frac{4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{77 \sqrt{4+3 x^2+x^4}}-\frac{\left (2+x^2\right ) \sqrt{\frac{4+3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{12 \sqrt{2} \sqrt{4+3 x^2+x^4}}+\frac{425 \left (2+x^2\right ) \sqrt{\frac{4+3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{3696 \sqrt{2} \sqrt{4+3 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.540215, size = 483, normalized size = 1.7 \[ \frac{\sqrt{2} \left (2 \sqrt{7}+7 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{\sqrt{7}-3 i}} x\right ),\frac{-\sqrt{7}+3 i}{\sqrt{7}+3 i}\right )-8 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x^3-2 \sqrt{2} \left (\sqrt{7}+3 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-25 i \sqrt{2} \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} \Pi \left (\frac{5}{14} \left (3+i \sqrt{7}\right );i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-26 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x}{616 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 409, normalized size = 1.4 \begin{align*} -2\,{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}} \left ({\frac{{x}^{3}}{154}}+{\frac{13\,x}{616}} \right ) }-{\frac{1}{77\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-{\frac{32}{77\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}+{\frac{32}{77\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticE} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}+{\frac{25}{308\,\sqrt{-3/8+i/8\sqrt{7}}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticPi} \left ( \sqrt{-{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7}}x,-{\frac{5}{-{\frac{21}{8}}+{\frac{7\,i}{8}}\sqrt{7}}},{\frac{\sqrt{-{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7}}}{\sqrt{-{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7}}}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{4} + 3 \, x^{2} + 4}}{5 \, x^{10} + 37 \, x^{8} + 127 \, x^{6} + 239 \, x^{4} + 248 \, x^{2} + 112}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac{3}{2}} \left (5 x^{2} + 7\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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