Optimal. Leaf size=312 \[ -\frac{2 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ),\frac{1}{8}\right )}{231 \sqrt{x^4+3 x^2+4}}-\frac{199 \sqrt{x^4+3 x^2+4} x}{27104 \left (x^2+2\right )}+\frac{625 \sqrt{x^4+3 x^2+4} x}{27104 \left (5 x^2+7\right )}+\frac{\left (37 x^2+24\right ) x}{13552 \sqrt{x^4+3 x^2+4}}+\frac{575 \sqrt{\frac{5}{77}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right )}{108416}+\frac{199 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{13552 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{9775 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{2276736 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]
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Rubi [A] time = 0.504976, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {1228, 1178, 1197, 1103, 1195, 1223, 1714, 1708, 1706, 1216} \[ -\frac{199 \sqrt{x^4+3 x^2+4} x}{27104 \left (x^2+2\right )}+\frac{625 \sqrt{x^4+3 x^2+4} x}{27104 \left (5 x^2+7\right )}+\frac{\left (37 x^2+24\right ) x}{13552 \sqrt{x^4+3 x^2+4}}+\frac{575 \sqrt{\frac{5}{77}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right )}{108416}-\frac{2 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{231 \sqrt{x^4+3 x^2+4}}+\frac{199 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{13552 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{9775 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{2276736 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1228
Rule 1178
Rule 1197
Rule 1103
Rule 1195
Rule 1223
Rule 1714
Rule 1708
Rule 1706
Rule 1216
Rubi steps
\begin{align*} \int \frac{1}{\left (7+5 x^2\right )^2 \left (4+3 x^2+x^4\right )^{3/2}} \, dx &=\int \left (\frac{-36+5 x^2}{1936 \left (4+3 x^2+x^4\right )^{3/2}}+\frac{25}{44 \left (7+5 x^2\right )^2 \sqrt{4+3 x^2+x^4}}-\frac{25}{1936 \left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}}\right ) \, dx\\ &=\frac{\int \frac{-36+5 x^2}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx}{1936}-\frac{25 \int \frac{1}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{1936}+\frac{25}{44} \int \frac{1}{\left (7+5 x^2\right )^2 \sqrt{4+3 x^2+x^4}} \, dx\\ &=\frac{x \left (24+37 x^2\right )}{13552 \sqrt{4+3 x^2+x^4}}+\frac{625 x \sqrt{4+3 x^2+x^4}}{27104 \left (7+5 x^2\right )}+\frac{\int \frac{-348-148 x^2}{\sqrt{4+3 x^2+x^4}} \, dx}{54208}-\frac{25 \int \frac{12+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{27104}+\frac{25 \int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx}{5808}-\frac{125 \int \frac{1+\frac{x^2}{2}}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{2904}\\ &=\frac{x \left (24+37 x^2\right )}{13552 \sqrt{4+3 x^2+x^4}}+\frac{625 x \sqrt{4+3 x^2+x^4}}{27104 \left (7+5 x^2\right )}-\frac{25 \sqrt{\frac{5}{77}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{4+3 x^2+x^4}}\right )}{7744}+\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 )}{11616 \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 )}{162624 \sqrt{2} \sqrt{4+3 x^2+x^4}}-\frac{5 \int \frac{410+425 x^2}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{27104}+\frac{37 \int \frac{1-\frac{x^2}{2}}{\sqrt{4+3 x^2+x^4}} \, dx}{6776}+\frac{125 \int \frac{1-\frac{x^2}{2}}{\sqrt{4+3 x^2+x^4}} \, dx}{13552}-\frac{23 \int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx}{1936}\\ &=\frac{x \left (24+37 x^2\right )}{13552 \sqrt{4+3 x^2+x^4}}-\frac{199 x \sqrt{4+3 x^2+x^4}}{27104 \left (2+x^2\right )}+\frac{625 x \sqrt{4+3 x^2+x^4}}{27104 \left (7+5 x^2\right )}-\frac{25 \sqrt{\frac{5}{77}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{4+3 x^2+x^4}}\right )}{7744}+\frac{199 \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 )}{13552 \sqrt{2} \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 )}{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 )}{162624 \sqrt{2} \sqrt{4+3 x^2+x^4}}-\frac{25}{924} \int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx+\frac{4625 \int \frac{1+\frac{x^2}{2}}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{40656}\\ &=\frac{x \left (24+37 x^2\right )}{13552 \sqrt{4+3 x^2+x^4}}-\frac{199 x \sqrt{4+3 x^2+x^4}}{27104 \left (2+x^2\right )}+\frac{625 x \sqrt{4+3 x^2+x^4}}{27104 \left (7+5 x^2\right )}+\frac{575 \sqrt{\frac{5}{77}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{4+3 x^2+x^4}}\right )}{108416}+\frac{199 \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 )}{13552 \sqrt{2} \sqrt{4+3 x^2+x^4}}-\frac{2 \sqrt{2} \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 )}{231 \sqrt{4+3 x^2+x^4}}+\frac{9775 \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 )}{2276736 \sqrt{2} \sqrt{4+3 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.593117, size = 311, normalized size = 1. \[ \frac{28 x \left (995 x^4+2633 x^2+2836\right )+i \sqrt{6+2 i \sqrt{7}} \left (5 x^2+7\right ) \sqrt{1-\frac{2 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left (7 \left (101+199 i \sqrt{7}\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{\sqrt{7}-3 i}} x\right ),\frac{-\sqrt{7}+3 i}{\sqrt{7}+3 i}\right )+1393 \left (3-i \sqrt{7}\right ) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-1150 \Pi \left (\frac{5}{14} \left (3+i \sqrt{7}\right );i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )\right )}{758912 \left (5 x^2+7\right ) \sqrt{x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.025, size = 433, normalized size = 1.4 \begin{align*} -2\,{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}} \left ( -{\frac{37\,{x}^{3}}{27104}}-{\frac{3\,x}{3388}} \right ) }+{\frac{625\,x}{135520\,{x}^{2}+189728}\sqrt{{x}^{4}+3\,{x}^{2}+4}}-{\frac{349}{6776\,\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{199}{847\,\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{199}{847\,\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{575}{189728\,\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 )}^{2}}\,{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}}{25 \, x^{12} + 220 \, x^{10} + 894 \, x^{8} + 2084 \, x^{6} + 2913 \, x^{4} + 2296 \, x^{2} + 784}, 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 )^{2}}\, 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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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