Optimal. Leaf size=312 \[ -\frac{23 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ),\frac{1}{8}\right )}{2940 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{139 \sqrt{x^4+3 x^2+4} x}{86240 \left (x^2+2\right )}+\frac{139 \sqrt{x^4+3 x^2+4} x}{17248 \left (5 x^2+7\right )}+\frac{\sqrt{x^4+3 x^2+4} x}{28 \left (5 x^2+7\right )^2}+\frac{14999 \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right )}{344960 \sqrt{385}}+\frac{139 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{43120 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{254983 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{36220800 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]
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Rubi [A] time = 0.710943, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {1228, 1223, 1696, 1714, 1195, 1708, 1103, 1706, 1216} \[ -\frac{139 \sqrt{x^4+3 x^2+4} x}{86240 \left (x^2+2\right )}+\frac{139 \sqrt{x^4+3 x^2+4} x}{17248 \left (5 x^2+7\right )}+\frac{\sqrt{x^4+3 x^2+4} x}{28 \left (5 x^2+7\right )^2}+\frac{14999 \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right )}{344960 \sqrt{385}}-\frac{23 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{2940 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{139 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{43120 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{254983 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{36220800 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1228
Rule 1223
Rule 1696
Rule 1714
Rule 1195
Rule 1708
Rule 1103
Rule 1706
Rule 1216
Rubi steps
\begin{align*} \int \frac{\sqrt{4+3 x^2+x^4}}{\left (7+5 x^2\right )^3} \, dx &=\int \left (\frac{44}{25 \left (7+5 x^2\right )^3 \sqrt{4+3 x^2+x^4}}+\frac{1}{25 \left (7+5 x^2\right )^2 \sqrt{4+3 x^2+x^4}}+\frac{1}{25 \left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}}\right ) \, dx\\ &=\frac{1}{25} \int \frac{1}{\left (7+5 x^2\right )^2 \sqrt{4+3 x^2+x^4}} \, dx+\frac{1}{25} \int \frac{1}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx+\frac{44}{25} \int \frac{1}{\left (7+5 x^2\right )^3 \sqrt{4+3 x^2+x^4}} \, dx\\ &=\frac{x \sqrt{4+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac{x \sqrt{4+3 x^2+x^4}}{616 \left (7+5 x^2\right )}-\frac{\int \frac{12+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{15400}-\frac{1}{700} \int \frac{-76-10 x^2-25 x^4}{\left (7+5 x^2\right )^2 \sqrt{4+3 x^2+x^4}} \, dx-\frac{1}{75} \int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx+\frac{2}{15} \int \frac{1+\frac{x^2}{2}}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx\\ &=\frac{x \sqrt{4+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac{139 x \sqrt{4+3 x^2+x^4}}{17248 \left (7+5 x^2\right )}+\frac{\tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{4+3 x^2+x^4}}\right )}{20 \sqrt{385}}-\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 )}{150 \sqrt{2} \sqrt{4+3 x^2+x^4}}+\frac{17 \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 )}{2100 \sqrt{2} \sqrt{4+3 x^2+x^4}}+\frac{\int \frac{-4412-4690 x^2-2775 x^4}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{431200}-\frac{\int \frac{410+425 x^2}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{77000}+\frac{\int \frac{1-\frac{x^2}{2}}{\sqrt{4+3 x^2+x^4}} \, dx}{1540}\\ &=-\frac{x \sqrt{4+3 x^2+x^4}}{3080 \left (2+x^2\right )}+\frac{x \sqrt{4+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac{139 x \sqrt{4+3 x^2+x^4}}{17248 \left (7+5 x^2\right )}+\frac{\tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{4+3 x^2+x^4}}\right )}{20 \sqrt{385}}+\frac{\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 )}{1540 \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 )}{150 \sqrt{2} \sqrt{4+3 x^2+x^4}}+\frac{17 \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 )}{2100 \sqrt{2} \sqrt{4+3 x^2+x^4}}+\frac{\int \frac{-60910-31775 x^2}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{2156000}-\frac{1}{525} \int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx+\frac{111 \int \frac{1-\frac{x^2}{2}}{\sqrt{4+3 x^2+x^4}} \, dx}{43120}+\frac{37 \int \frac{1+\frac{x^2}{2}}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{4620}\\ &=-\frac{139 x \sqrt{4+3 x^2+x^4}}{86240 \left (2+x^2\right )}+\frac{x \sqrt{4+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac{139 x \sqrt{4+3 x^2+x^4}}{17248 \left (7+5 x^2\right )}+\frac{653 \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{4+3 x^2+x^4}}\right )}{12320 \sqrt{385}}+\frac{139 \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 )}{43120 \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 )}{525 \sqrt{4+3 x^2+x^4}}+\frac{11101 \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 )}{1293600 \sqrt{2} \sqrt{4+3 x^2+x^4}}-\frac{\int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx}{2450}-\frac{219 \int \frac{1+\frac{x^2}{2}}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{8624}\\ &=-\frac{139 x \sqrt{4+3 x^2+x^4}}{86240 \left (2+x^2\right )}+\frac{x \sqrt{4+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac{139 x \sqrt{4+3 x^2+x^4}}{17248 \left (7+5 x^2\right )}+\frac{14999 \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{4+3 x^2+x^4}}\right )}{344960 \sqrt{385}}+\frac{139 \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 )}{43120 \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 )}{4900 \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 )}{525 \sqrt{4+3 x^2+x^4}}+\frac{254983 \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 )}{36220800 \sqrt{2} \sqrt{4+3 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.670891, size = 308, normalized size = 0.99 \[ \frac{\frac{700 x \left (695 x^2+1589\right ) \left (x^4+3 x^2+4\right )}{\left (5 x^2+7\right )^2}+i \sqrt{6+2 i \sqrt{7}} \sqrt{1-\frac{2 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left (\left (-9597+4865 i \sqrt{7}\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{\sqrt{7}-3 i}} x\right ),\frac{-\sqrt{7}+3 i}{\sqrt{7}+3 i}\right )+4865 \left (3-i \sqrt{7}\right ) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-29998 \Pi \left (\frac{5}{14} \left (3+i \sqrt{7}\right );i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )\right )}{12073600 \sqrt{x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.024, size = 434, normalized size = 1.4 \begin{align*}{\frac{x}{28\, \left ( 5\,{x}^{2}+7 \right ) ^{2}}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{139\,x}{86240\,{x}^{2}+120736}\sqrt{{x}^{4}+3\,{x}^{2}+4}}-{\frac{51}{15400\,\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{139}{2695\,\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{139}{2695\,\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{14999}{3018400\,\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{\sqrt{x^{4} + 3 \, x^{2} + 4}}{{\left (5 \, x^{2} + 7\right )}^{3}}\,{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}}{125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343}, 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{\sqrt{\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )}}{\left (5 x^{2} + 7\right )^{3}}\, 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{\sqrt{x^{4} + 3 \, x^{2} + 4}}{{\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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