Optimal. Leaf size=305 \[ \frac{4 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ),\frac{1}{8}\right )}{175 \sqrt{x^4+3 x^2+4}}+\frac{4 \sqrt{x^4+3 x^2+4} x}{175 \left (x^2+2\right )}+\frac{22 \sqrt{x^4+3 x^2+4} x}{175 \left (5 x^2+7\right )}+\frac{1}{75} \sqrt{x^4+3 x^2+4} x+\frac{13}{350} \sqrt{\frac{11}{35}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right )-\frac{4 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{175 \sqrt{x^4+3 x^2+4}}+\frac{2431 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{36750 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]
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Rubi [A] time = 0.534379, antiderivative size = 372, normalized size of antiderivative = 1.22, number of steps used = 19, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {1228, 1103, 1139, 1195, 1122, 1197, 1223, 1714, 1708, 1706, 1216} \[ \frac{4 \sqrt{x^4+3 x^2+4} x}{175 \left (x^2+2\right )}+\frac{22 \sqrt{x^4+3 x^2+4} x}{175 \left (5 x^2+7\right )}+\frac{1}{75} \sqrt{x^4+3 x^2+4} x+\frac{13}{350} \sqrt{\frac{11}{35}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right )+\frac{4 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{175 \sqrt{x^4+3 x^2+4}}-\frac{4 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{175 \sqrt{x^4+3 x^2+4}}+\frac{187 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{13125 \sqrt{x^4+3 x^2+4}}+\frac{6919 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{183750 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1228
Rule 1103
Rule 1139
Rule 1195
Rule 1122
Rule 1197
Rule 1223
Rule 1714
Rule 1708
Rule 1706
Rule 1216
Rubi steps
\begin{align*} \int \frac{\left (4+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx &=\int \left (\frac{152}{625 \sqrt{4+3 x^2+x^4}}+\frac{16 x^2}{125 \sqrt{4+3 x^2+x^4}}+\frac{x^4}{25 \sqrt{4+3 x^2+x^4}}+\frac{1936}{625 \left (7+5 x^2\right )^2 \sqrt{4+3 x^2+x^4}}+\frac{88}{625 \left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}}\right ) \, dx\\ &=\frac{1}{25} \int \frac{x^4}{\sqrt{4+3 x^2+x^4}} \, dx+\frac{16}{125} \int \frac{x^2}{\sqrt{4+3 x^2+x^4}} \, dx+\frac{88}{625} \int \frac{1}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx+\frac{152}{625} \int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx+\frac{1936}{625} \int \frac{1}{\left (7+5 x^2\right )^2 \sqrt{4+3 x^2+x^4}} \, dx\\ &=\frac{1}{75} x \sqrt{4+3 x^2+x^4}+\frac{22 x \sqrt{4+3 x^2+x^4}}{175 \left (7+5 x^2\right )}+\frac{38 \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 )}{625 \sqrt{4+3 x^2+x^4}}-\frac{22 \int \frac{12+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{4375}-\frac{1}{75} \int \frac{4+6 x^2}{\sqrt{4+3 x^2+x^4}} \, dx-\frac{88 \int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx}{1875}+\frac{32}{125} \int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx-\frac{32}{125} \int \frac{1-\frac{x^2}{2}}{\sqrt{4+3 x^2+x^4}} \, dx+\frac{176}{375} \int \frac{1+\frac{x^2}{2}}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx\\ &=\frac{1}{75} x \sqrt{4+3 x^2+x^4}+\frac{16 x \sqrt{4+3 x^2+x^4}}{125 \left (2+x^2\right )}+\frac{22 x \sqrt{4+3 x^2+x^4}}{175 \left (7+5 x^2\right )}+\frac{2}{125} \sqrt{\frac{11}{35}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{4+3 x^2+x^4}}\right )-\frac{16 \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 )}{125 \sqrt{4+3 x^2+x^4}}+\frac{212 \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 )}{1875 \sqrt{4+3 x^2+x^4}}+\frac{187 \sqrt{2} \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 )}{13125 \sqrt{4+3 x^2+x^4}}-\frac{22 \int \frac{410+425 x^2}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{21875}+\frac{44}{875} \int \frac{1-\frac{x^2}{2}}{\sqrt{4+3 x^2+x^4}} \, dx+\frac{4}{25} \int \frac{1-\frac{x^2}{2}}{\sqrt{4+3 x^2+x^4}} \, dx-\frac{16}{75} \int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx\\ &=\frac{1}{75} x \sqrt{4+3 x^2+x^4}+\frac{4 x \sqrt{4+3 x^2+x^4}}{175 \left (2+x^2\right )}+\frac{22 x \sqrt{4+3 x^2+x^4}}{175 \left (7+5 x^2\right )}+\frac{2}{125} \sqrt{\frac{11}{35}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{4+3 x^2+x^4}}\right )-\frac{4 \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 )}{175 \sqrt{4+3 x^2+x^4}}+\frac{112 \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 )}{1875 \sqrt{4+3 x^2+x^4}}+\frac{187 \sqrt{2} \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 )}{13125 \sqrt{4+3 x^2+x^4}}-\frac{1936 \int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx}{13125}+\frac{1628 \int \frac{1+\frac{x^2}{2}}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{2625}\\ &=\frac{1}{75} x \sqrt{4+3 x^2+x^4}+\frac{4 x \sqrt{4+3 x^2+x^4}}{175 \left (2+x^2\right )}+\frac{22 x \sqrt{4+3 x^2+x^4}}{175 \left (7+5 x^2\right )}+\frac{13}{350} \sqrt{\frac{11}{35}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{4+3 x^2+x^4}}\right )-\frac{4 \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 )}{175 \sqrt{4+3 x^2+x^4}}+\frac{4 \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 )}{175 \sqrt{4+3 x^2+x^4}}+\frac{6919 \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 )}{183750 \sqrt{2} \sqrt{4+3 x^2+x^4}}+\frac{187 \sqrt{2} \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 )}{13125 \sqrt{4+3 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.585088, size = 309, normalized size = 1.01 \[ \frac{\frac{175 x \left (7 x^2+23\right ) \left (x^4+3 x^2+4\right )}{5 x^2+7}-i \sqrt{6+2 i \sqrt{7}} \sqrt{1-\frac{2 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left (7 \left (158+15 i \sqrt{7}\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{\sqrt{7}-3 i}} x\right ),\frac{-\sqrt{7}+3 i}{\sqrt{7}+3 i}\right )+105 \left (3-i \sqrt{7}\right ) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )+429 \Pi \left (\frac{5}{14} \left (3+i \sqrt{7}\right );i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )\right )}{18375 \sqrt{x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 425, normalized size = 1.4 \begin{align*}{\frac{22\,x}{875\,{x}^{2}+1225}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{x}{75}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{232}{375\,\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{128}{175\,\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{128}{175\,\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{286}{6125\,\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{{\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{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac{3}{2}}}{25 \, x^{4} + 70 \, x^{2} + 49}, 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{\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{{\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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