Optimal. Leaf size=231 \[ \frac{5 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{2 x^2+2}} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{784 \sqrt{x^4+3 x^2+2}}+\frac{17 \sqrt{x^4+3 x^2+2} x}{9800 \left (5 x^2+7\right )}-\frac{3 \sqrt{x^4+3 x^2+2} x}{350 \left (5 x^2+7\right )^2}+\frac{3 \left (x^2+2\right ) x}{392 \sqrt{x^4+3 x^2+2}}-\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{2 x^2+2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{196 \sqrt{x^4+3 x^2+2}}+\frac{141 \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{27440 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]
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Rubi [A] time = 0.667752, antiderivative size = 288, normalized size of antiderivative = 1.25, number of steps used = 27, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {1228, 1099, 1135, 1223, 1696, 1716, 1189, 1214, 1456, 539} \[ \frac{17 \sqrt{x^4+3 x^2+2} x}{9800 \left (5 x^2+7\right )}-\frac{3 \sqrt{x^4+3 x^2+2} x}{350 \left (5 x^2+7\right )^2}+\frac{3 \left (x^2+2\right ) x}{392 \sqrt{x^4+3 x^2+2}}+\frac{5 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{784 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{6 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{875 \sqrt{x^4+3 x^2+2}}-\frac{39 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{24500 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{141 \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{27440 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1228
Rule 1099
Rule 1135
Rule 1223
Rule 1696
Rule 1716
Rule 1189
Rule 1214
Rule 1456
Rule 539
Rubi steps
\begin{align*} \int \frac{\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^3} \, dx &=\int \left (\frac{9}{625 \sqrt{2+3 x^2+x^4}}+\frac{x^2}{125 \sqrt{2+3 x^2+x^4}}+\frac{36}{625 \left (7+5 x^2\right )^3 \sqrt{2+3 x^2+x^4}}-\frac{12}{625 \left (7+5 x^2\right )^2 \sqrt{2+3 x^2+x^4}}-\frac{11}{625 \left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}}\right ) \, dx\\ &=\frac{1}{125} \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{9}{625} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx-\frac{11}{625} \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx-\frac{12}{625} \int \frac{1}{\left (7+5 x^2\right )^2 \sqrt{2+3 x^2+x^4}} \, dx+\frac{36}{625} \int \frac{1}{\left (7+5 x^2\right )^3 \sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{x \left (2+x^2\right )}{125 \sqrt{2+3 x^2+x^4}}-\frac{3 x \sqrt{2+3 x^2+x^4}}{350 \left (7+5 x^2\right )^2}+\frac{x \sqrt{2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac{\sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{125 \sqrt{2+3 x^2+x^4}}+\frac{9 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{625 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{\int \frac{62+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{4375}+\frac{3 \int \frac{74-10 x^2-25 x^4}{\left (7+5 x^2\right )^2 \sqrt{2+3 x^2+x^4}} \, dx}{8750}-\frac{11 \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx}{1250}+\frac{11}{500} \int \frac{2+2 x^2}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{x \left (2+x^2\right )}{125 \sqrt{2+3 x^2+x^4}}-\frac{3 x \sqrt{2+3 x^2+x^4}}{350 \left (7+5 x^2\right )^2}+\frac{17 x \sqrt{2+3 x^2+x^4}}{9800 \left (7+5 x^2\right )}-\frac{\sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{125 \sqrt{2+3 x^2+x^4}}+\frac{7 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{1250 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{\int \frac{2838+2310 x^2+975 x^4}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{245000}+\frac{\int \frac{-175-125 x^2}{\sqrt{2+3 x^2+x^4}} \, dx}{109375}-\frac{13 \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{4375}+\frac{\left (11 \sqrt{1+\frac{x^2}{2}} \sqrt{2+2 x^2}\right ) \int \frac{\sqrt{2+2 x^2}}{\sqrt{1+\frac{x^2}{2}} \left (7+5 x^2\right )} \, dx}{500 \sqrt{2+3 x^2+x^4}}\\ &=\frac{x \left (2+x^2\right )}{125 \sqrt{2+3 x^2+x^4}}-\frac{3 x \sqrt{2+3 x^2+x^4}}{350 \left (7+5 x^2\right )^2}+\frac{17 x \sqrt{2+3 x^2+x^4}}{9800 \left (7+5 x^2\right )}-\frac{\sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{125 \sqrt{2+3 x^2+x^4}}+\frac{7 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{1250 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{11 \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{1750 \sqrt{2} \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}-\frac{\int \frac{-4725-4875 x^2}{\sqrt{2+3 x^2+x^4}} \, dx}{6125000}-\frac{1}{875} \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx-\frac{13 \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx}{8750}-\frac{1}{625} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{13 \int \frac{2+2 x^2}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{3500}+\frac{303 \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{49000}\\ &=\frac{6 x \left (2+x^2\right )}{875 \sqrt{2+3 x^2+x^4}}-\frac{3 x \sqrt{2+3 x^2+x^4}}{350 \left (7+5 x^2\right )^2}+\frac{17 x \sqrt{2+3 x^2+x^4}}{9800 \left (7+5 x^2\right )}-\frac{6 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{875 \sqrt{2+3 x^2+x^4}}+\frac{11 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{4375 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{11 \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{1750 \sqrt{2} \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}+\frac{27 \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx}{35000}+\frac{39 \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx}{49000}+\frac{303 \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx}{98000}-\frac{303 \int \frac{2+2 x^2}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{39200}+\frac{\left (13 \sqrt{1+\frac{x^2}{2}} \sqrt{2+2 x^2}\right ) \int \frac{\sqrt{2+2 x^2}}{\sqrt{1+\frac{x^2}{2}} \left (7+5 x^2\right )} \, dx}{3500 \sqrt{2+3 x^2+x^4}}\\ &=\frac{3 x \left (2+x^2\right )}{392 \sqrt{2+3 x^2+x^4}}-\frac{3 x \sqrt{2+3 x^2+x^4}}{350 \left (7+5 x^2\right )^2}+\frac{17 x \sqrt{2+3 x^2+x^4}}{9800 \left (7+5 x^2\right )}-\frac{39 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{24500 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{6 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{875 \sqrt{2+3 x^2+x^4}}+\frac{5 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{784 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{9 \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{1225 \sqrt{2} \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}-\frac{\left (303 \sqrt{1+\frac{x^2}{2}} \sqrt{2+2 x^2}\right ) \int \frac{\sqrt{2+2 x^2}}{\sqrt{1+\frac{x^2}{2}} \left (7+5 x^2\right )} \, dx}{39200 \sqrt{2+3 x^2+x^4}}\\ &=\frac{3 x \left (2+x^2\right )}{392 \sqrt{2+3 x^2+x^4}}-\frac{3 x \sqrt{2+3 x^2+x^4}}{350 \left (7+5 x^2\right )^2}+\frac{17 x \sqrt{2+3 x^2+x^4}}{9800 \left (7+5 x^2\right )}-\frac{39 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{24500 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{6 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{875 \sqrt{2+3 x^2+x^4}}+\frac{5 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{784 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{141 \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{27440 \sqrt{2} \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.375343, size = 174, normalized size = 0.75 \[ \frac{-406 i \sqrt{x^2+1} \sqrt{x^2+2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ),2\right )+\frac{119 x \left (x^4+3 x^2+2\right )}{5 x^2+7}-\frac{588 x \left (x^4+3 x^2+2\right )}{\left (5 x^2+7\right )^2}-525 i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+141 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left (\frac{10}{7};\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )}{68600 \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 186, normalized size = 0.8 \begin{align*} -{\frac{3\,x}{350\, \left ( 5\,{x}^{2}+7 \right ) ^{2}}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{17\,x}{49000\,{x}^{2}+68600}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{29\,i}{9800}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{{\frac{3\,i}{784}}\sqrt{2}{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{141\,i}{68600}}\sqrt{2}\sqrt{1+{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},{\frac{10}{7}},\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \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} + 2\right )}^{\frac{3}{2}}}{{\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{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{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{\left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac{3}{2}}}{\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{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{{\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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