Optimal. Leaf size=102 \[ -\frac{1251 \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )}{24500}+\frac{563 \sqrt{-x^4+x^2+2} x}{9800 \left (5 x^2+7\right )}-\frac{17 \sqrt{-x^4+x^2+2} x}{350 \left (5 x^2+7\right )^2}+\frac{191 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{9800}+\frac{9879 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{343000} \]
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Rubi [A] time = 0.497346, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 13, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.542, Rules used = {1228, 1095, 419, 1132, 493, 424, 1223, 1696, 1716, 1180, 524, 1212, 537} \[ \frac{563 \sqrt{-x^4+x^2+2} x}{9800 \left (5 x^2+7\right )}-\frac{17 \sqrt{-x^4+x^2+2} x}{350 \left (5 x^2+7\right )^2}-\frac{1251 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{24500}+\frac{191 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{9800}+\frac{9879 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{343000} \]
Antiderivative was successfully verified.
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Rule 1228
Rule 1095
Rule 419
Rule 1132
Rule 493
Rule 424
Rule 1223
Rule 1696
Rule 1716
Rule 1180
Rule 524
Rule 1212
Rule 537
Rubi steps
\begin{align*} \int \frac{\left (2+x^2-x^4\right )^{3/2}}{\left (7+5 x^2\right )^3} \, dx &=\int \left (-\frac{31}{625 \sqrt{2+x^2-x^4}}+\frac{x^2}{125 \sqrt{2+x^2-x^4}}+\frac{1156}{625 \left (7+5 x^2\right )^3 \sqrt{2+x^2-x^4}}-\frac{1292}{625 \left (7+5 x^2\right )^2 \sqrt{2+x^2-x^4}}+\frac{429}{625 \left (7+5 x^2\right ) \sqrt{2+x^2-x^4}}\right ) \, dx\\ &=\frac{1}{125} \int \frac{x^2}{\sqrt{2+x^2-x^4}} \, dx-\frac{31}{625} \int \frac{1}{\sqrt{2+x^2-x^4}} \, dx+\frac{429}{625} \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+x^2-x^4}} \, dx+\frac{1156}{625} \int \frac{1}{\left (7+5 x^2\right )^3 \sqrt{2+x^2-x^4}} \, dx-\frac{1292}{625} \int \frac{1}{\left (7+5 x^2\right )^2 \sqrt{2+x^2-x^4}} \, dx\\ &=-\frac{17 x \sqrt{2+x^2-x^4}}{350 \left (7+5 x^2\right )^2}+\frac{19 x \sqrt{2+x^2-x^4}}{175 \left (7+5 x^2\right )}+\frac{17 \int \frac{186-190 x^2+25 x^4}{\left (7+5 x^2\right )^2 \sqrt{2+x^2-x^4}} \, dx}{8750}-\frac{19 \int \frac{118-70 x^2-25 x^4}{\left (7+5 x^2\right ) \sqrt{2+x^2-x^4}} \, dx}{4375}+\frac{2}{125} \int \frac{x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx-\frac{62}{625} \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx+\frac{858}{625} \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2} \left (7+5 x^2\right )} \, dx\\ &=-\frac{17 x \sqrt{2+x^2-x^4}}{350 \left (7+5 x^2\right )^2}+\frac{563 x \sqrt{2+x^2-x^4}}{9800 \left (7+5 x^2\right )}-\frac{31}{625} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{429 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{4375}+\frac{\int \frac{37698-32690 x^2-12525 x^4}{\left (7+5 x^2\right ) \sqrt{2+x^2-x^4}} \, dx}{245000}+\frac{19 \int \frac{175+125 x^2}{\sqrt{2+x^2-x^4}} \, dx}{109375}+\frac{1}{125} \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx-\frac{2}{125} \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx-\frac{3173 \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+x^2-x^4}} \, dx}{4375}\\ &=-\frac{17 x \sqrt{2+x^2-x^4}}{350 \left (7+5 x^2\right )^2}+\frac{563 x \sqrt{2+x^2-x^4}}{9800 \left (7+5 x^2\right )}+\frac{1}{125} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{36}{625} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{429 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{4375}-\frac{\int \frac{75775+62625 x^2}{\sqrt{2+x^2-x^4}} \, dx}{6125000}+\frac{38 \int \frac{175+125 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx}{109375}+\frac{11783 \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+x^2-x^4}} \, dx}{49000}-\frac{6346 \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2} \left (7+5 x^2\right )} \, dx}{4375}\\ &=-\frac{17 x \sqrt{2+x^2-x^4}}{350 \left (7+5 x^2\right )^2}+\frac{563 x \sqrt{2+x^2-x^4}}{9800 \left (7+5 x^2\right )}+\frac{1}{125} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{36}{625} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{34 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{6125}-\frac{\int \frac{75775+62625 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx}{3062500}+\frac{76 \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx}{4375}+\frac{19}{875} \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx+\frac{11783 \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2} \left (7+5 x^2\right )} \, dx}{24500}\\ &=-\frac{17 x \sqrt{2+x^2-x^4}}{350 \left (7+5 x^2\right )^2}+\frac{563 x \sqrt{2+x^2-x^4}}{9800 \left (7+5 x^2\right )}+\frac{26}{875} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{214 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{4375}+\frac{9879 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{343000}-\frac{263 \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx}{61250}-\frac{501 \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx}{49000}\\ &=-\frac{17 x \sqrt{2+x^2-x^4}}{350 \left (7+5 x^2\right )^2}+\frac{563 x \sqrt{2+x^2-x^4}}{9800 \left (7+5 x^2\right )}+\frac{191 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{9800}-\frac{1251 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{24500}+\frac{9879 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{343000}\\ \end{align*}
Mathematica [C] time = 0.422615, size = 244, normalized size = 2.39 \[ \frac{-2541 i \sqrt{2} \left (5 x^2+7\right )^2 \sqrt{-x^4+x^2+2} \text{EllipticF}\left (i \sinh ^{-1}(x),-\frac{1}{2}\right )-197050 x^7-45500 x^5+636650 x^3+13370 i \sqrt{2} \left (5 x^2+7\right )^2 \sqrt{-x^4+x^2+2} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-246975 i \sqrt{2} \sqrt{-x^4+x^2+2} x^4 \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )-691530 i \sqrt{2} \sqrt{-x^4+x^2+2} x^2 \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )-484071 i \sqrt{2} \sqrt{-x^4+x^2+2} \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )+485100 x}{686000 \left (5 x^2+7\right )^2 \sqrt{-x^4+x^2+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 189, normalized size = 1.9 \begin{align*} -{\frac{17\,x}{350\, \left ( 5\,{x}^{2}+7 \right ) ^{2}}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{563\,x}{49000\,{x}^{2}+68600}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{1251\,\sqrt{2}}{49000}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{191\,\sqrt{2}}{19600}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticE} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{9879\,\sqrt{2}}{343000}\sqrt{1-{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{x\sqrt{2}}{2}},-{\frac{10}{7}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{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} + 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} + 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} - 2\right ) \left (x^{2} + 1\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} + 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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