Optimal. Leaf size=116 \[ -\frac{539419}{77} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )-\frac{625}{11} \left (-x^4+x^2+2\right )^{3/2} x^5-\frac{14500}{33} \left (-x^4+x^2+2\right )^{3/2} x^3-\frac{116100}{77} \left (-x^4+x^2+2\right )^{3/2} x+\frac{1}{231} \left (717372 x^2+177953\right ) \sqrt{-x^4+x^2+2} x+\frac{3764813}{231} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
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Rubi [A] time = 0.111749, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {1206, 1679, 1176, 1180, 524, 424, 419} \[ -\frac{625}{11} \left (-x^4+x^2+2\right )^{3/2} x^5-\frac{14500}{33} \left (-x^4+x^2+2\right )^{3/2} x^3-\frac{116100}{77} \left (-x^4+x^2+2\right )^{3/2} x+\frac{1}{231} \left (717372 x^2+177953\right ) \sqrt{-x^4+x^2+2} x-\frac{539419}{77} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{3764813}{231} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
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Rule 1206
Rule 1679
Rule 1176
Rule 1180
Rule 524
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \left (7+5 x^2\right )^4 \sqrt{2+x^2-x^4} \, dx &=-\frac{625}{11} x^5 \left (2+x^2-x^4\right )^{3/2}-\frac{1}{11} \int \sqrt{2+x^2-x^4} \left (-26411-75460 x^2-87100 x^4-43500 x^6\right ) \, dx\\ &=-\frac{14500}{33} x^3 \left (2+x^2-x^4\right )^{3/2}-\frac{625}{11} x^5 \left (2+x^2-x^4\right )^{3/2}+\frac{1}{99} \int \sqrt{2+x^2-x^4} \left (237699+940140 x^2+1044900 x^4\right ) \, dx\\ &=-\frac{116100}{77} x \left (2+x^2-x^4\right )^{3/2}-\frac{14500}{33} x^3 \left (2+x^2-x^4\right )^{3/2}-\frac{625}{11} x^5 \left (2+x^2-x^4\right )^{3/2}-\frac{1}{693} \int \left (-3753693-10760580 x^2\right ) \sqrt{2+x^2-x^4} \, dx\\ &=\frac{1}{231} x \left (177953+717372 x^2\right ) \sqrt{2+x^2-x^4}-\frac{116100}{77} x \left (2+x^2-x^4\right )^{3/2}-\frac{14500}{33} x^3 \left (2+x^2-x^4\right )^{3/2}-\frac{625}{11} x^5 \left (2+x^2-x^4\right )^{3/2}+\frac{\int \frac{96595020+169416585 x^2}{\sqrt{2+x^2-x^4}} \, dx}{10395}\\ &=\frac{1}{231} x \left (177953+717372 x^2\right ) \sqrt{2+x^2-x^4}-\frac{116100}{77} x \left (2+x^2-x^4\right )^{3/2}-\frac{14500}{33} x^3 \left (2+x^2-x^4\right )^{3/2}-\frac{625}{11} x^5 \left (2+x^2-x^4\right )^{3/2}+\frac{2 \int \frac{96595020+169416585 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx}{10395}\\ &=\frac{1}{231} x \left (177953+717372 x^2\right ) \sqrt{2+x^2-x^4}-\frac{116100}{77} x \left (2+x^2-x^4\right )^{3/2}-\frac{14500}{33} x^3 \left (2+x^2-x^4\right )^{3/2}-\frac{625}{11} x^5 \left (2+x^2-x^4\right )^{3/2}-\frac{1078838}{77} \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx+\frac{3764813}{231} \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx\\ &=\frac{1}{231} x \left (177953+717372 x^2\right ) \sqrt{2+x^2-x^4}-\frac{116100}{77} x \left (2+x^2-x^4\right )^{3/2}-\frac{14500}{33} x^3 \left (2+x^2-x^4\right )^{3/2}-\frac{625}{11} x^5 \left (2+x^2-x^4\right )^{3/2}+\frac{3764813}{231} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{539419}{77} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )\\ \end{align*}
Mathematica [C] time = 0.125528, size = 112, normalized size = 0.97 \[ \frac{-4838091 i \sqrt{-2 x^4+2 x^2+4} \text{EllipticF}\left (i \sinh ^{-1}(x),-\frac{1}{2}\right )-13125 x^{13}-75250 x^{11}-105925 x^9+231228 x^7+1125819 x^5-186503 x^3+3764813 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-1037294 x}{231 \sqrt{-x^4+x^2+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 193, normalized size = 1.7 \begin{align*} -{\frac{518647\,x}{231}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{20050\,{x}^{5}}{21}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{166072\,{x}^{3}}{231}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{625\,{x}^{9}}{11}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{12625\,{x}^{7}}{33}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{3764813\,\sqrt{2}}{462}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{1073278\,\sqrt{2}}{231}\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}}}} \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 \sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}^{4}\,{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 ({\left (625 \, x^{8} + 3500 \, x^{6} + 7350 \, x^{4} + 6860 \, x^{2} + 2401\right )} \sqrt{-x^{4} + x^{2} + 2}, 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 \sqrt{- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )} \left (5 x^{2} + 7\right )^{4}\, 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 \sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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