Optimal. Leaf size=72 \[ -\frac{178}{625} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )+\frac{1}{75} x \sqrt{-x^4+x^2+2} \left (13-3 x^2\right )+\frac{92}{375} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{1156 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{4375} \]
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Rubi [A] time = 0.136052, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1208, 1176, 1180, 524, 424, 419, 1212, 537} \[ \frac{1}{75} x \sqrt{-x^4+x^2+2} \left (13-3 x^2\right )-\frac{178}{625} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{92}{375} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{1156 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{4375} \]
Antiderivative was successfully verified.
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Rule 1208
Rule 1176
Rule 1180
Rule 524
Rule 424
Rule 419
Rule 1212
Rule 537
Rubi steps
\begin{align*} \int \frac{\left (2+x^2-x^4\right )^{3/2}}{7+5 x^2} \, dx &=-\left (\frac{1}{25} \int \left (-12+5 x^2\right ) \sqrt{2+x^2-x^4} \, dx\right )-\frac{34}{25} \int \frac{\sqrt{2+x^2-x^4}}{7+5 x^2} \, dx\\ &=\frac{1}{75} x \left (13-3 x^2\right ) \sqrt{2+x^2-x^4}+\frac{1}{375} \int \frac{230-10 x^2}{\sqrt{2+x^2-x^4}} \, dx+\frac{34}{625} \int \frac{-12+5 x^2}{\sqrt{2+x^2-x^4}} \, dx+\frac{1156}{625} \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+x^2-x^4}} \, dx\\ &=\frac{1}{75} x \left (13-3 x^2\right ) \sqrt{2+x^2-x^4}+\frac{2}{375} \int \frac{230-10 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx+\frac{68}{625} \int \frac{-12+5 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx+\frac{2312}{625} \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2} \left (7+5 x^2\right )} \, dx\\ &=\frac{1}{75} x \left (13-3 x^2\right ) \sqrt{2+x^2-x^4}+\frac{1156 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{4375}-\frac{2}{75} \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx+\frac{34}{125} \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx+\frac{32}{25} \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx-\frac{1156}{625} \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx\\ &=\frac{1}{75} x \left (13-3 x^2\right ) \sqrt{2+x^2-x^4}+\frac{92}{375} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{178}{625} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{1156 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{4375}\\ \end{align*}
Mathematica [C] time = 0.195584, size = 130, normalized size = 1.81 \[ \frac{-2961 i \sqrt{-2 x^4+2 x^2+4} \text{EllipticF}\left (i \sinh ^{-1}(x),-\frac{1}{2}\right )+525 x^7-2800 x^5+1225 x^3+3220 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-1734 i \sqrt{-2 x^4+2 x^2+4} \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )+4550 x}{13125 \sqrt{-x^4+x^2+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 173, normalized size = 2.4 \begin{align*} -{\frac{{x}^{3}}{25}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{13\,x}{75}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{89\,\sqrt{2}}{625}\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{46\,\sqrt{2}}{375}\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{1156\,\sqrt{2}}{4375}\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}}}{5 \, x^{2} + 7}\,{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}}}{5 \, x^{2} + 7}, 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}}}{5 x^{2} + 7}\, 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}}}{5 \, x^{2} + 7}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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