Optimal. Leaf size=168 \[ \frac {25}{7} x \left (x^4+3 x^2+2\right )^{3/2}+\frac {1}{21} x \left (114 x^2+407\right ) \sqrt {x^4+3 x^2+2}+\frac {31 x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}+\frac {472 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{21 \sqrt {x^4+3 x^2+2}}-\frac {31 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {x^4+3 x^2+2}} \]
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Rubi [A] time = 0.07, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1206, 1176, 1189, 1099, 1135} \[ \frac {25}{7} x \left (x^4+3 x^2+2\right )^{3/2}+\frac {1}{21} x \left (114 x^2+407\right ) \sqrt {x^4+3 x^2+2}+\frac {31 x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}+\frac {472 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{21 \sqrt {x^4+3 x^2+2}}-\frac {31 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1099
Rule 1135
Rule 1176
Rule 1189
Rule 1206
Rubi steps
\begin {align*} \int \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4} \, dx &=\frac {25}{7} x \left (2+3 x^2+x^4\right )^{3/2}+\frac {1}{7} \int \left (293+190 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx\\ &=\frac {1}{21} x \left (407+114 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {25}{7} x \left (2+3 x^2+x^4\right )^{3/2}+\frac {1}{105} \int \frac {4720+3255 x^2}{\sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {1}{21} x \left (407+114 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {25}{7} x \left (2+3 x^2+x^4\right )^{3/2}+31 \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {944}{21} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {31 x \left (2+x^2\right )}{\sqrt {2+3 x^2+x^4}}+\frac {1}{21} x \left (407+114 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {25}{7} x \left (2+3 x^2+x^4\right )^{3/2}-\frac {31 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2+3 x^2+x^4}}+\frac {472 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{21 \sqrt {2+3 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 114, normalized size = 0.68 \[ \frac {75 x^9+564 x^7+1724 x^5+2349 x^3-293 i \sqrt {x^2+1} \sqrt {x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-651 i \sqrt {x^2+1} \sqrt {x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+1114 x}{21 \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (25 \, x^{4} + 70 \, x^{2} + 49\right )} \sqrt {x^{4} + 3 \, x^{2} + 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 155, normalized size = 0.92 \[ \frac {25 \sqrt {x^{4}+3 x^{2}+2}\, x^{5}}{7}+\frac {113 \sqrt {x^{4}+3 x^{2}+2}\, x^{3}}{7}+\frac {557 \sqrt {x^{4}+3 x^{2}+2}\, x}{21}-\frac {472 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{21 \sqrt {x^{4}+3 x^{2}+2}}+\frac {31 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+\EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{2 \sqrt {x^{4}+3 x^{2}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (5\,x^2+7\right )}^2\,\sqrt {x^4+3\,x^2+2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\left (x^{2} + 1\right ) \left (x^{2} + 2\right )} \left (5 x^{2} + 7\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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