Optimal. Leaf size=193 \[ \frac {275}{7} \left (x^4+3 x^2+2\right )^{3/2} x+\frac {1}{21} \left (757 x^2+2608\right ) \sqrt {x^4+3 x^2+2} x+\frac {577 \left (x^2+2\right ) x}{3 \sqrt {x^4+3 x^2+2}}+\frac {2945 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{21 \sqrt {x^4+3 x^2+2}}-\frac {577 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {x^4+3 x^2+2}}+\frac {125}{9} \left (x^4+3 x^2+2\right )^{3/2} x^3 \]
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Rubi [A] time = 0.10, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1206, 1679, 1176, 1189, 1099, 1135} \[ \frac {125}{9} \left (x^4+3 x^2+2\right )^{3/2} x^3+\frac {275}{7} \left (x^4+3 x^2+2\right )^{3/2} x+\frac {1}{21} \left (757 x^2+2608\right ) \sqrt {x^4+3 x^2+2} x+\frac {577 \left (x^2+2\right ) x}{3 \sqrt {x^4+3 x^2+2}}+\frac {2945 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{21 \sqrt {x^4+3 x^2+2}}-\frac {577 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1099
Rule 1135
Rule 1176
Rule 1189
Rule 1206
Rule 1679
Rubi steps
\begin {align*} \int \left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4} \, dx &=\frac {125}{9} x^3 \left (2+3 x^2+x^4\right )^{3/2}+\frac {1}{9} \int \sqrt {2+3 x^2+x^4} \left (3087+5865 x^2+2475 x^4\right ) \, dx\\ &=\frac {275}{7} x \left (2+3 x^2+x^4\right )^{3/2}+\frac {125}{9} x^3 \left (2+3 x^2+x^4\right )^{3/2}+\frac {1}{63} \int \left (16659+11355 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx\\ &=\frac {1}{21} x \left (2608+757 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {275}{7} x \left (2+3 x^2+x^4\right )^{3/2}+\frac {125}{9} x^3 \left (2+3 x^2+x^4\right )^{3/2}+\frac {1}{945} \int \frac {265050+181755 x^2}{\sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {1}{21} x \left (2608+757 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {275}{7} x \left (2+3 x^2+x^4\right )^{3/2}+\frac {125}{9} x^3 \left (2+3 x^2+x^4\right )^{3/2}+\frac {577}{3} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {5890}{21} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {577 x \left (2+x^2\right )}{3 \sqrt {2+3 x^2+x^4}}+\frac {1}{21} x \left (2608+757 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {275}{7} x \left (2+3 x^2+x^4\right )^{3/2}+\frac {125}{9} x^3 \left (2+3 x^2+x^4\right )^{3/2}-\frac {577 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {2+3 x^2+x^4}}+\frac {2945 \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 = 0.10, size = 119, normalized size = 0.62 \[ \frac {875 x^{11}+7725 x^9+28496 x^7+57312 x^5+61214 x^3-5553 i \sqrt {x^2+1} \sqrt {x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-12117 i \sqrt {x^2+1} \sqrt {x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+25548 x}{63 \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343\right )} \sqrt {x^{4} + 3 \, x^{2} + 2}, x\right ) \]
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 \[ \int \sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 172, normalized size = 0.89 \[ \frac {125 \sqrt {x^{4}+3 x^{2}+2}\, x^{7}}{9}+\frac {1700 \sqrt {x^{4}+3 x^{2}+2}\, x^{5}}{21}+\frac {11446 \sqrt {x^{4}+3 x^{2}+2}\, x^{3}}{63}+\frac {4258 \sqrt {x^{4}+3 x^{2}+2}\, x}{21}-\frac {2945 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{21 \sqrt {x^{4}+3 x^{2}+2}}+\frac {577 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+\EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{6 \sqrt {x^{4}+3 x^{2}+2}} \]
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 \[ \int \sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{3}\,{d x} \]
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 \[ \int {\left (5\,x^2+7\right )}^3\,\sqrt {x^4+3\,x^2+2} \,d x \]
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 \[ \int \sqrt {\left (x^{2} + 1\right ) \left (x^{2} + 2\right )} \left (5 x^{2} + 7\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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