Optimal. Leaf size=198 \[ \frac {1}{7} x \left (x^4+3 x^2+4\right )^{3/2}+\frac {1}{35} x \left (9 x^2+49\right ) \sqrt {x^4+3 x^2+4}+\frac {138 x \sqrt {x^4+3 x^2+4}}{35 \left (x^2+2\right )}+\frac {4 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {138 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{35 \sqrt {x^4+3 x^2+4}} \]
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Rubi [A] time = 0.07, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1091, 1176, 1197, 1103, 1195} \[ \frac {1}{7} x \left (x^4+3 x^2+4\right )^{3/2}+\frac {1}{35} x \left (9 x^2+49\right ) \sqrt {x^4+3 x^2+4}+\frac {138 x \sqrt {x^4+3 x^2+4}}{35 \left (x^2+2\right )}+\frac {4 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {138 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{35 \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1091
Rule 1103
Rule 1176
Rule 1195
Rule 1197
Rubi steps
\begin {align*} \int \left (4+3 x^2+x^4\right )^{3/2} \, dx &=\frac {1}{7} x \left (4+3 x^2+x^4\right )^{3/2}+\frac {3}{7} \int \left (8+3 x^2\right ) \sqrt {4+3 x^2+x^4} \, dx\\ &=\frac {1}{35} x \left (49+9 x^2\right ) \sqrt {4+3 x^2+x^4}+\frac {1}{7} x \left (4+3 x^2+x^4\right )^{3/2}+\frac {1}{35} \int \frac {284+138 x^2}{\sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {1}{35} x \left (49+9 x^2\right ) \sqrt {4+3 x^2+x^4}+\frac {1}{7} x \left (4+3 x^2+x^4\right )^{3/2}-\frac {276}{35} \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx+16 \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {138 x \sqrt {4+3 x^2+x^4}}{35 \left (2+x^2\right )}+\frac {1}{35} x \left (49+9 x^2\right ) \sqrt {4+3 x^2+x^4}+\frac {1}{7} x \left (4+3 x^2+x^4\right )^{3/2}-\frac {138 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{35 \sqrt {4+3 x^2+x^4}}+\frac {4 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {4+3 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.42, size = 343, normalized size = 1.73 \[ \frac {\sqrt {2} \left (69 \sqrt {7}-77 i\right ) \sqrt {\frac {-2 i x^2+\sqrt {7}-3 i}{\sqrt {7}-3 i}} \sqrt {\frac {2 i x^2+\sqrt {7}+3 i}{\sqrt {7}+3 i}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )-69 \sqrt {2} \left (\sqrt {7}+3 i\right ) \sqrt {\frac {-2 i x^2+\sqrt {7}-3 i}{\sqrt {7}-3 i}} \sqrt {\frac {2 i x^2+\sqrt {7}+3 i}{\sqrt {7}+3 i}} E\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+2 \sqrt {-\frac {i}{\sqrt {7}-3 i}} x \left (5 x^8+39 x^6+161 x^4+303 x^2+276\right )}{70 \sqrt {-\frac {i}{\sqrt {7}-3 i}} \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}, x\right ) \]
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 \[ \int {\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.00, size = 258, normalized size = 1.30 \[ \frac {\sqrt {x^{4}+3 x^{2}+4}\, x^{5}}{7}+\frac {24 \sqrt {x^{4}+3 x^{2}+4}\, x^{3}}{35}+\frac {69 \sqrt {x^{4}+3 x^{2}+4}\, x}{35}+\frac {1136 \sqrt {-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{35 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {4416 \sqrt {-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )+\EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{35 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (i \sqrt {7}+3\right )} \]
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 \[ \int {\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (x^4+3\,x^2+4\right )}^{3/2} \,d x \]
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 \[ \int \left (x^{4} + 3 x^{2} + 4\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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