Optimal. Leaf size=249 \[ \frac {\sqrt [4]{\frac {a x^{10}+4 a x^8+6 a x^6+4 a x^4+a x^2+x^8+4 x^6+6 x^4+4 x^2+1}{x^2}} \left (\frac {\sqrt {x} \tan ^{-1}\left (\frac {\sqrt [4]{a x^2+1}}{\sqrt [4]{a} \sqrt {x}}\right )}{4 a^{3/4}}+\frac {\sqrt {x} \tanh ^{-1}\left (\frac {\sqrt [4]{a x^2+1}}{\sqrt [4]{a} \sqrt {x}}\right )}{4 a^{3/4}}+\frac {1}{2} x^2 \sqrt [4]{a x^2+1}-2 \sqrt [4]{a x^2+1}+\sqrt [4]{a} \sqrt {x} \tan ^{-1}\left (\frac {\sqrt [4]{a x^2+1}}{\sqrt [4]{a} \sqrt {x}}\right )+\sqrt [4]{a} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt [4]{a x^2+1}}{\sqrt [4]{a} \sqrt {x}}\right )\right )}{\left (x^2+1\right ) \sqrt [4]{a x^2+1}} \]
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Rubi [A] time = 0.41, antiderivative size = 244, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 9, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6688, 6718, 453, 279, 329, 331, 298, 203, 206} \begin {gather*} -\frac {(4 a+1) \sqrt {x} \sqrt [4]{\frac {\left (x^2+1\right )^4 \left (a x^2+1\right )}{x^2}} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+1}}\right )}{4 a^{3/4} \left (x^2+1\right ) \sqrt [4]{a x^2+1}}+\frac {(4 a+1) \sqrt {x} \sqrt [4]{\frac {\left (x^2+1\right )^4 \left (a x^2+1\right )}{x^2}} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+1}}\right )}{4 a^{3/4} \left (x^2+1\right ) \sqrt [4]{a x^2+1}}+\frac {(4 a+1) x^2 \sqrt [4]{\frac {\left (x^2+1\right )^4 \left (a x^2+1\right )}{x^2}}}{2 \left (x^2+1\right )}-\frac {2 \left (a x^2+1\right ) \sqrt [4]{\frac {\left (x^2+1\right )^4 \left (a x^2+1\right )}{x^2}}}{x^2+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 279
Rule 298
Rule 329
Rule 331
Rule 453
Rule 6688
Rule 6718
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{\frac {1+4 x^2+a x^2+6 x^4+4 a x^4+4 x^6+6 a x^6+x^8+4 a x^8+a x^{10}}{x^2}}}{x} \, dx &=\int \frac {\sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}}{x} \, dx\\ &=\frac {\left (\sqrt {x} \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}\right ) \int \frac {\left (1+x^2\right ) \sqrt [4]{1+a x^2}}{x^{3/2}} \, dx}{\left (1+x^2\right ) \sqrt [4]{1+a x^2}}\\ &=-\frac {2 \left (1+a x^2\right ) \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}}{1+x^2}-\frac {\left ((-1-4 a) \sqrt {x} \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}\right ) \int \sqrt {x} \sqrt [4]{1+a x^2} \, dx}{\left (1+x^2\right ) \sqrt [4]{1+a x^2}}\\ &=\frac {(1+4 a) x^2 \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}}{2 \left (1+x^2\right )}-\frac {2 \left (1+a x^2\right ) \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}}{1+x^2}-\frac {\left ((-1-4 a) \sqrt {x} \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}\right ) \int \frac {\sqrt {x}}{\left (1+a x^2\right )^{3/4}} \, dx}{4 \left (1+x^2\right ) \sqrt [4]{1+a x^2}}\\ &=\frac {(1+4 a) x^2 \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}}{2 \left (1+x^2\right )}-\frac {2 \left (1+a x^2\right ) \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}}{1+x^2}-\frac {\left ((-1-4 a) \sqrt {x} \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \left (1+x^2\right ) \sqrt [4]{1+a x^2}}\\ &=\frac {(1+4 a) x^2 \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}}{2 \left (1+x^2\right )}-\frac {2 \left (1+a x^2\right ) \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}}{1+x^2}-\frac {\left ((-1-4 a) \sqrt {x} \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+a x^2}}\right )}{2 \left (1+x^2\right ) \sqrt [4]{1+a x^2}}\\ &=\frac {(1+4 a) x^2 \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}}{2 \left (1+x^2\right )}-\frac {2 \left (1+a x^2\right ) \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}}{1+x^2}-\frac {\left ((-1-4 a) \sqrt {x} \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+a x^2}}\right )}{4 \sqrt {a} \left (1+x^2\right ) \sqrt [4]{1+a x^2}}+\frac {\left ((-1-4 a) \sqrt {x} \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+a x^2}}\right )}{4 \sqrt {a} \left (1+x^2\right ) \sqrt [4]{1+a x^2}}\\ &=\frac {(1+4 a) x^2 \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}}{2 \left (1+x^2\right )}-\frac {2 \left (1+a x^2\right ) \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}}}{1+x^2}-\frac {(1+4 a) \sqrt {x} \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{1+a x^2}}\right )}{4 a^{3/4} \left (1+x^2\right ) \sqrt [4]{1+a x^2}}+\frac {(1+4 a) \sqrt {x} \sqrt [4]{\frac {\left (1+x^2\right )^4 \left (1+a x^2\right )}{x^2}} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{1+a x^2}}\right )}{4 a^{3/4} \left (1+x^2\right ) \sqrt [4]{1+a x^2}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 84, normalized size = 0.34 \begin {gather*} -\frac {2 \sqrt [4]{\frac {\left (x^2+1\right )^4 \left (a x^2+1\right )}{x^2}} \left (3 \left (a x^2+1\right )^{5/4}-(4 a+1) x^2 \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-a x^2\right )\right )}{3 \left (x^2+1\right ) \sqrt [4]{a x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.29, size = 237, normalized size = 0.95 \begin {gather*} \frac {\left (-4+x^2\right ) \sqrt [4]{\frac {1+4 x^2+a x^2+6 x^4+4 a x^4+4 x^6+6 a x^6+x^8+4 a x^8+a x^{10}}{x^2}}}{2 \left (1+x^2\right )}+\frac {(-1-4 a) \tan ^{-1}\left (\frac {\sqrt [4]{a} \left (1+x^2\right )}{\sqrt [4]{\frac {1+4 x^2+a x^2+6 x^4+4 a x^4+4 x^6+6 a x^6+x^8+4 a x^8+a x^{10}}{x^2}}}\right )}{4 a^{3/4}}+\frac {(1+4 a) \tanh ^{-1}\left (\frac {\sqrt [4]{a} \left (1+x^2\right )}{\sqrt [4]{\frac {1+4 x^2+a x^2+6 x^4+4 a x^4+4 x^6+6 a x^6+x^8+4 a x^8+a x^{10}}{x^2}}}\right )}{4 a^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {\left (\frac {a x^{10} + 4 \, a x^{8} + x^{8} + 6 \, a x^{6} + 4 \, x^{6} + 4 \, a x^{4} + 6 \, x^{4} + a x^{2} + 4 \, x^{2} + 1}{x^{2}}\right )^{\frac {1}{4}}}{x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (\frac {a \,x^{10}+4 a \,x^{8}+x^{8}+6 a \,x^{6}+4 x^{6}+4 a \,x^{4}+6 x^{4}+a \,x^{2}+4 x^{2}+1}{x^{2}}\right )^{\frac {1}{4}}}{x}\, dx\]
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 \begin {gather*} \int \frac {\left (\frac {a x^{10} + 4 \, a x^{8} + x^{8} + 6 \, a x^{6} + 4 \, x^{6} + 4 \, a x^{4} + 6 \, x^{4} + a x^{2} + 4 \, x^{2} + 1}{x^{2}}\right )^{\frac {1}{4}}}{x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (\frac {a\,x^2+4\,a\,x^4+6\,a\,x^6+4\,a\,x^8+a\,x^{10}+4\,x^2+6\,x^4+4\,x^6+x^8+1}{x^2}\right )}^{1/4}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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