Optimal. Leaf size=131 \[ -\frac {693 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 b^{13/2}}-\frac {231 x^3}{256 b^5 \left (a+b x^2\right )}-\frac {231 x^5}{640 b^4 \left (a+b x^2\right )^2}-\frac {33 x^7}{160 b^3 \left (a+b x^2\right )^3}-\frac {11 x^9}{80 b^2 \left (a+b x^2\right )^4}-\frac {x^{11}}{10 b \left (a+b x^2\right )^5}+\frac {693 x}{256 b^6} \]
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Rubi [A] time = 0.08, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 288, 321, 205} \begin {gather*} -\frac {11 x^9}{80 b^2 \left (a+b x^2\right )^4}-\frac {33 x^7}{160 b^3 \left (a+b x^2\right )^3}-\frac {231 x^5}{640 b^4 \left (a+b x^2\right )^2}-\frac {231 x^3}{256 b^5 \left (a+b x^2\right )}-\frac {693 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 b^{13/2}}-\frac {x^{11}}{10 b \left (a+b x^2\right )^5}+\frac {693 x}{256 b^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 205
Rule 288
Rule 321
Rubi steps
\begin {align*} \int \frac {x^{12}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {x^{12}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {x^{11}}{10 b \left (a+b x^2\right )^5}+\frac {1}{10} \left (11 b^4\right ) \int \frac {x^{10}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {x^{11}}{10 b \left (a+b x^2\right )^5}-\frac {11 x^9}{80 b^2 \left (a+b x^2\right )^4}+\frac {1}{80} \left (99 b^2\right ) \int \frac {x^8}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {x^{11}}{10 b \left (a+b x^2\right )^5}-\frac {11 x^9}{80 b^2 \left (a+b x^2\right )^4}-\frac {33 x^7}{160 b^3 \left (a+b x^2\right )^3}+\frac {231}{160} \int \frac {x^6}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {x^{11}}{10 b \left (a+b x^2\right )^5}-\frac {11 x^9}{80 b^2 \left (a+b x^2\right )^4}-\frac {33 x^7}{160 b^3 \left (a+b x^2\right )^3}-\frac {231 x^5}{640 b^4 \left (a+b x^2\right )^2}+\frac {231 \int \frac {x^4}{\left (a b+b^2 x^2\right )^2} \, dx}{128 b^2}\\ &=-\frac {x^{11}}{10 b \left (a+b x^2\right )^5}-\frac {11 x^9}{80 b^2 \left (a+b x^2\right )^4}-\frac {33 x^7}{160 b^3 \left (a+b x^2\right )^3}-\frac {231 x^5}{640 b^4 \left (a+b x^2\right )^2}-\frac {231 x^3}{256 b^5 \left (a+b x^2\right )}+\frac {693 \int \frac {x^2}{a b+b^2 x^2} \, dx}{256 b^4}\\ &=\frac {693 x}{256 b^6}-\frac {x^{11}}{10 b \left (a+b x^2\right )^5}-\frac {11 x^9}{80 b^2 \left (a+b x^2\right )^4}-\frac {33 x^7}{160 b^3 \left (a+b x^2\right )^3}-\frac {231 x^5}{640 b^4 \left (a+b x^2\right )^2}-\frac {231 x^3}{256 b^5 \left (a+b x^2\right )}-\frac {(693 a) \int \frac {1}{a b+b^2 x^2} \, dx}{256 b^5}\\ &=\frac {693 x}{256 b^6}-\frac {x^{11}}{10 b \left (a+b x^2\right )^5}-\frac {11 x^9}{80 b^2 \left (a+b x^2\right )^4}-\frac {33 x^7}{160 b^3 \left (a+b x^2\right )^3}-\frac {231 x^5}{640 b^4 \left (a+b x^2\right )^2}-\frac {231 x^3}{256 b^5 \left (a+b x^2\right )}-\frac {693 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 b^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 100, normalized size = 0.76 \begin {gather*} \frac {\frac {\sqrt {b} x \left (3465 a^5+16170 a^4 b x^2+29568 a^3 b^2 x^4+26070 a^2 b^3 x^6+10615 a b^4 x^8+1280 b^5 x^{10}\right )}{\left (a+b x^2\right )^5}-3465 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{1280 b^{13/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{12}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.79, size = 400, normalized size = 3.05 \begin {gather*} \left [\frac {2560 \, b^{5} x^{11} + 21230 \, a b^{4} x^{9} + 52140 \, a^{2} b^{3} x^{7} + 59136 \, a^{3} b^{2} x^{5} + 32340 \, a^{4} b x^{3} + 6930 \, a^{5} x + 3465 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{2560 \, {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}}, \frac {1280 \, b^{5} x^{11} + 10615 \, a b^{4} x^{9} + 26070 \, a^{2} b^{3} x^{7} + 29568 \, a^{3} b^{2} x^{5} + 16170 \, a^{4} b x^{3} + 3465 \, a^{5} x - 3465 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{1280 \, {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 87, normalized size = 0.66 \begin {gather*} -\frac {693 \, a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} b^{6}} + \frac {x}{b^{6}} + \frac {4215 \, a b^{4} x^{9} + 13270 \, a^{2} b^{3} x^{7} + 16768 \, a^{3} b^{2} x^{5} + 9770 \, a^{4} b x^{3} + 2185 \, a^{5} x}{1280 \, {\left (b x^{2} + a\right )}^{5} b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 123, normalized size = 0.94 \begin {gather*} \frac {843 a \,x^{9}}{256 \left (b \,x^{2}+a \right )^{5} b^{2}}+\frac {1327 a^{2} x^{7}}{128 \left (b \,x^{2}+a \right )^{5} b^{3}}+\frac {131 a^{3} x^{5}}{10 \left (b \,x^{2}+a \right )^{5} b^{4}}+\frac {977 a^{4} x^{3}}{128 \left (b \,x^{2}+a \right )^{5} b^{5}}+\frac {437 a^{5} x}{256 \left (b \,x^{2}+a \right )^{5} b^{6}}-\frac {693 a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \sqrt {a b}\, b^{6}}+\frac {x}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 134, normalized size = 1.02 \begin {gather*} \frac {4215 \, a b^{4} x^{9} + 13270 \, a^{2} b^{3} x^{7} + 16768 \, a^{3} b^{2} x^{5} + 9770 \, a^{4} b x^{3} + 2185 \, a^{5} x}{1280 \, {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}} - \frac {693 \, a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} b^{6}} + \frac {x}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 130, normalized size = 0.99 \begin {gather*} \frac {\frac {437\,a^5\,x}{256}+\frac {977\,a^4\,b\,x^3}{128}+\frac {131\,a^3\,b^2\,x^5}{10}+\frac {1327\,a^2\,b^3\,x^7}{128}+\frac {843\,a\,b^4\,x^9}{256}}{a^5\,b^6+5\,a^4\,b^7\,x^2+10\,a^3\,b^8\,x^4+10\,a^2\,b^9\,x^6+5\,a\,b^{10}\,x^8+b^{11}\,x^{10}}+\frac {x}{b^6}-\frac {693\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,b^{13/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.95, size = 178, normalized size = 1.36 \begin {gather*} \frac {693 \sqrt {- \frac {a}{b^{13}}} \log {\left (- b^{6} \sqrt {- \frac {a}{b^{13}}} + x \right )}}{512} - \frac {693 \sqrt {- \frac {a}{b^{13}}} \log {\left (b^{6} \sqrt {- \frac {a}{b^{13}}} + x \right )}}{512} + \frac {2185 a^{5} x + 9770 a^{4} b x^{3} + 16768 a^{3} b^{2} x^{5} + 13270 a^{2} b^{3} x^{7} + 4215 a b^{4} x^{9}}{1280 a^{5} b^{6} + 6400 a^{4} b^{7} x^{2} + 12800 a^{3} b^{8} x^{4} + 12800 a^{2} b^{9} x^{6} + 6400 a b^{10} x^{8} + 1280 b^{11} x^{10}} + \frac {x}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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