Optimal. Leaf size=117 \[ -\frac {231 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 b^{13/2}}+\frac {231 a^2 x}{16 b^6}-\frac {77 a x^3}{16 b^5}-\frac {33 x^7}{16 b^3 \left (a+b x^2\right )}-\frac {11 x^9}{24 b^2 \left (a+b x^2\right )^2}-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}+\frac {231 x^5}{80 b^4} \]
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Rubi [A] time = 0.07, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 288, 302, 205} \[ \frac {231 a^2 x}{16 b^6}-\frac {231 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 b^{13/2}}-\frac {11 x^9}{24 b^2 \left (a+b x^2\right )^2}-\frac {33 x^7}{16 b^3 \left (a+b x^2\right )}-\frac {77 a x^3}{16 b^5}-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}+\frac {231 x^5}{80 b^4} \]
Antiderivative was successfully verified.
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Rule 28
Rule 205
Rule 288
Rule 302
Rubi steps
\begin {align*} \int \frac {x^{12}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {x^{12}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}+\frac {1}{6} \left (11 b^2\right ) \int \frac {x^{10}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}-\frac {11 x^9}{24 b^2 \left (a+b x^2\right )^2}+\frac {33}{8} \int \frac {x^8}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}-\frac {11 x^9}{24 b^2 \left (a+b x^2\right )^2}-\frac {33 x^7}{16 b^3 \left (a+b x^2\right )}+\frac {231 \int \frac {x^6}{a b+b^2 x^2} \, dx}{16 b^2}\\ &=-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}-\frac {11 x^9}{24 b^2 \left (a+b x^2\right )^2}-\frac {33 x^7}{16 b^3 \left (a+b x^2\right )}+\frac {231 \int \left (\frac {a^2}{b^4}-\frac {a x^2}{b^3}+\frac {x^4}{b^2}-\frac {a^3}{b^3 \left (a b+b^2 x^2\right )}\right ) \, dx}{16 b^2}\\ &=\frac {231 a^2 x}{16 b^6}-\frac {77 a x^3}{16 b^5}+\frac {231 x^5}{80 b^4}-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}-\frac {11 x^9}{24 b^2 \left (a+b x^2\right )^2}-\frac {33 x^7}{16 b^3 \left (a+b x^2\right )}-\frac {\left (231 a^3\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{16 b^5}\\ &=\frac {231 a^2 x}{16 b^6}-\frac {77 a x^3}{16 b^5}+\frac {231 x^5}{80 b^4}-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}-\frac {11 x^9}{24 b^2 \left (a+b x^2\right )^2}-\frac {33 x^7}{16 b^3 \left (a+b x^2\right )}-\frac {231 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 b^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 99, normalized size = 0.85 \[ \frac {3465 a^5 x+9240 a^4 b x^3+7623 a^3 b^2 x^5+1584 a^2 b^3 x^7-176 a b^4 x^9+48 b^5 x^{11}}{240 b^6 \left (a+b x^2\right )^3}-\frac {231 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 b^{13/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 322, normalized size = 2.75 \[ \left [\frac {96 \, b^{5} x^{11} - 352 \, a b^{4} x^{9} + 3168 \, a^{2} b^{3} x^{7} + 15246 \, a^{3} b^{2} x^{5} + 18480 \, a^{4} b x^{3} + 6930 \, a^{5} x + 3465 \, {\left (a^{2} b^{3} x^{6} + 3 \, a^{3} b^{2} x^{4} + 3 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{480 \, {\left (b^{9} x^{6} + 3 \, a b^{8} x^{4} + 3 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}}, \frac {48 \, b^{5} x^{11} - 176 \, a b^{4} x^{9} + 1584 \, a^{2} b^{3} x^{7} + 7623 \, a^{3} b^{2} x^{5} + 9240 \, a^{4} b x^{3} + 3465 \, a^{5} x - 3465 \, {\left (a^{2} b^{3} x^{6} + 3 \, a^{3} b^{2} x^{4} + 3 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{240 \, {\left (b^{9} x^{6} + 3 \, a b^{8} x^{4} + 3 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 96, normalized size = 0.82 \[ -\frac {231 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} b^{6}} + \frac {267 \, a^{3} b^{2} x^{5} + 472 \, a^{4} b x^{3} + 213 \, a^{5} x}{48 \, {\left (b x^{2} + a\right )}^{3} b^{6}} + \frac {3 \, b^{16} x^{5} - 20 \, a b^{15} x^{3} + 150 \, a^{2} b^{14} x}{15 \, b^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 108, normalized size = 0.92 \[ \frac {89 a^{3} x^{5}}{16 \left (b \,x^{2}+a \right )^{3} b^{4}}+\frac {59 a^{4} x^{3}}{6 \left (b \,x^{2}+a \right )^{3} b^{5}}+\frac {x^{5}}{5 b^{4}}+\frac {71 a^{5} x}{16 \left (b \,x^{2}+a \right )^{3} b^{6}}-\frac {4 a \,x^{3}}{3 b^{5}}-\frac {231 a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \sqrt {a b}\, b^{6}}+\frac {10 a^{2} x}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 116, normalized size = 0.99 \[ \frac {267 \, a^{3} b^{2} x^{5} + 472 \, a^{4} b x^{3} + 213 \, a^{5} x}{48 \, {\left (b^{9} x^{6} + 3 \, a b^{8} x^{4} + 3 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}} - \frac {231 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} b^{6}} + \frac {3 \, b^{2} x^{5} - 20 \, a b x^{3} + 150 \, a^{2} x}{15 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 109, normalized size = 0.93 \[ \frac {\frac {71\,a^5\,x}{16}+\frac {59\,a^4\,b\,x^3}{6}+\frac {89\,a^3\,b^2\,x^5}{16}}{a^3\,b^6+3\,a^2\,b^7\,x^2+3\,a\,b^8\,x^4+b^9\,x^6}+\frac {x^5}{5\,b^4}-\frac {4\,a\,x^3}{3\,b^5}+\frac {10\,a^2\,x}{b^6}-\frac {231\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{16\,b^{13/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.66, size = 172, normalized size = 1.47 \[ \frac {10 a^{2} x}{b^{6}} - \frac {4 a x^{3}}{3 b^{5}} + \frac {231 \sqrt {- \frac {a^{5}}{b^{13}}} \log {\left (x - \frac {b^{6} \sqrt {- \frac {a^{5}}{b^{13}}}}{a^{2}} \right )}}{32} - \frac {231 \sqrt {- \frac {a^{5}}{b^{13}}} \log {\left (x + \frac {b^{6} \sqrt {- \frac {a^{5}}{b^{13}}}}{a^{2}} \right )}}{32} + \frac {213 a^{5} x + 472 a^{4} b x^{3} + 267 a^{3} b^{2} x^{5}}{48 a^{3} b^{6} + 144 a^{2} b^{7} x^{2} + 144 a b^{8} x^{4} + 48 b^{9} x^{6}} + \frac {x^{5}}{5 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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