Optimal. Leaf size=155 \[ -\frac {9009 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 b^{17/2}}+\frac {9009 a^2 x}{256 b^8}-\frac {3003 a x^3}{256 b^7}-\frac {1287 x^7}{256 b^5 \left (a+b x^2\right )}-\frac {143 x^9}{128 b^4 \left (a+b x^2\right )^2}-\frac {13 x^{11}}{32 b^3 \left (a+b x^2\right )^3}-\frac {3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}+\frac {9009 x^5}{1280 b^6} \]
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Rubi [A] time = 0.11, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 288, 302, 205} \begin {gather*} \frac {9009 a^2 x}{256 b^8}-\frac {9009 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 b^{17/2}}-\frac {3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}-\frac {13 x^{11}}{32 b^3 \left (a+b x^2\right )^3}-\frac {143 x^9}{128 b^4 \left (a+b x^2\right )^2}-\frac {1287 x^7}{256 b^5 \left (a+b x^2\right )}-\frac {3003 a x^3}{256 b^7}-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}+\frac {9009 x^5}{1280 b^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 205
Rule 288
Rule 302
Rubi steps
\begin {align*} \int \frac {x^{16}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {x^{16}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}+\frac {1}{2} \left (3 b^4\right ) \int \frac {x^{14}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}-\frac {3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}+\frac {1}{16} \left (39 b^2\right ) \int \frac {x^{12}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}-\frac {3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}-\frac {13 x^{11}}{32 b^3 \left (a+b x^2\right )^3}+\frac {143}{32} \int \frac {x^{10}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}-\frac {3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}-\frac {13 x^{11}}{32 b^3 \left (a+b x^2\right )^3}-\frac {143 x^9}{128 b^4 \left (a+b x^2\right )^2}+\frac {1287 \int \frac {x^8}{\left (a b+b^2 x^2\right )^2} \, dx}{128 b^2}\\ &=-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}-\frac {3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}-\frac {13 x^{11}}{32 b^3 \left (a+b x^2\right )^3}-\frac {143 x^9}{128 b^4 \left (a+b x^2\right )^2}-\frac {1287 x^7}{256 b^5 \left (a+b x^2\right )}+\frac {9009 \int \frac {x^6}{a b+b^2 x^2} \, dx}{256 b^4}\\ &=-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}-\frac {3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}-\frac {13 x^{11}}{32 b^3 \left (a+b x^2\right )^3}-\frac {143 x^9}{128 b^4 \left (a+b x^2\right )^2}-\frac {1287 x^7}{256 b^5 \left (a+b x^2\right )}+\frac {9009 \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}{256 b^4}\\ &=\frac {9009 a^2 x}{256 b^8}-\frac {3003 a x^3}{256 b^7}+\frac {9009 x^5}{1280 b^6}-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}-\frac {3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}-\frac {13 x^{11}}{32 b^3 \left (a+b x^2\right )^3}-\frac {143 x^9}{128 b^4 \left (a+b x^2\right )^2}-\frac {1287 x^7}{256 b^5 \left (a+b x^2\right )}-\frac {\left (9009 a^3\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{256 b^7}\\ &=\frac {9009 a^2 x}{256 b^8}-\frac {3003 a x^3}{256 b^7}+\frac {9009 x^5}{1280 b^6}-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}-\frac {3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}-\frac {13 x^{11}}{32 b^3 \left (a+b x^2\right )^3}-\frac {143 x^9}{128 b^4 \left (a+b x^2\right )^2}-\frac {1287 x^7}{256 b^5 \left (a+b x^2\right )}-\frac {9009 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 b^{17/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 122, normalized size = 0.79 \begin {gather*} \frac {\frac {\sqrt {b} x \left (45045 a^7+210210 a^6 b x^2+384384 a^5 b^2 x^4+338910 a^4 b^3 x^6+137995 a^3 b^4 x^8+16640 a^2 b^5 x^{10}-1280 a b^6 x^{12}+256 b^7 x^{14}\right )}{\left (a+b x^2\right )^5}-45045 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{1280 b^{17/2}} \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^{16}}{\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 = 0.82, size = 454, normalized size = 2.93 \begin {gather*} \left [\frac {512 \, b^{7} x^{15} - 2560 \, a b^{6} x^{13} + 33280 \, a^{2} b^{5} x^{11} + 275990 \, a^{3} b^{4} x^{9} + 677820 \, a^{4} b^{3} x^{7} + 768768 \, a^{5} b^{2} x^{5} + 420420 \, a^{6} b x^{3} + 90090 \, a^{7} x + 45045 \, {\left (a^{2} b^{5} x^{10} + 5 \, a^{3} b^{4} x^{8} + 10 \, a^{4} b^{3} x^{6} + 10 \, a^{5} b^{2} x^{4} + 5 \, a^{6} b x^{2} + a^{7}\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^{13} x^{10} + 5 \, a b^{12} x^{8} + 10 \, a^{2} b^{11} x^{6} + 10 \, a^{3} b^{10} x^{4} + 5 \, a^{4} b^{9} x^{2} + a^{5} b^{8}\right )}}, \frac {256 \, b^{7} x^{15} - 1280 \, a b^{6} x^{13} + 16640 \, a^{2} b^{5} x^{11} + 137995 \, a^{3} b^{4} x^{9} + 338910 \, a^{4} b^{3} x^{7} + 384384 \, a^{5} b^{2} x^{5} + 210210 \, a^{6} b x^{3} + 45045 \, a^{7} x - 45045 \, {\left (a^{2} b^{5} x^{10} + 5 \, a^{3} b^{4} x^{8} + 10 \, a^{4} b^{3} x^{6} + 10 \, a^{5} b^{2} x^{4} + 5 \, a^{6} b x^{2} + a^{7}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{1280 \, {\left (b^{13} x^{10} + 5 \, a b^{12} x^{8} + 10 \, a^{2} b^{11} x^{6} + 10 \, a^{3} b^{10} x^{4} + 5 \, a^{4} b^{9} x^{2} + a^{5} b^{8}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 117, normalized size = 0.75 \begin {gather*} -\frac {9009 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} b^{8}} + \frac {26635 \, a^{3} b^{4} x^{9} + 94430 \, a^{4} b^{3} x^{7} + 128128 \, a^{5} b^{2} x^{5} + 78370 \, a^{6} b x^{3} + 18165 \, a^{7} x}{1280 \, {\left (b x^{2} + a\right )}^{5} b^{8}} + \frac {b^{24} x^{5} - 10 \, a b^{23} x^{3} + 105 \, a^{2} b^{22} x}{5 \, b^{30}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 148, normalized size = 0.95 \begin {gather*} \frac {5327 a^{3} x^{9}}{256 \left (b \,x^{2}+a \right )^{5} b^{4}}+\frac {9443 a^{4} x^{7}}{128 \left (b \,x^{2}+a \right )^{5} b^{5}}+\frac {1001 a^{5} x^{5}}{10 \left (b \,x^{2}+a \right )^{5} b^{6}}+\frac {7837 a^{6} x^{3}}{128 \left (b \,x^{2}+a \right )^{5} b^{7}}+\frac {3633 a^{7} x}{256 \left (b \,x^{2}+a \right )^{5} b^{8}}+\frac {x^{5}}{5 b^{6}}-\frac {2 a \,x^{3}}{b^{7}}-\frac {9009 a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \sqrt {a b}\, b^{8}}+\frac {21 a^{2} x}{b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.89, size = 159, normalized size = 1.03 \begin {gather*} \frac {26635 \, a^{3} b^{4} x^{9} + 94430 \, a^{4} b^{3} x^{7} + 128128 \, a^{5} b^{2} x^{5} + 78370 \, a^{6} b x^{3} + 18165 \, a^{7} x}{1280 \, {\left (b^{13} x^{10} + 5 \, a b^{12} x^{8} + 10 \, a^{2} b^{11} x^{6} + 10 \, a^{3} b^{10} x^{4} + 5 \, a^{4} b^{9} x^{2} + a^{5} b^{8}\right )}} - \frac {9009 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} b^{8}} + \frac {b^{2} x^{5} - 10 \, a b x^{3} + 105 \, a^{2} x}{5 \, b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 153, normalized size = 0.99 \begin {gather*} \frac {\frac {3633\,a^7\,x}{256}+\frac {7837\,a^6\,b\,x^3}{128}+\frac {1001\,a^5\,b^2\,x^5}{10}+\frac {9443\,a^4\,b^3\,x^7}{128}+\frac {5327\,a^3\,b^4\,x^9}{256}}{a^5\,b^8+5\,a^4\,b^9\,x^2+10\,a^3\,b^{10}\,x^4+10\,a^2\,b^{11}\,x^6+5\,a\,b^{12}\,x^8+b^{13}\,x^{10}}+\frac {x^5}{5\,b^6}-\frac {2\,a\,x^3}{b^7}+\frac {21\,a^2\,x}{b^8}-\frac {9009\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,b^{17/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.09, size = 218, normalized size = 1.41 \begin {gather*} \frac {21 a^{2} x}{b^{8}} - \frac {2 a x^{3}}{b^{7}} + \frac {9009 \sqrt {- \frac {a^{5}}{b^{17}}} \log {\left (x - \frac {b^{8} \sqrt {- \frac {a^{5}}{b^{17}}}}{a^{2}} \right )}}{512} - \frac {9009 \sqrt {- \frac {a^{5}}{b^{17}}} \log {\left (x + \frac {b^{8} \sqrt {- \frac {a^{5}}{b^{17}}}}{a^{2}} \right )}}{512} + \frac {18165 a^{7} x + 78370 a^{6} b x^{3} + 128128 a^{5} b^{2} x^{5} + 94430 a^{4} b^{3} x^{7} + 26635 a^{3} b^{4} x^{9}}{1280 a^{5} b^{8} + 6400 a^{4} b^{9} x^{2} + 12800 a^{3} b^{10} x^{4} + 12800 a^{2} b^{11} x^{6} + 6400 a b^{12} x^{8} + 1280 b^{13} x^{10}} + \frac {x^{5}}{5 b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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