Optimal. Leaf size=157 \[ -\frac {9009 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{17/2}}-\frac {9009 b^2}{256 a^8 x}+\frac {3003 b}{256 a^7 x^3}-\frac {9009}{1280 a^6 x^5}+\frac {1287}{256 a^5 x^5 \left (a+b x^2\right )}+\frac {143}{128 a^4 x^5 \left (a+b x^2\right )^2}+\frac {13}{32 a^3 x^5 \left (a+b x^2\right )^3}+\frac {3}{16 a^2 x^5 \left (a+b x^2\right )^4}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.12, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 290, 325, 205} \[ -\frac {9009 b^2}{256 a^8 x}-\frac {9009 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{17/2}}+\frac {3003 b}{256 a^7 x^3}+\frac {1287}{256 a^5 x^5 \left (a+b x^2\right )}+\frac {143}{128 a^4 x^5 \left (a+b x^2\right )^2}+\frac {13}{32 a^3 x^5 \left (a+b x^2\right )^3}+\frac {3}{16 a^2 x^5 \left (a+b x^2\right )^4}-\frac {9009}{1280 a^6 x^5}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5} \]
Antiderivative was successfully verified.
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Rule 28
Rule 205
Rule 290
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {1}{x^6 \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {1}{10 a x^5 \left (a+b x^2\right )^5}+\frac {\left (3 b^5\right ) \int \frac {1}{x^6 \left (a b+b^2 x^2\right )^5} \, dx}{2 a}\\ &=\frac {1}{10 a x^5 \left (a+b x^2\right )^5}+\frac {3}{16 a^2 x^5 \left (a+b x^2\right )^4}+\frac {\left (39 b^4\right ) \int \frac {1}{x^6 \left (a b+b^2 x^2\right )^4} \, dx}{16 a^2}\\ &=\frac {1}{10 a x^5 \left (a+b x^2\right )^5}+\frac {3}{16 a^2 x^5 \left (a+b x^2\right )^4}+\frac {13}{32 a^3 x^5 \left (a+b x^2\right )^3}+\frac {\left (143 b^3\right ) \int \frac {1}{x^6 \left (a b+b^2 x^2\right )^3} \, dx}{32 a^3}\\ &=\frac {1}{10 a x^5 \left (a+b x^2\right )^5}+\frac {3}{16 a^2 x^5 \left (a+b x^2\right )^4}+\frac {13}{32 a^3 x^5 \left (a+b x^2\right )^3}+\frac {143}{128 a^4 x^5 \left (a+b x^2\right )^2}+\frac {\left (1287 b^2\right ) \int \frac {1}{x^6 \left (a b+b^2 x^2\right )^2} \, dx}{128 a^4}\\ &=\frac {1}{10 a x^5 \left (a+b x^2\right )^5}+\frac {3}{16 a^2 x^5 \left (a+b x^2\right )^4}+\frac {13}{32 a^3 x^5 \left (a+b x^2\right )^3}+\frac {143}{128 a^4 x^5 \left (a+b x^2\right )^2}+\frac {1287}{256 a^5 x^5 \left (a+b x^2\right )}+\frac {(9009 b) \int \frac {1}{x^6 \left (a b+b^2 x^2\right )} \, dx}{256 a^5}\\ &=-\frac {9009}{1280 a^6 x^5}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}+\frac {3}{16 a^2 x^5 \left (a+b x^2\right )^4}+\frac {13}{32 a^3 x^5 \left (a+b x^2\right )^3}+\frac {143}{128 a^4 x^5 \left (a+b x^2\right )^2}+\frac {1287}{256 a^5 x^5 \left (a+b x^2\right )}-\frac {\left (9009 b^2\right ) \int \frac {1}{x^4 \left (a b+b^2 x^2\right )} \, dx}{256 a^6}\\ &=-\frac {9009}{1280 a^6 x^5}+\frac {3003 b}{256 a^7 x^3}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}+\frac {3}{16 a^2 x^5 \left (a+b x^2\right )^4}+\frac {13}{32 a^3 x^5 \left (a+b x^2\right )^3}+\frac {143}{128 a^4 x^5 \left (a+b x^2\right )^2}+\frac {1287}{256 a^5 x^5 \left (a+b x^2\right )}+\frac {\left (9009 b^3\right ) \int \frac {1}{x^2 \left (a b+b^2 x^2\right )} \, dx}{256 a^7}\\ &=-\frac {9009}{1280 a^6 x^5}+\frac {3003 b}{256 a^7 x^3}-\frac {9009 b^2}{256 a^8 x}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}+\frac {3}{16 a^2 x^5 \left (a+b x^2\right )^4}+\frac {13}{32 a^3 x^5 \left (a+b x^2\right )^3}+\frac {143}{128 a^4 x^5 \left (a+b x^2\right )^2}+\frac {1287}{256 a^5 x^5 \left (a+b x^2\right )}-\frac {\left (9009 b^4\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{256 a^8}\\ &=-\frac {9009}{1280 a^6 x^5}+\frac {3003 b}{256 a^7 x^3}-\frac {9009 b^2}{256 a^8 x}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}+\frac {3}{16 a^2 x^5 \left (a+b x^2\right )^4}+\frac {13}{32 a^3 x^5 \left (a+b x^2\right )^3}+\frac {143}{128 a^4 x^5 \left (a+b x^2\right )^2}+\frac {1287}{256 a^5 x^5 \left (a+b x^2\right )}-\frac {9009 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{17/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 123, normalized size = 0.78 \[ -\frac {9009 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{17/2}}-\frac {256 a^7-1280 a^6 b x^2+16640 a^5 b^2 x^4+137995 a^4 b^3 x^6+338910 a^3 b^4 x^8+384384 a^2 b^5 x^{10}+210210 a b^6 x^{12}+45045 b^7 x^{14}}{1280 a^8 x^5 \left (a+b x^2\right )^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 462, normalized size = 2.94 \[ \left [-\frac {90090 \, b^{7} x^{14} + 420420 \, a b^{6} x^{12} + 768768 \, a^{2} b^{5} x^{10} + 677820 \, a^{3} b^{4} x^{8} + 275990 \, a^{4} b^{3} x^{6} + 33280 \, a^{5} b^{2} x^{4} - 2560 \, a^{6} b x^{2} + 512 \, a^{7} - 45045 \, {\left (b^{7} x^{15} + 5 \, a b^{6} x^{13} + 10 \, a^{2} b^{5} x^{11} + 10 \, a^{3} b^{4} x^{9} + 5 \, a^{4} b^{3} x^{7} + a^{5} b^{2} x^{5}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{2560 \, {\left (a^{8} b^{5} x^{15} + 5 \, a^{9} b^{4} x^{13} + 10 \, a^{10} b^{3} x^{11} + 10 \, a^{11} b^{2} x^{9} + 5 \, a^{12} b x^{7} + a^{13} x^{5}\right )}}, -\frac {45045 \, b^{7} x^{14} + 210210 \, a b^{6} x^{12} + 384384 \, a^{2} b^{5} x^{10} + 338910 \, a^{3} b^{4} x^{8} + 137995 \, a^{4} b^{3} x^{6} + 16640 \, a^{5} b^{2} x^{4} - 1280 \, a^{6} b x^{2} + 256 \, a^{7} + 45045 \, {\left (b^{7} x^{15} + 5 \, a b^{6} x^{13} + 10 \, a^{2} b^{5} x^{11} + 10 \, a^{3} b^{4} x^{9} + 5 \, a^{4} b^{3} x^{7} + a^{5} b^{2} x^{5}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{1280 \, {\left (a^{8} b^{5} x^{15} + 5 \, a^{9} b^{4} x^{13} + 10 \, a^{10} b^{3} x^{11} + 10 \, a^{11} b^{2} x^{9} + 5 \, a^{12} b x^{7} + a^{13} x^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 115, normalized size = 0.73 \[ -\frac {9009 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{8}} - \frac {45045 \, b^{7} x^{14} + 210210 \, a b^{6} x^{12} + 384384 \, a^{2} b^{5} x^{10} + 338910 \, a^{3} b^{4} x^{8} + 137995 \, a^{4} b^{3} x^{6} + 16640 \, a^{5} b^{2} x^{4} - 1280 \, a^{6} b x^{2} + 256 \, a^{7}}{1280 \, {\left (b x^{3} + a x\right )}^{5} a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 150, normalized size = 0.96 \[ -\frac {3633 b^{7} x^{9}}{256 \left (b \,x^{2}+a \right )^{5} a^{8}}-\frac {7837 b^{6} x^{7}}{128 \left (b \,x^{2}+a \right )^{5} a^{7}}-\frac {1001 b^{5} x^{5}}{10 \left (b \,x^{2}+a \right )^{5} a^{6}}-\frac {9443 b^{4} x^{3}}{128 \left (b \,x^{2}+a \right )^{5} a^{5}}-\frac {5327 b^{3} x}{256 \left (b \,x^{2}+a \right )^{5} a^{4}}-\frac {9009 b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \sqrt {a b}\, a^{8}}-\frac {21 b^{2}}{a^{8} x}+\frac {2 b}{a^{7} x^{3}}-\frac {1}{5 a^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.11, size = 163, normalized size = 1.04 \[ -\frac {45045 \, b^{7} x^{14} + 210210 \, a b^{6} x^{12} + 384384 \, a^{2} b^{5} x^{10} + 338910 \, a^{3} b^{4} x^{8} + 137995 \, a^{4} b^{3} x^{6} + 16640 \, a^{5} b^{2} x^{4} - 1280 \, a^{6} b x^{2} + 256 \, a^{7}}{1280 \, {\left (a^{8} b^{5} x^{15} + 5 \, a^{9} b^{4} x^{13} + 10 \, a^{10} b^{3} x^{11} + 10 \, a^{11} b^{2} x^{9} + 5 \, a^{12} b x^{7} + a^{13} x^{5}\right )}} - \frac {9009 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.65, size = 158, normalized size = 1.01 \[ -\frac {\frac {1}{5\,a}-\frac {b\,x^2}{a^2}+\frac {13\,b^2\,x^4}{a^3}+\frac {27599\,b^3\,x^6}{256\,a^4}+\frac {33891\,b^4\,x^8}{128\,a^5}+\frac {3003\,b^5\,x^{10}}{10\,a^6}+\frac {21021\,b^6\,x^{12}}{128\,a^7}+\frac {9009\,b^7\,x^{14}}{256\,a^8}}{a^5\,x^5+5\,a^4\,b\,x^7+10\,a^3\,b^2\,x^9+10\,a^2\,b^3\,x^{11}+5\,a\,b^4\,x^{13}+b^5\,x^{15}}-\frac {9009\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,a^{17/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.95, size = 221, normalized size = 1.41 \[ \frac {9009 \sqrt {- \frac {b^{5}}{a^{17}}} \log {\left (- \frac {a^{9} \sqrt {- \frac {b^{5}}{a^{17}}}}{b^{3}} + x \right )}}{512} - \frac {9009 \sqrt {- \frac {b^{5}}{a^{17}}} \log {\left (\frac {a^{9} \sqrt {- \frac {b^{5}}{a^{17}}}}{b^{3}} + x \right )}}{512} + \frac {- 256 a^{7} + 1280 a^{6} b x^{2} - 16640 a^{5} b^{2} x^{4} - 137995 a^{4} b^{3} x^{6} - 338910 a^{3} b^{4} x^{8} - 384384 a^{2} b^{5} x^{10} - 210210 a b^{6} x^{12} - 45045 b^{7} x^{14}}{1280 a^{13} x^{5} + 6400 a^{12} b x^{7} + 12800 a^{11} b^{2} x^{9} + 12800 a^{10} b^{3} x^{11} + 6400 a^{9} b^{4} x^{13} + 1280 a^{8} b^{5} x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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