Optimal. Leaf size=133 \[ -\frac {693 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{13/2}}-\frac {693}{256 a^6 x}+\frac {231}{256 a^5 x \left (a+b x^2\right )}+\frac {231}{640 a^4 x \left (a+b x^2\right )^2}+\frac {33}{160 a^3 x \left (a+b x^2\right )^3}+\frac {11}{80 a^2 x \left (a+b x^2\right )^4}+\frac {1}{10 a x \left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.09, antiderivative size = 133, 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, 290, 325, 205} \begin {gather*} \frac {231}{256 a^5 x \left (a+b x^2\right )}+\frac {231}{640 a^4 x \left (a+b x^2\right )^2}+\frac {33}{160 a^3 x \left (a+b x^2\right )^3}+\frac {11}{80 a^2 x \left (a+b x^2\right )^4}-\frac {693 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{13/2}}-\frac {693}{256 a^6 x}+\frac {1}{10 a x \left (a+b x^2\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 205
Rule 290
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {1}{x^2 \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {1}{10 a x \left (a+b x^2\right )^5}+\frac {\left (11 b^5\right ) \int \frac {1}{x^2 \left (a b+b^2 x^2\right )^5} \, dx}{10 a}\\ &=\frac {1}{10 a x \left (a+b x^2\right )^5}+\frac {11}{80 a^2 x \left (a+b x^2\right )^4}+\frac {\left (99 b^4\right ) \int \frac {1}{x^2 \left (a b+b^2 x^2\right )^4} \, dx}{80 a^2}\\ &=\frac {1}{10 a x \left (a+b x^2\right )^5}+\frac {11}{80 a^2 x \left (a+b x^2\right )^4}+\frac {33}{160 a^3 x \left (a+b x^2\right )^3}+\frac {\left (231 b^3\right ) \int \frac {1}{x^2 \left (a b+b^2 x^2\right )^3} \, dx}{160 a^3}\\ &=\frac {1}{10 a x \left (a+b x^2\right )^5}+\frac {11}{80 a^2 x \left (a+b x^2\right )^4}+\frac {33}{160 a^3 x \left (a+b x^2\right )^3}+\frac {231}{640 a^4 x \left (a+b x^2\right )^2}+\frac {\left (231 b^2\right ) \int \frac {1}{x^2 \left (a b+b^2 x^2\right )^2} \, dx}{128 a^4}\\ &=\frac {1}{10 a x \left (a+b x^2\right )^5}+\frac {11}{80 a^2 x \left (a+b x^2\right )^4}+\frac {33}{160 a^3 x \left (a+b x^2\right )^3}+\frac {231}{640 a^4 x \left (a+b x^2\right )^2}+\frac {231}{256 a^5 x \left (a+b x^2\right )}+\frac {(693 b) \int \frac {1}{x^2 \left (a b+b^2 x^2\right )} \, dx}{256 a^5}\\ &=-\frac {693}{256 a^6 x}+\frac {1}{10 a x \left (a+b x^2\right )^5}+\frac {11}{80 a^2 x \left (a+b x^2\right )^4}+\frac {33}{160 a^3 x \left (a+b x^2\right )^3}+\frac {231}{640 a^4 x \left (a+b x^2\right )^2}+\frac {231}{256 a^5 x \left (a+b x^2\right )}-\frac {\left (693 b^2\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{256 a^6}\\ &=-\frac {693}{256 a^6 x}+\frac {1}{10 a x \left (a+b x^2\right )^5}+\frac {11}{80 a^2 x \left (a+b x^2\right )^4}+\frac {33}{160 a^3 x \left (a+b x^2\right )^3}+\frac {231}{640 a^4 x \left (a+b x^2\right )^2}+\frac {231}{256 a^5 x \left (a+b x^2\right )}-\frac {693 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 101, normalized size = 0.76 \begin {gather*} -\frac {693 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{13/2}}-\frac {1280 a^5+10615 a^4 b x^2+26070 a^3 b^2 x^4+29568 a^2 b^3 x^6+16170 a b^4 x^8+3465 b^5 x^{10}}{1280 a^6 x \left (a+b x^2\right )^5} \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 {1}{x^2 \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 = 3.13, size = 400, normalized size = 3.01 \begin {gather*} \left [-\frac {6930 \, b^{5} x^{10} + 32340 \, a b^{4} x^{8} + 59136 \, a^{2} b^{3} x^{6} + 52140 \, a^{3} b^{2} x^{4} + 21230 \, a^{4} b x^{2} + 2560 \, a^{5} - 3465 \, {\left (b^{5} x^{11} + 5 \, a b^{4} x^{9} + 10 \, a^{2} b^{3} x^{7} + 10 \, a^{3} b^{2} x^{5} + 5 \, a^{4} b x^{3} + a^{5} x\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^{6} b^{5} x^{11} + 5 \, a^{7} b^{4} x^{9} + 10 \, a^{8} b^{3} x^{7} + 10 \, a^{9} b^{2} x^{5} + 5 \, a^{10} b x^{3} + a^{11} x\right )}}, -\frac {3465 \, b^{5} x^{10} + 16170 \, a b^{4} x^{8} + 29568 \, a^{2} b^{3} x^{6} + 26070 \, a^{3} b^{2} x^{4} + 10615 \, a^{4} b x^{2} + 1280 \, a^{5} + 3465 \, {\left (b^{5} x^{11} + 5 \, a b^{4} x^{9} + 10 \, a^{2} b^{3} x^{7} + 10 \, a^{3} b^{2} x^{5} + 5 \, a^{4} b x^{3} + a^{5} x\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{1280 \, {\left (a^{6} b^{5} x^{11} + 5 \, a^{7} b^{4} x^{9} + 10 \, a^{8} b^{3} x^{7} + 10 \, a^{9} b^{2} x^{5} + 5 \, a^{10} b x^{3} + a^{11} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 90, normalized size = 0.68 \begin {gather*} -\frac {693 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{6}} - \frac {1}{a^{6} x} - \frac {2185 \, b^{5} x^{9} + 9770 \, a b^{4} x^{7} + 16768 \, a^{2} b^{3} x^{5} + 13270 \, a^{3} b^{2} x^{3} + 4215 \, a^{4} b x}{1280 \, {\left (b x^{2} + a\right )}^{5} a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 126, normalized size = 0.95 \begin {gather*} -\frac {437 b^{5} x^{9}}{256 \left (b \,x^{2}+a \right )^{5} a^{6}}-\frac {977 b^{4} x^{7}}{128 \left (b \,x^{2}+a \right )^{5} a^{5}}-\frac {131 b^{3} x^{5}}{10 \left (b \,x^{2}+a \right )^{5} a^{4}}-\frac {1327 b^{2} x^{3}}{128 \left (b \,x^{2}+a \right )^{5} a^{3}}-\frac {843 b x}{256 \left (b \,x^{2}+a \right )^{5} a^{2}}-\frac {693 b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \sqrt {a b}\, a^{6}}-\frac {1}{a^{6} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.12, size = 137, normalized size = 1.03 \begin {gather*} -\frac {3465 \, b^{5} x^{10} + 16170 \, a b^{4} x^{8} + 29568 \, a^{2} b^{3} x^{6} + 26070 \, a^{3} b^{2} x^{4} + 10615 \, a^{4} b x^{2} + 1280 \, a^{5}}{1280 \, {\left (a^{6} b^{5} x^{11} + 5 \, a^{7} b^{4} x^{9} + 10 \, a^{8} b^{3} x^{7} + 10 \, a^{9} b^{2} x^{5} + 5 \, a^{10} b x^{3} + a^{11} x\right )}} - \frac {693 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.58, size = 132, normalized size = 0.99 \begin {gather*} -\frac {\frac {1}{a}+\frac {2123\,b\,x^2}{256\,a^2}+\frac {2607\,b^2\,x^4}{128\,a^3}+\frac {231\,b^3\,x^6}{10\,a^4}+\frac {1617\,b^4\,x^8}{128\,a^5}+\frac {693\,b^5\,x^{10}}{256\,a^6}}{a^5\,x+5\,a^4\,b\,x^3+10\,a^3\,b^2\,x^5+10\,a^2\,b^3\,x^7+5\,a\,b^4\,x^9+b^5\,x^{11}}-\frac {693\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,a^{13/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.83, size = 187, normalized size = 1.41 \begin {gather*} \frac {693 \sqrt {- \frac {b}{a^{13}}} \log {\left (- \frac {a^{7} \sqrt {- \frac {b}{a^{13}}}}{b} + x \right )}}{512} - \frac {693 \sqrt {- \frac {b}{a^{13}}} \log {\left (\frac {a^{7} \sqrt {- \frac {b}{a^{13}}}}{b} + x \right )}}{512} + \frac {- 1280 a^{5} - 10615 a^{4} b x^{2} - 26070 a^{3} b^{2} x^{4} - 29568 a^{2} b^{3} x^{6} - 16170 a b^{4} x^{8} - 3465 b^{5} x^{10}}{1280 a^{11} x + 6400 a^{10} b x^{3} + 12800 a^{9} b^{2} x^{5} + 12800 a^{8} b^{3} x^{7} + 6400 a^{7} b^{4} x^{9} + 1280 a^{6} b^{5} x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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