Optimal. Leaf size=119 \[ -\frac {231 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 a^{13/2}}-\frac {231 b^2}{16 a^6 x}+\frac {77 b}{16 a^5 x^3}-\frac {231}{80 a^4 x^5}+\frac {33}{16 a^3 x^5 \left (a+b x^2\right )}+\frac {11}{24 a^2 x^5 \left (a+b x^2\right )^2}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.08, antiderivative size = 119, 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 b^2}{16 a^6 x}-\frac {231 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 a^{13/2}}+\frac {77 b}{16 a^5 x^3}+\frac {33}{16 a^3 x^5 \left (a+b x^2\right )}+\frac {11}{24 a^2 x^5 \left (a+b x^2\right )^2}-\frac {231}{80 a^4 x^5}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3} \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^6 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {1}{x^6 \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac {1}{6 a x^5 \left (a+b x^2\right )^3}+\frac {\left (11 b^3\right ) \int \frac {1}{x^6 \left (a b+b^2 x^2\right )^3} \, dx}{6 a}\\ &=\frac {1}{6 a x^5 \left (a+b x^2\right )^3}+\frac {11}{24 a^2 x^5 \left (a+b x^2\right )^2}+\frac {\left (33 b^2\right ) \int \frac {1}{x^6 \left (a b+b^2 x^2\right )^2} \, dx}{8 a^2}\\ &=\frac {1}{6 a x^5 \left (a+b x^2\right )^3}+\frac {11}{24 a^2 x^5 \left (a+b x^2\right )^2}+\frac {33}{16 a^3 x^5 \left (a+b x^2\right )}+\frac {(231 b) \int \frac {1}{x^6 \left (a b+b^2 x^2\right )} \, dx}{16 a^3}\\ &=-\frac {231}{80 a^4 x^5}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}+\frac {11}{24 a^2 x^5 \left (a+b x^2\right )^2}+\frac {33}{16 a^3 x^5 \left (a+b x^2\right )}-\frac {\left (231 b^2\right ) \int \frac {1}{x^4 \left (a b+b^2 x^2\right )} \, dx}{16 a^4}\\ &=-\frac {231}{80 a^4 x^5}+\frac {77 b}{16 a^5 x^3}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}+\frac {11}{24 a^2 x^5 \left (a+b x^2\right )^2}+\frac {33}{16 a^3 x^5 \left (a+b x^2\right )}+\frac {\left (231 b^3\right ) \int \frac {1}{x^2 \left (a b+b^2 x^2\right )} \, dx}{16 a^5}\\ &=-\frac {231}{80 a^4 x^5}+\frac {77 b}{16 a^5 x^3}-\frac {231 b^2}{16 a^6 x}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}+\frac {11}{24 a^2 x^5 \left (a+b x^2\right )^2}+\frac {33}{16 a^3 x^5 \left (a+b x^2\right )}-\frac {\left (231 b^4\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{16 a^6}\\ &=-\frac {231}{80 a^4 x^5}+\frac {77 b}{16 a^5 x^3}-\frac {231 b^2}{16 a^6 x}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}+\frac {11}{24 a^2 x^5 \left (a+b x^2\right )^2}+\frac {33}{16 a^3 x^5 \left (a+b x^2\right )}-\frac {231 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 a^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 101, normalized size = 0.85 \begin {gather*} -\frac {231 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 a^{13/2}}-\frac {48 a^5-176 a^4 b x^2+1584 a^3 b^2 x^4+7623 a^2 b^3 x^6+9240 a b^4 x^8+3465 b^5 x^{10}}{240 a^6 x^5 \left (a+b x^2\right )^3} \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^6 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.87, size = 330, normalized size = 2.77 \begin {gather*} \left [-\frac {6930 \, b^{5} x^{10} + 18480 \, a b^{4} x^{8} + 15246 \, a^{2} b^{3} x^{6} + 3168 \, a^{3} b^{2} x^{4} - 352 \, a^{4} b x^{2} + 96 \, a^{5} - 3465 \, {\left (b^{5} x^{11} + 3 \, a b^{4} x^{9} + 3 \, a^{2} b^{3} x^{7} + a^{3} 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 )}{480 \, {\left (a^{6} b^{3} x^{11} + 3 \, a^{7} b^{2} x^{9} + 3 \, a^{8} b x^{7} + a^{9} x^{5}\right )}}, -\frac {3465 \, b^{5} x^{10} + 9240 \, a b^{4} x^{8} + 7623 \, a^{2} b^{3} x^{6} + 1584 \, a^{3} b^{2} x^{4} - 176 \, a^{4} b x^{2} + 48 \, a^{5} + 3465 \, {\left (b^{5} x^{11} + 3 \, a b^{4} x^{9} + 3 \, a^{2} b^{3} x^{7} + a^{3} b^{2} x^{5}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{240 \, {\left (a^{6} b^{3} x^{11} + 3 \, a^{7} b^{2} x^{9} + 3 \, a^{8} b x^{7} + a^{9} x^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 93, normalized size = 0.78 \begin {gather*} -\frac {231 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} a^{6}} - \frac {213 \, b^{5} x^{5} + 472 \, a b^{4} x^{3} + 267 \, a^{2} b^{3} x}{48 \, {\left (b x^{2} + a\right )}^{3} a^{6}} - \frac {150 \, b^{2} x^{4} - 20 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{6} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 110, normalized size = 0.92 \begin {gather*} -\frac {71 b^{5} x^{5}}{16 \left (b \,x^{2}+a \right )^{3} a^{6}}-\frac {59 b^{4} x^{3}}{6 \left (b \,x^{2}+a \right )^{3} a^{5}}-\frac {89 b^{3} x}{16 \left (b \,x^{2}+a \right )^{3} a^{4}}-\frac {231 b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \sqrt {a b}\, a^{6}}-\frac {10 b^{2}}{a^{6} x}+\frac {4 b}{3 a^{5} x^{3}}-\frac {1}{5 a^{4} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.01, size = 119, normalized size = 1.00 \begin {gather*} -\frac {3465 \, b^{5} x^{10} + 9240 \, a b^{4} x^{8} + 7623 \, a^{2} b^{3} x^{6} + 1584 \, a^{3} b^{2} x^{4} - 176 \, a^{4} b x^{2} + 48 \, a^{5}}{240 \, {\left (a^{6} b^{3} x^{11} + 3 \, a^{7} b^{2} x^{9} + 3 \, a^{8} b x^{7} + a^{9} x^{5}\right )}} - \frac {231 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.46, size = 114, normalized size = 0.96 \begin {gather*} -\frac {\frac {1}{5\,a}-\frac {11\,b\,x^2}{15\,a^2}+\frac {33\,b^2\,x^4}{5\,a^3}+\frac {2541\,b^3\,x^6}{80\,a^4}+\frac {77\,b^4\,x^8}{2\,a^5}+\frac {231\,b^5\,x^{10}}{16\,a^6}}{a^3\,x^5+3\,a^2\,b\,x^7+3\,a\,b^2\,x^9+b^3\,x^{11}}-\frac {231\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{16\,a^{13/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.71, size = 173, normalized size = 1.45 \begin {gather*} \frac {231 \sqrt {- \frac {b^{5}}{a^{13}}} \log {\left (- \frac {a^{7} \sqrt {- \frac {b^{5}}{a^{13}}}}{b^{3}} + x \right )}}{32} - \frac {231 \sqrt {- \frac {b^{5}}{a^{13}}} \log {\left (\frac {a^{7} \sqrt {- \frac {b^{5}}{a^{13}}}}{b^{3}} + x \right )}}{32} + \frac {- 48 a^{5} + 176 a^{4} b x^{2} - 1584 a^{3} b^{2} x^{4} - 7623 a^{2} b^{3} x^{6} - 9240 a b^{4} x^{8} - 3465 b^{5} x^{10}}{240 a^{9} x^{5} + 720 a^{8} b x^{7} + 720 a^{7} b^{2} x^{9} + 240 a^{6} b^{3} x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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