Optimal. Leaf size=104 \[ \frac {105 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 b^{11/2}}-\frac {105 a x}{16 b^5}-\frac {21 x^5}{16 b^3 \left (a+b x^2\right )}-\frac {3 x^7}{8 b^2 \left (a+b x^2\right )^2}-\frac {x^9}{6 b \left (a+b x^2\right )^3}+\frac {35 x^3}{16 b^4} \]
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Rubi [A] time = 0.06, antiderivative size = 104, 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} \[ \frac {105 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 b^{11/2}}-\frac {3 x^7}{8 b^2 \left (a+b x^2\right )^2}-\frac {21 x^5}{16 b^3 \left (a+b x^2\right )}-\frac {105 a x}{16 b^5}-\frac {x^9}{6 b \left (a+b x^2\right )^3}+\frac {35 x^3}{16 b^4} \]
Antiderivative was successfully verified.
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Rule 28
Rule 205
Rule 288
Rule 302
Rubi steps
\begin {align*} \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {x^{10}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {x^9}{6 b \left (a+b x^2\right )^3}+\frac {1}{2} \left (3 b^2\right ) \int \frac {x^8}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {x^9}{6 b \left (a+b x^2\right )^3}-\frac {3 x^7}{8 b^2 \left (a+b x^2\right )^2}+\frac {21}{8} \int \frac {x^6}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac {x^9}{6 b \left (a+b x^2\right )^3}-\frac {3 x^7}{8 b^2 \left (a+b x^2\right )^2}-\frac {21 x^5}{16 b^3 \left (a+b x^2\right )}+\frac {105 \int \frac {x^4}{a b+b^2 x^2} \, dx}{16 b^2}\\ &=-\frac {x^9}{6 b \left (a+b x^2\right )^3}-\frac {3 x^7}{8 b^2 \left (a+b x^2\right )^2}-\frac {21 x^5}{16 b^3 \left (a+b x^2\right )}+\frac {105 \int \left (-\frac {a}{b^3}+\frac {x^2}{b^2}+\frac {a^2}{b^2 \left (a b+b^2 x^2\right )}\right ) \, dx}{16 b^2}\\ &=-\frac {105 a x}{16 b^5}+\frac {35 x^3}{16 b^4}-\frac {x^9}{6 b \left (a+b x^2\right )^3}-\frac {3 x^7}{8 b^2 \left (a+b x^2\right )^2}-\frac {21 x^5}{16 b^3 \left (a+b x^2\right )}+\frac {\left (105 a^2\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{16 b^4}\\ &=-\frac {105 a x}{16 b^5}+\frac {35 x^3}{16 b^4}-\frac {x^9}{6 b \left (a+b x^2\right )^3}-\frac {3 x^7}{8 b^2 \left (a+b x^2\right )^2}-\frac {21 x^5}{16 b^3 \left (a+b x^2\right )}+\frac {105 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 b^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 89, normalized size = 0.86 \[ \frac {315 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+\frac {\sqrt {b} x \left (-315 a^4-840 a^3 b x^2-693 a^2 b^2 x^4-144 a b^3 x^6+16 b^4 x^8\right )}{\left (a+b x^2\right )^3}}{48 b^{11/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 296, normalized size = 2.85 \[ \left [\frac {32 \, b^{4} x^{9} - 288 \, a b^{3} x^{7} - 1386 \, a^{2} b^{2} x^{5} - 1680 \, a^{3} b x^{3} - 630 \, a^{4} x + 315 \, {\left (a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} + 3 \, a^{3} b x^{2} + a^{4}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{96 \, {\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}}, \frac {16 \, b^{4} x^{9} - 144 \, a b^{3} x^{7} - 693 \, a^{2} b^{2} x^{5} - 840 \, a^{3} b x^{3} - 315 \, a^{4} x + 315 \, {\left (a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} + 3 \, a^{3} b x^{2} + a^{4}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{48 \, {\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 84, normalized size = 0.81 \[ \frac {105 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} b^{5}} - \frac {165 \, a^{2} b^{2} x^{5} + 280 \, a^{3} b x^{3} + 123 \, a^{4} x}{48 \, {\left (b x^{2} + a\right )}^{3} b^{5}} + \frac {b^{8} x^{3} - 12 \, a b^{7} x}{3 \, b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 97, normalized size = 0.93 \[ -\frac {55 a^{2} x^{5}}{16 \left (b \,x^{2}+a \right )^{3} b^{3}}-\frac {35 a^{3} x^{3}}{6 \left (b \,x^{2}+a \right )^{3} b^{4}}-\frac {41 a^{4} x}{16 \left (b \,x^{2}+a \right )^{3} b^{5}}+\frac {x^{3}}{3 b^{4}}+\frac {105 a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \sqrt {a b}\, b^{5}}-\frac {4 a x}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.01, size = 104, normalized size = 1.00 \[ -\frac {165 \, a^{2} b^{2} x^{5} + 280 \, a^{3} b x^{3} + 123 \, a^{4} x}{48 \, {\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}} + \frac {105 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} b^{5}} + \frac {b x^{3} - 12 \, a x}{3 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.36, size = 99, normalized size = 0.95 \[ \frac {x^3}{3\,b^4}-\frac {\frac {41\,a^4\,x}{16}+\frac {35\,a^3\,b\,x^3}{6}+\frac {55\,a^2\,b^2\,x^5}{16}}{a^3\,b^5+3\,a^2\,b^6\,x^2+3\,a\,b^7\,x^4+b^8\,x^6}+\frac {105\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{16\,b^{11/2}}-\frac {4\,a\,x}{b^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.63, size = 156, normalized size = 1.50 \[ - \frac {4 a x}{b^{5}} - \frac {105 \sqrt {- \frac {a^{3}}{b^{11}}} \log {\left (x - \frac {b^{5} \sqrt {- \frac {a^{3}}{b^{11}}}}{a} \right )}}{32} + \frac {105 \sqrt {- \frac {a^{3}}{b^{11}}} \log {\left (x + \frac {b^{5} \sqrt {- \frac {a^{3}}{b^{11}}}}{a} \right )}}{32} + \frac {- 123 a^{4} x - 280 a^{3} b x^{3} - 165 a^{2} b^{2} x^{5}}{48 a^{3} b^{5} + 144 a^{2} b^{6} x^{2} + 144 a b^{7} x^{4} + 48 b^{8} x^{6}} + \frac {x^{3}}{3 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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