Optimal. Leaf size=92 \[ \frac {9 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 a^{11/2}}-\frac {9 b^3 x}{2 a^5}+\frac {3 b^2 x^3}{2 a^4}-\frac {9 b x^5}{10 a^3}+\frac {9 x^7}{14 a^2}-\frac {x^9}{2 a \left (a x^2+b\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {263, 288, 302, 205} \[ \frac {3 b^2 x^3}{2 a^4}-\frac {9 b^3 x}{2 a^5}+\frac {9 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 a^{11/2}}-\frac {9 b x^5}{10 a^3}+\frac {9 x^7}{14 a^2}-\frac {x^9}{2 a \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
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Rule 205
Rule 263
Rule 288
Rule 302
Rubi steps
\begin {align*} \int \frac {x^6}{\left (a+\frac {b}{x^2}\right )^2} \, dx &=\int \frac {x^{10}}{\left (b+a x^2\right )^2} \, dx\\ &=-\frac {x^9}{2 a \left (b+a x^2\right )}+\frac {9 \int \frac {x^8}{b+a x^2} \, dx}{2 a}\\ &=-\frac {x^9}{2 a \left (b+a x^2\right )}+\frac {9 \int \left (-\frac {b^3}{a^4}+\frac {b^2 x^2}{a^3}-\frac {b x^4}{a^2}+\frac {x^6}{a}+\frac {b^4}{a^4 \left (b+a x^2\right )}\right ) \, dx}{2 a}\\ &=-\frac {9 b^3 x}{2 a^5}+\frac {3 b^2 x^3}{2 a^4}-\frac {9 b x^5}{10 a^3}+\frac {9 x^7}{14 a^2}-\frac {x^9}{2 a \left (b+a x^2\right )}+\frac {\left (9 b^4\right ) \int \frac {1}{b+a x^2} \, dx}{2 a^5}\\ &=-\frac {9 b^3 x}{2 a^5}+\frac {3 b^2 x^3}{2 a^4}-\frac {9 b x^5}{10 a^3}+\frac {9 x^7}{14 a^2}-\frac {x^9}{2 a \left (b+a x^2\right )}+\frac {9 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 a^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 82, normalized size = 0.89 \[ \frac {9 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 a^{11/2}}+\frac {x \left (10 a^3 x^6-28 a^2 b x^4-\frac {35 b^4}{a x^2+b}+70 a b^2 x^2-280 b^3\right )}{70 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 212, normalized size = 2.30 \[ \left [\frac {20 \, a^{4} x^{9} - 36 \, a^{3} b x^{7} + 84 \, a^{2} b^{2} x^{5} - 420 \, a b^{3} x^{3} - 630 \, b^{4} x + 315 \, {\left (a b^{3} x^{2} + b^{4}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {a x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - b}{a x^{2} + b}\right )}{140 \, {\left (a^{6} x^{2} + a^{5} b\right )}}, \frac {10 \, a^{4} x^{9} - 18 \, a^{3} b x^{7} + 42 \, a^{2} b^{2} x^{5} - 210 \, a b^{3} x^{3} - 315 \, b^{4} x + 315 \, {\left (a b^{3} x^{2} + b^{4}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a x \sqrt {\frac {b}{a}}}{b}\right )}{70 \, {\left (a^{6} x^{2} + a^{5} b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 84, normalized size = 0.91 \[ \frac {9 \, b^{4} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{5}} - \frac {b^{4} x}{2 \, {\left (a x^{2} + b\right )} a^{5}} + \frac {5 \, a^{12} x^{7} - 14 \, a^{11} b x^{5} + 35 \, a^{10} b^{2} x^{3} - 140 \, a^{9} b^{3} x}{35 \, a^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 78, normalized size = 0.85 \[ \frac {x^{7}}{7 a^{2}}-\frac {2 b \,x^{5}}{5 a^{3}}+\frac {b^{2} x^{3}}{a^{4}}-\frac {b^{4} x}{2 \left (a \,x^{2}+b \right ) a^{5}}+\frac {9 b^{4} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{5}}-\frac {4 b^{3} x}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.00, size = 82, normalized size = 0.89 \[ -\frac {b^{4} x}{2 \, {\left (a^{6} x^{2} + a^{5} b\right )}} + \frac {9 \, b^{4} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{5}} + \frac {5 \, a^{3} x^{7} - 14 \, a^{2} b x^{5} + 35 \, a b^{2} x^{3} - 140 \, b^{3} x}{35 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 77, normalized size = 0.84 \[ \frac {x^7}{7\,a^2}-\frac {2\,b\,x^5}{5\,a^3}-\frac {4\,b^3\,x}{a^5}+\frac {9\,b^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{2\,a^{11/2}}+\frac {b^2\,x^3}{a^4}-\frac {b^4\,x}{2\,\left (a^6\,x^2+b\,a^5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 134, normalized size = 1.46 \[ - \frac {b^{4} x}{2 a^{6} x^{2} + 2 a^{5} b} - \frac {9 \sqrt {- \frac {b^{7}}{a^{11}}} \log {\left (- \frac {a^{5} \sqrt {- \frac {b^{7}}{a^{11}}}}{b^{3}} + x \right )}}{4} + \frac {9 \sqrt {- \frac {b^{7}}{a^{11}}} \log {\left (\frac {a^{5} \sqrt {- \frac {b^{7}}{a^{11}}}}{b^{3}} + x \right )}}{4} + \frac {x^{7}}{7 a^{2}} - \frac {2 b x^{5}}{5 a^{3}} + \frac {b^{2} x^{3}}{a^{4}} - \frac {4 b^{3} x}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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