Optimal. Leaf size=58 \[ -\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2}}-\frac {b^2}{a^3 x}+\frac {b}{3 a^2 x^3}-\frac {1}{5 a x^5} \]
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Rubi [A] time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {325, 205} \[ -\frac {b^2}{a^3 x}-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2}}+\frac {b}{3 a^2 x^3}-\frac {1}{5 a x^5} \]
Antiderivative was successfully verified.
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Rule 205
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (a+b x^2\right )} \, dx &=-\frac {1}{5 a x^5}-\frac {b \int \frac {1}{x^4 \left (a+b x^2\right )} \, dx}{a}\\ &=-\frac {1}{5 a x^5}+\frac {b}{3 a^2 x^3}+\frac {b^2 \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{a^2}\\ &=-\frac {1}{5 a x^5}+\frac {b}{3 a^2 x^3}-\frac {b^2}{a^3 x}-\frac {b^3 \int \frac {1}{a+b x^2} \, dx}{a^3}\\ &=-\frac {1}{5 a x^5}+\frac {b}{3 a^2 x^3}-\frac {b^2}{a^3 x}-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 58, normalized size = 1.00 \[ -\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2}}-\frac {b^2}{a^3 x}+\frac {b}{3 a^2 x^3}-\frac {1}{5 a x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 132, normalized size = 2.28 \[ \left [\frac {15 \, b^{2} x^{5} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 30 \, b^{2} x^{4} + 10 \, a b x^{2} - 6 \, a^{2}}{30 \, a^{3} x^{5}}, -\frac {15 \, b^{2} x^{5} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 15 \, b^{2} x^{4} - 5 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{3} x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 52, normalized size = 0.90 \[ -\frac {b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {15 \, b^{2} x^{4} - 5 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.90 \[ -\frac {b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{3}}-\frac {b^{2}}{a^{3} x}+\frac {b}{3 a^{2} x^{3}}-\frac {1}{5 a \,x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.89, size = 52, normalized size = 0.90 \[ -\frac {b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {15 \, b^{2} x^{4} - 5 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 48, normalized size = 0.83 \[ -\frac {\frac {1}{5\,a}-\frac {b\,x^2}{3\,a^2}+\frac {b^2\,x^4}{a^3}}{x^5}-\frac {b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{a^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.26, size = 100, normalized size = 1.72 \[ \frac {\sqrt {- \frac {b^{5}}{a^{7}}} \log {\left (- \frac {a^{4} \sqrt {- \frac {b^{5}}{a^{7}}}}{b^{3}} + x \right )}}{2} - \frac {\sqrt {- \frac {b^{5}}{a^{7}}} \log {\left (\frac {a^{4} \sqrt {- \frac {b^{5}}{a^{7}}}}{b^{3}} + x \right )}}{2} + \frac {- 3 a^{2} + 5 a b x^{2} - 15 b^{2} x^{4}}{15 a^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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