Optimal. Leaf size=57 \[ -\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2}}-\frac {3}{2 a^2 x}+\frac {1}{2 a x \left (a+b x^2\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2}}-\frac {3}{2 a^2 x}+\frac {1}{2 a x \left (a+b x^2\right )} \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 )} \, dx &=b^2 \int \frac {1}{x^2 \left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac {1}{2 a x \left (a+b x^2\right )}+\frac {(3 b) \int \frac {1}{x^2 \left (a b+b^2 x^2\right )} \, dx}{2 a}\\ &=-\frac {3}{2 a^2 x}+\frac {1}{2 a x \left (a+b x^2\right )}-\frac {\left (3 b^2\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{2 a^2}\\ &=-\frac {3}{2 a^2 x}+\frac {1}{2 a x \left (a+b x^2\right )}-\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 54, normalized size = 0.95 \begin {gather*} -\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2}}-\frac {b x}{2 a^2 \left (a+b x^2\right )}-\frac {1}{a^2 x} \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 )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.87, size = 136, normalized size = 2.39 \begin {gather*} \left [-\frac {6 \, b x^{2} - 3 \, {\left (b x^{3} + a 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 ) + 4 \, a}{4 \, {\left (a^{2} b x^{3} + a^{3} x\right )}}, -\frac {3 \, b x^{2} + 3 \, {\left (b x^{3} + a x\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 2 \, a}{2 \, {\left (a^{2} b x^{3} + a^{3} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 47, normalized size = 0.82 \begin {gather*} -\frac {3 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2}} - \frac {3 \, b x^{2} + 2 \, a}{2 \, {\left (b x^{3} + a x\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 46, normalized size = 0.81 \begin {gather*} -\frac {b x}{2 \left (b \,x^{2}+a \right ) a^{2}}-\frac {3 b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{2}}-\frac {1}{a^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.95, size = 49, normalized size = 0.86 \begin {gather*} -\frac {3 \, b x^{2} + 2 \, a}{2 \, {\left (a^{2} b x^{3} + a^{3} x\right )}} - \frac {3 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.49, size = 44, normalized size = 0.77 \begin {gather*} -\frac {\frac {1}{a}+\frac {3\,b\,x^2}{2\,a^2}}{b\,x^3+a\,x}-\frac {3\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,a^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 92, normalized size = 1.61 \begin {gather*} \frac {3 \sqrt {- \frac {b}{a^{5}}} \log {\left (- \frac {a^{3} \sqrt {- \frac {b}{a^{5}}}}{b} + x \right )}}{4} - \frac {3 \sqrt {- \frac {b}{a^{5}}} \log {\left (\frac {a^{3} \sqrt {- \frac {b}{a^{5}}}}{b} + x \right )}}{4} + \frac {- 2 a - 3 b x^{2}}{2 a^{3} x + 2 a^{2} b x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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