Optimal. Leaf size=79 \[ \frac {5 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 a^{7/2} \sqrt {b}}+\frac {5 x}{16 a^3 \left (a+b x^2\right )}+\frac {5 x}{24 a^2 \left (a+b x^2\right )^2}+\frac {x}{6 a \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.04, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {28, 199, 205} \begin {gather*} \frac {5 x}{16 a^3 \left (a+b x^2\right )}+\frac {5 x}{24 a^2 \left (a+b x^2\right )^2}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 a^{7/2} \sqrt {b}}+\frac {x}{6 a \left (a+b x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 199
Rule 205
Rubi steps
\begin {align*} \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {1}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac {x}{6 a \left (a+b x^2\right )^3}+\frac {\left (5 b^3\right ) \int \frac {1}{\left (a b+b^2 x^2\right )^3} \, dx}{6 a}\\ &=\frac {x}{6 a \left (a+b x^2\right )^3}+\frac {5 x}{24 a^2 \left (a+b x^2\right )^2}+\frac {\left (5 b^2\right ) \int \frac {1}{\left (a b+b^2 x^2\right )^2} \, dx}{8 a^2}\\ &=\frac {x}{6 a \left (a+b x^2\right )^3}+\frac {5 x}{24 a^2 \left (a+b x^2\right )^2}+\frac {5 x}{16 a^3 \left (a+b x^2\right )}+\frac {(5 b) \int \frac {1}{a b+b^2 x^2} \, dx}{16 a^3}\\ &=\frac {x}{6 a \left (a+b x^2\right )^3}+\frac {5 x}{24 a^2 \left (a+b x^2\right )^2}+\frac {5 x}{16 a^3 \left (a+b x^2\right )}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 a^{7/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 66, normalized size = 0.84 \begin {gather*} \frac {5 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 a^{7/2} \sqrt {b}}+\frac {33 a^2 x+40 a b x^3+15 b^2 x^5}{48 a^3 \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}{\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 = 0.80, size = 254, normalized size = 3.22 \begin {gather*} \left [\frac {30 \, a b^{3} x^{5} + 80 \, a^{2} b^{2} x^{3} + 66 \, a^{3} b x - 15 \, {\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{96 \, {\left (a^{4} b^{4} x^{6} + 3 \, a^{5} b^{3} x^{4} + 3 \, a^{6} b^{2} x^{2} + a^{7} b\right )}}, \frac {15 \, a b^{3} x^{5} + 40 \, a^{2} b^{2} x^{3} + 33 \, a^{3} b x + 15 \, {\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{48 \, {\left (a^{4} b^{4} x^{6} + 3 \, a^{5} b^{3} x^{4} + 3 \, a^{6} b^{2} x^{2} + a^{7} b\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 56, normalized size = 0.71 \begin {gather*} \frac {5 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} a^{3}} + \frac {15 \, b^{2} x^{5} + 40 \, a b x^{3} + 33 \, a^{2} x}{48 \, {\left (b x^{2} + a\right )}^{3} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 66, normalized size = 0.84 \begin {gather*} \frac {x}{6 \left (b \,x^{2}+a \right )^{3} a}+\frac {5 x}{24 \left (b \,x^{2}+a \right )^{2} a^{2}}+\frac {5 x}{16 \left (b \,x^{2}+a \right ) a^{3}}+\frac {5 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \sqrt {a b}\, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.01, size = 80, normalized size = 1.01 \begin {gather*} \frac {15 \, b^{2} x^{5} + 40 \, a b x^{3} + 33 \, a^{2} x}{48 \, {\left (a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6}\right )}} + \frac {5 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.36, size = 77, normalized size = 0.97 \begin {gather*} \frac {\frac {11\,x}{16\,a}+\frac {5\,b\,x^3}{6\,a^2}+\frac {5\,b^2\,x^5}{16\,a^3}}{a^3+3\,a^2\,b\,x^2+3\,a\,b^2\,x^4+b^3\,x^6}+\frac {5\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{16\,a^{7/2}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 129, normalized size = 1.63 \begin {gather*} - \frac {5 \sqrt {- \frac {1}{a^{7} b}} \log {\left (- a^{4} \sqrt {- \frac {1}{a^{7} b}} + x \right )}}{32} + \frac {5 \sqrt {- \frac {1}{a^{7} b}} \log {\left (a^{4} \sqrt {- \frac {1}{a^{7} b}} + x \right )}}{32} + \frac {33 a^{2} x + 40 a b x^{3} + 15 b^{2} x^{5}}{48 a^{6} + 144 a^{5} b x^{2} + 144 a^{4} b^{2} x^{4} + 48 a^{3} b^{3} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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