Optimal. Leaf size=66 \[ \frac {5 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{7/2}}-\frac {5 a x}{2 b^3}-\frac {x^5}{2 b \left (a+b x^2\right )}+\frac {5 x^3}{6 b^2} \]
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Rubi [A] time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 288, 302, 205} \begin {gather*} \frac {5 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{7/2}}-\frac {5 a x}{2 b^3}-\frac {x^5}{2 b \left (a+b x^2\right )}+\frac {5 x^3}{6 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 205
Rule 288
Rule 302
Rubi steps
\begin {align*} \int \frac {x^6}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac {x^6}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac {x^5}{2 b \left (a+b x^2\right )}+\frac {5}{2} \int \frac {x^4}{a b+b^2 x^2} \, dx\\ &=-\frac {x^5}{2 b \left (a+b x^2\right )}+\frac {5}{2} \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\\ &=-\frac {5 a x}{2 b^3}+\frac {5 x^3}{6 b^2}-\frac {x^5}{2 b \left (a+b x^2\right )}+\frac {\left (5 a^2\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{2 b^2}\\ &=-\frac {5 a x}{2 b^3}+\frac {5 x^3}{6 b^2}-\frac {x^5}{2 b \left (a+b x^2\right )}+\frac {5 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 60, normalized size = 0.91 \begin {gather*} \frac {5 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{7/2}}+\frac {x \left (-\frac {3 a^2}{a+b x^2}-12 a+2 b x^2\right )}{6 b^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 {x^6}{a^2+2 a b x^2+b^2 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.66, size = 164, normalized size = 2.48 \begin {gather*} \left [\frac {4 \, b^{2} x^{5} - 20 \, a b x^{3} - 30 \, a^{2} x + 15 \, {\left (a b x^{2} + a^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{12 \, {\left (b^{4} x^{2} + a b^{3}\right )}}, \frac {2 \, b^{2} x^{5} - 10 \, a b x^{3} - 15 \, a^{2} x + 15 \, {\left (a b x^{2} + a^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{6 \, {\left (b^{4} x^{2} + a b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 61, normalized size = 0.92 \begin {gather*} \frac {5 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{3}} - \frac {a^{2} x}{2 \, {\left (b x^{2} + a\right )} b^{3}} + \frac {b^{4} x^{3} - 6 \, a b^{3} x}{3 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 57, normalized size = 0.86 \begin {gather*} \frac {x^{3}}{3 b^{2}}-\frac {a^{2} x}{2 \left (b \,x^{2}+a \right ) b^{3}}+\frac {5 a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{3}}-\frac {2 a x}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.13, size = 59, normalized size = 0.89 \begin {gather*} -\frac {a^{2} x}{2 \, {\left (b^{4} x^{2} + a b^{3}\right )}} + \frac {5 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{3}} + \frac {b x^{3} - 6 \, a x}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 56, normalized size = 0.85 \begin {gather*} \frac {x^3}{3\,b^2}+\frac {5\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,b^{7/2}}-\frac {a^2\,x}{2\,\left (b^4\,x^2+a\,b^3\right )}-\frac {2\,a\,x}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 107, normalized size = 1.62 \begin {gather*} - \frac {a^{2} x}{2 a b^{3} + 2 b^{4} x^{2}} - \frac {2 a x}{b^{3}} - \frac {5 \sqrt {- \frac {a^{3}}{b^{7}}} \log {\left (x - \frac {b^{3} \sqrt {- \frac {a^{3}}{b^{7}}}}{a} \right )}}{4} + \frac {5 \sqrt {- \frac {a^{3}}{b^{7}}} \log {\left (x + \frac {b^{3} \sqrt {- \frac {a^{3}}{b^{7}}}}{a} \right )}}{4} + \frac {x^{3}}{3 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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