Optimal. Leaf size=79 \[ -\frac {7 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}}+\frac {7 a^2 x}{2 b^4}-\frac {7 a x^3}{6 b^3}-\frac {x^7}{2 b \left (a+b x^2\right )}+\frac {7 x^5}{10 b^2} \]
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Rubi [A] time = 0.05, antiderivative size = 79, 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 {7 a^2 x}{2 b^4}-\frac {7 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}}-\frac {7 a x^3}{6 b^3}-\frac {x^7}{2 b \left (a+b x^2\right )}+\frac {7 x^5}{10 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^8}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac {x^8}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac {x^7}{2 b \left (a+b x^2\right )}+\frac {7}{2} \int \frac {x^6}{a b+b^2 x^2} \, dx\\ &=-\frac {x^7}{2 b \left (a+b x^2\right )}+\frac {7}{2} \int \left (\frac {a^2}{b^4}-\frac {a x^2}{b^3}+\frac {x^4}{b^2}-\frac {a^3}{b^3 \left (a b+b^2 x^2\right )}\right ) \, dx\\ &=\frac {7 a^2 x}{2 b^4}-\frac {7 a x^3}{6 b^3}+\frac {7 x^5}{10 b^2}-\frac {x^7}{2 b \left (a+b x^2\right )}-\frac {\left (7 a^3\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{2 b^3}\\ &=\frac {7 a^2 x}{2 b^4}-\frac {7 a x^3}{6 b^3}+\frac {7 x^5}{10 b^2}-\frac {x^7}{2 b \left (a+b x^2\right )}-\frac {7 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 71, normalized size = 0.90 \begin {gather*} \frac {x \left (\frac {15 a^3}{a+b x^2}+90 a^2-20 a b x^2+6 b^2 x^4\right )}{30 b^4}-\frac {7 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}} \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^8}{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 = 0.89, size = 190, normalized size = 2.41 \begin {gather*} \left [\frac {12 \, b^{3} x^{7} - 28 \, a b^{2} x^{5} + 140 \, a^{2} b x^{3} + 210 \, a^{3} x + 105 \, {\left (a^{2} b x^{2} + a^{3}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{60 \, {\left (b^{5} x^{2} + a b^{4}\right )}}, \frac {6 \, b^{3} x^{7} - 14 \, a b^{2} x^{5} + 70 \, a^{2} b x^{3} + 105 \, a^{3} x - 105 \, {\left (a^{2} b x^{2} + a^{3}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{30 \, {\left (b^{5} x^{2} + a b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 73, normalized size = 0.92 \begin {gather*} -\frac {7 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{4}} + \frac {a^{3} x}{2 \, {\left (b x^{2} + a\right )} b^{4}} + \frac {3 \, b^{8} x^{5} - 10 \, a b^{7} x^{3} + 45 \, a^{2} b^{6} x}{15 \, b^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 68, normalized size = 0.86 \begin {gather*} \frac {x^{5}}{5 b^{2}}-\frac {2 a \,x^{3}}{3 b^{3}}+\frac {a^{3} x}{2 \left (b \,x^{2}+a \right ) b^{4}}-\frac {7 a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{4}}+\frac {3 a^{2} x}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.95, size = 71, normalized size = 0.90 \begin {gather*} \frac {a^{3} x}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} - \frac {7 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{4}} + \frac {3 \, b^{2} x^{5} - 10 \, a b x^{3} + 45 \, a^{2} x}{15 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.27, size = 66, normalized size = 0.84 \begin {gather*} \frac {x^5}{5\,b^2}-\frac {2\,a\,x^3}{3\,b^3}+\frac {3\,a^2\,x}{b^4}-\frac {7\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,b^{9/2}}+\frac {a^3\,x}{2\,\left (b^5\,x^2+a\,b^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 124, normalized size = 1.57 \begin {gather*} \frac {a^{3} x}{2 a b^{4} + 2 b^{5} x^{2}} + \frac {3 a^{2} x}{b^{4}} - \frac {2 a x^{3}}{3 b^{3}} + \frac {7 \sqrt {- \frac {a^{5}}{b^{9}}} \log {\left (x - \frac {b^{4} \sqrt {- \frac {a^{5}}{b^{9}}}}{a^{2}} \right )}}{4} - \frac {7 \sqrt {- \frac {a^{5}}{b^{9}}} \log {\left (x + \frac {b^{4} \sqrt {- \frac {a^{5}}{b^{9}}}}{a^{2}} \right )}}{4} + \frac {x^{5}}{5 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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