Optimal. Leaf size=79 \[ \frac{7 a^2 x}{2 b^4}-\frac{7 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{9/2}}-\frac{7 a x^3}{6 b^3}-\frac{x^7}{2 b \left (a+b x^2\right )}+\frac{7 x^5}{10 b^2} \]
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Rubi [A] time = 0.0452181, antiderivative size = 79, normalized size of antiderivative = 1., 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} \[ \frac{7 a^2 x}{2 b^4}-\frac{7 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{9/2}}-\frac{7 a x^3}{6 b^3}-\frac{x^7}{2 b \left (a+b x^2\right )}+\frac{7 x^5}{10 b^2} \]
Antiderivative was successfully verified.
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Rule 28
Rule 288
Rule 302
Rule 205
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*}
Mathematica [A] time = 0.045999, size = 71, normalized size = 0.9 \[ \frac{x \left (\frac{15 a^3}{a+b x^2}+90 a^2-20 a b x^2+6 b^2 x^4\right )}{30 b^4}-\frac{7 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 68, normalized size = 0.9 \begin{align*}{\frac{{x}^{5}}{5\,{b}^{2}}}-{\frac{2\,a{x}^{3}}{3\,{b}^{3}}}+3\,{\frac{{a}^{2}x}{{b}^{4}}}+{\frac{x{a}^{3}}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{7\,{a}^{3}}{2\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76173, size = 409, normalized size = 5.18 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.461368, size = 124, normalized size = 1.57 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18509, size = 99, normalized size = 1.25 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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