Optimal. Leaf size=68 \[ \frac{b^2 x^3}{3 a^3}-\frac{b^3 x}{a^4}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{9/2}}-\frac{b x^5}{5 a^2}+\frac{x^7}{7 a} \]
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Rubi [A] time = 0.0279305, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 302, 205} \[ \frac{b^2 x^3}{3 a^3}-\frac{b^3 x}{a^4}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{9/2}}-\frac{b x^5}{5 a^2}+\frac{x^7}{7 a} \]
Antiderivative was successfully verified.
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Rule 263
Rule 302
Rule 205
Rubi steps
\begin{align*} \int \frac{x^6}{a+\frac{b}{x^2}} \, dx &=\int \frac{x^8}{b+a x^2} \, dx\\ &=\int \left (-\frac{b^3}{a^4}+\frac{b^2 x^2}{a^3}-\frac{b x^4}{a^2}+\frac{x^6}{a}+\frac{b^4}{a^4 \left (b+a x^2\right )}\right ) \, dx\\ &=-\frac{b^3 x}{a^4}+\frac{b^2 x^3}{3 a^3}-\frac{b x^5}{5 a^2}+\frac{x^7}{7 a}+\frac{b^4 \int \frac{1}{b+a x^2} \, dx}{a^4}\\ &=-\frac{b^3 x}{a^4}+\frac{b^2 x^3}{3 a^3}-\frac{b x^5}{5 a^2}+\frac{x^7}{7 a}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0251687, size = 68, normalized size = 1. \[ \frac{b^2 x^3}{3 a^3}-\frac{b^3 x}{a^4}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{9/2}}-\frac{b x^5}{5 a^2}+\frac{x^7}{7 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 60, normalized size = 0.9 \begin{align*}{\frac{{x}^{7}}{7\,a}}-{\frac{b{x}^{5}}{5\,{a}^{2}}}+{\frac{{b}^{2}{x}^{3}}{3\,{a}^{3}}}-{\frac{{b}^{3}x}{{a}^{4}}}+{\frac{{b}^{4}}{{a}^{4}}\arctan \left ({ax{\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 = 1.49121, size = 336, normalized size = 4.94 \begin{align*} \left [\frac{30 \, a^{3} x^{7} - 42 \, a^{2} b x^{5} + 70 \, a b^{2} x^{3} + 105 \, b^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right ) - 210 \, b^{3} x}{210 \, a^{4}}, \frac{15 \, a^{3} x^{7} - 21 \, a^{2} b x^{5} + 35 \, a b^{2} x^{3} + 105 \, b^{3} \sqrt{\frac{b}{a}} \arctan \left (\frac{a x \sqrt{\frac{b}{a}}}{b}\right ) - 105 \, b^{3} x}{105 \, a^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.328377, size = 107, normalized size = 1.57 \begin{align*} - \frac{\sqrt{- \frac{b^{7}}{a^{9}}} \log{\left (- \frac{a^{4} \sqrt{- \frac{b^{7}}{a^{9}}}}{b^{3}} + x \right )}}{2} + \frac{\sqrt{- \frac{b^{7}}{a^{9}}} \log{\left (\frac{a^{4} \sqrt{- \frac{b^{7}}{a^{9}}}}{b^{3}} + x \right )}}{2} + \frac{x^{7}}{7 a} - \frac{b x^{5}}{5 a^{2}} + \frac{b^{2} x^{3}}{3 a^{3}} - \frac{b^{3} x}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18998, size = 88, normalized size = 1.29 \begin{align*} \frac{b^{4} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} + \frac{15 \, a^{6} x^{7} - 21 \, a^{5} b x^{5} + 35 \, a^{4} b^{2} x^{3} - 105 \, a^{3} b^{3} x}{105 \, a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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