Optimal. Leaf size=55 \[ -\frac {a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}}+\frac {a^2 x}{b^3}-\frac {a x^3}{3 b^2}+\frac {x^5}{5 b} \]
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Rubi [A] time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {302, 205} \[ \frac {a^2 x}{b^3}-\frac {a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}}-\frac {a x^3}{3 b^2}+\frac {x^5}{5 b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 302
Rubi steps
\begin {align*} \int \frac {x^6}{a+b x^2} \, dx &=\int \left (\frac {a^2}{b^3}-\frac {a x^2}{b^2}+\frac {x^4}{b}-\frac {a^3}{b^3 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {a^2 x}{b^3}-\frac {a x^3}{3 b^2}+\frac {x^5}{5 b}-\frac {a^3 \int \frac {1}{a+b x^2} \, dx}{b^3}\\ &=\frac {a^2 x}{b^3}-\frac {a x^3}{3 b^2}+\frac {x^5}{5 b}-\frac {a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 55, normalized size = 1.00 \[ -\frac {a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}}+\frac {a^2 x}{b^3}-\frac {a x^3}{3 b^2}+\frac {x^5}{5 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 126, normalized size = 2.29 \[ \left [\frac {6 \, b^{2} x^{5} - 10 \, a b x^{3} + 15 \, a^{2} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 30 \, a^{2} x}{30 \, b^{3}}, \frac {3 \, b^{2} x^{5} - 5 \, a b x^{3} - 15 \, a^{2} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 15 \, a^{2} x}{15 \, b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 55, normalized size = 1.00 \[ -\frac {a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {3 \, b^{4} x^{5} - 5 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x}{15 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 49, normalized size = 0.89 \[ \frac {x^{5}}{5 b}-\frac {a \,x^{3}}{3 b^{2}}-\frac {a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{3}}+\frac {a^{2} x}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.84, size = 50, normalized size = 0.91 \[ -\frac {a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {3 \, b^{2} x^{5} - 5 \, a b x^{3} + 15 \, a^{2} x}{15 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 43, normalized size = 0.78 \[ \frac {x^5}{5\,b}-\frac {a\,x^3}{3\,b^2}+\frac {a^2\,x}{b^3}-\frac {a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{b^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 95, normalized size = 1.73 \[ \frac {a^{2} x}{b^{3}} - \frac {a x^{3}}{3 b^{2}} + \frac {\sqrt {- \frac {a^{5}}{b^{7}}} \log {\left (x - \frac {b^{3} \sqrt {- \frac {a^{5}}{b^{7}}}}{a^{2}} \right )}}{2} - \frac {\sqrt {- \frac {a^{5}}{b^{7}}} \log {\left (x + \frac {b^{3} \sqrt {- \frac {a^{5}}{b^{7}}}}{a^{2}} \right )}}{2} + \frac {x^{5}}{5 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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