Optimal. Leaf size=98 \[ \frac {b^{5/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{9/2}}-\frac {b^2 x (b B-A c)}{c^4}+\frac {b x^3 (b B-A c)}{3 c^3}-\frac {x^5 (b B-A c)}{5 c^2}+\frac {B x^7}{7 c} \]
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Rubi [A] time = 0.08, antiderivative size = 98, 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 = {1584, 459, 302, 205} \begin {gather*} -\frac {b^2 x (b B-A c)}{c^4}+\frac {b^{5/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{9/2}}-\frac {x^5 (b B-A c)}{5 c^2}+\frac {b x^3 (b B-A c)}{3 c^3}+\frac {B x^7}{7 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 302
Rule 459
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^8 \left (A+B x^2\right )}{b x^2+c x^4} \, dx &=\int \frac {x^6 \left (A+B x^2\right )}{b+c x^2} \, dx\\ &=\frac {B x^7}{7 c}-\frac {(7 b B-7 A c) \int \frac {x^6}{b+c x^2} \, dx}{7 c}\\ &=\frac {B x^7}{7 c}-\frac {(7 b B-7 A c) \int \left (\frac {b^2}{c^3}-\frac {b x^2}{c^2}+\frac {x^4}{c}-\frac {b^3}{c^3 \left (b+c x^2\right )}\right ) \, dx}{7 c}\\ &=-\frac {b^2 (b B-A c) x}{c^4}+\frac {b (b B-A c) x^3}{3 c^3}-\frac {(b B-A c) x^5}{5 c^2}+\frac {B x^7}{7 c}+\frac {\left (b^3 (b B-A c)\right ) \int \frac {1}{b+c x^2} \, dx}{c^4}\\ &=-\frac {b^2 (b B-A c) x}{c^4}+\frac {b (b B-A c) x^3}{3 c^3}-\frac {(b B-A c) x^5}{5 c^2}+\frac {B x^7}{7 c}+\frac {b^{5/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 98, normalized size = 1.00 \begin {gather*} \frac {b^{5/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{9/2}}-\frac {b^2 x (b B-A c)}{c^4}+\frac {b x^3 (b B-A c)}{3 c^3}+\frac {x^5 (A c-b B)}{5 c^2}+\frac {B x^7}{7 c} \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 \left (A+B x^2\right )}{b x^2+c x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 228, normalized size = 2.33 \begin {gather*} \left [\frac {30 \, B c^{3} x^{7} - 42 \, {\left (B b c^{2} - A c^{3}\right )} x^{5} + 70 \, {\left (B b^{2} c - A b c^{2}\right )} x^{3} - 105 \, {\left (B b^{3} - A b^{2} c\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} - 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right ) - 210 \, {\left (B b^{3} - A b^{2} c\right )} x}{210 \, c^{4}}, \frac {15 \, B c^{3} x^{7} - 21 \, {\left (B b c^{2} - A c^{3}\right )} x^{5} + 35 \, {\left (B b^{2} c - A b c^{2}\right )} x^{3} + 105 \, {\left (B b^{3} - A b^{2} c\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right ) - 105 \, {\left (B b^{3} - A b^{2} c\right )} x}{105 \, c^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 108, normalized size = 1.10 \begin {gather*} \frac {{\left (B b^{4} - A b^{3} c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c^{4}} + \frac {15 \, B c^{6} x^{7} - 21 \, B b c^{5} x^{5} + 21 \, A c^{6} x^{5} + 35 \, B b^{2} c^{4} x^{3} - 35 \, A b c^{5} x^{3} - 105 \, B b^{3} c^{3} x + 105 \, A b^{2} c^{4} x}{105 \, c^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 116, normalized size = 1.18 \begin {gather*} \frac {B \,x^{7}}{7 c}+\frac {A \,x^{5}}{5 c}-\frac {B b \,x^{5}}{5 c^{2}}-\frac {A b \,x^{3}}{3 c^{2}}+\frac {B \,b^{2} x^{3}}{3 c^{3}}-\frac {A \,b^{3} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{3}}+\frac {B \,b^{4} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{4}}+\frac {A \,b^{2} x}{c^{3}}-\frac {B \,b^{3} x}{c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 100, normalized size = 1.02 \begin {gather*} \frac {{\left (B b^{4} - A b^{3} c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c^{4}} + \frac {15 \, B c^{3} x^{7} - 21 \, {\left (B b c^{2} - A c^{3}\right )} x^{5} + 35 \, {\left (B b^{2} c - A b c^{2}\right )} x^{3} - 105 \, {\left (B b^{3} - A b^{2} c\right )} x}{105 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 118, normalized size = 1.20 \begin {gather*} x^5\,\left (\frac {A}{5\,c}-\frac {B\,b}{5\,c^2}\right )+\frac {B\,x^7}{7\,c}+\frac {b^{5/2}\,\mathrm {atan}\left (\frac {b^{5/2}\,\sqrt {c}\,x\,\left (A\,c-B\,b\right )}{B\,b^4-A\,b^3\,c}\right )\,\left (A\,c-B\,b\right )}{c^{9/2}}-\frac {b\,x^3\,\left (\frac {A}{c}-\frac {B\,b}{c^2}\right )}{3\,c}+\frac {b^2\,x\,\left (\frac {A}{c}-\frac {B\,b}{c^2}\right )}{c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.41, size = 180, normalized size = 1.84 \begin {gather*} \frac {B x^{7}}{7 c} + x^{5} \left (\frac {A}{5 c} - \frac {B b}{5 c^{2}}\right ) + x^{3} \left (- \frac {A b}{3 c^{2}} + \frac {B b^{2}}{3 c^{3}}\right ) + x \left (\frac {A b^{2}}{c^{3}} - \frac {B b^{3}}{c^{4}}\right ) - \frac {\sqrt {- \frac {b^{5}}{c^{9}}} \left (- A c + B b\right ) \log {\left (- \frac {c^{4} \sqrt {- \frac {b^{5}}{c^{9}}} \left (- A c + B b\right )}{- A b^{2} c + B b^{3}} + x \right )}}{2} + \frac {\sqrt {- \frac {b^{5}}{c^{9}}} \left (- A c + B b\right ) \log {\left (\frac {c^{4} \sqrt {- \frac {b^{5}}{c^{9}}} \left (- A c + B b\right )}{- A b^{2} c + B b^{3}} + x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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