Optimal. Leaf size=89 \[ -\frac {(2 b B-A c) x}{c^3}+\frac {B x^3}{3 c^2}-\frac {b (b B-A c) x}{2 c^3 \left (b+c x^2\right )}+\frac {\sqrt {b} (5 b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{7/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 89, 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 = {1598, 466,
1167, 211} \begin {gather*} \frac {\sqrt {b} (5 b B-3 A c) \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{7/2}}-\frac {b x (b B-A c)}{2 c^3 \left (b+c x^2\right )}-\frac {x (2 b B-A c)}{c^3}+\frac {B x^3}{3 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 466
Rule 1167
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^8 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {x^4 \left (A+B x^2\right )}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac {b (b B-A c) x}{2 c^3 \left (b+c x^2\right )}-\frac {\int \frac {-b (b B-A c)+2 c (b B-A c) x^2-2 B c^2 x^4}{b+c x^2} \, dx}{2 c^3}\\ &=-\frac {b (b B-A c) x}{2 c^3 \left (b+c x^2\right )}-\frac {\int \left (2 (2 b B-A c)-2 B c x^2+\frac {-5 b^2 B+3 A b c}{b+c x^2}\right ) \, dx}{2 c^3}\\ &=-\frac {(2 b B-A c) x}{c^3}+\frac {B x^3}{3 c^2}-\frac {b (b B-A c) x}{2 c^3 \left (b+c x^2\right )}+\frac {(b (5 b B-3 A c)) \int \frac {1}{b+c x^2} \, dx}{2 c^3}\\ &=-\frac {(2 b B-A c) x}{c^3}+\frac {B x^3}{3 c^2}-\frac {b (b B-A c) x}{2 c^3 \left (b+c x^2\right )}+\frac {\sqrt {b} (5 b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 89, normalized size = 1.00 \begin {gather*} \frac {(-2 b B+A c) x}{c^3}+\frac {B x^3}{3 c^2}+\frac {\left (-b^2 B+A b c\right ) x}{2 c^3 \left (b+c x^2\right )}+\frac {\sqrt {b} (5 b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 75, normalized size = 0.84
method | result | size |
default | \(\frac {\frac {1}{3} B c \,x^{3}+A c x -2 b B x}{c^{3}}-\frac {b \left (\frac {\left (-\frac {A c}{2}+\frac {B b}{2}\right ) x}{c \,x^{2}+b}+\frac {\left (3 A c -5 B b \right ) \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}}\right )}{c^{3}}\) | \(75\) |
risch | \(\frac {B \,x^{3}}{3 c^{2}}+\frac {A x}{c^{2}}-\frac {2 b B x}{c^{3}}+\frac {\left (\frac {1}{2} A b c -\frac {1}{2} b^{2} B \right ) x}{c^{3} \left (c \,x^{2}+b \right )}+\frac {3 \sqrt {-b c}\, \ln \left (-\sqrt {-b c}\, x -b \right ) A}{4 c^{3}}-\frac {5 \sqrt {-b c}\, \ln \left (-\sqrt {-b c}\, x -b \right ) B b}{4 c^{4}}-\frac {3 \sqrt {-b c}\, \ln \left (\sqrt {-b c}\, x -b \right ) A}{4 c^{3}}+\frac {5 \sqrt {-b c}\, \ln \left (\sqrt {-b c}\, x -b \right ) B b}{4 c^{4}}\) | \(155\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 85, normalized size = 0.96 \begin {gather*} -\frac {{\left (B b^{2} - A b c\right )} x}{2 \, {\left (c^{4} x^{2} + b c^{3}\right )}} + \frac {{\left (5 \, B b^{2} - 3 \, A b c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} c^{3}} + \frac {B c x^{3} - 3 \, {\left (2 \, B b - A c\right )} x}{3 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.63, size = 240, normalized size = 2.70 \begin {gather*} \left [\frac {4 \, B c^{2} x^{5} - 4 \, {\left (5 \, B b c - 3 \, A c^{2}\right )} x^{3} - 3 \, {\left (5 \, B b^{2} - 3 \, A b c + {\left (5 \, B b c - 3 \, A c^{2}\right )} x^{2}\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} - 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right ) - 6 \, {\left (5 \, B b^{2} - 3 \, A b c\right )} x}{12 \, {\left (c^{4} x^{2} + b c^{3}\right )}}, \frac {2 \, B c^{2} x^{5} - 2 \, {\left (5 \, B b c - 3 \, A c^{2}\right )} x^{3} + 3 \, {\left (5 \, B b^{2} - 3 \, A b c + {\left (5 \, B b c - 3 \, A c^{2}\right )} x^{2}\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right ) - 3 \, {\left (5 \, B b^{2} - 3 \, A b c\right )} x}{6 \, {\left (c^{4} x^{2} + b c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.35, size = 129, normalized size = 1.45 \begin {gather*} \frac {B x^{3}}{3 c^{2}} + x \left (\frac {A}{c^{2}} - \frac {2 B b}{c^{3}}\right ) + \frac {x \left (A b c - B b^{2}\right )}{2 b c^{3} + 2 c^{4} x^{2}} - \frac {\sqrt {- \frac {b}{c^{7}}} \left (- 3 A c + 5 B b\right ) \log {\left (- c^{3} \sqrt {- \frac {b}{c^{7}}} + x \right )}}{4} + \frac {\sqrt {- \frac {b}{c^{7}}} \left (- 3 A c + 5 B b\right ) \log {\left (c^{3} \sqrt {- \frac {b}{c^{7}}} + x \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.70, size = 88, normalized size = 0.99 \begin {gather*} \frac {{\left (5 \, B b^{2} - 3 \, A b c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} c^{3}} - \frac {B b^{2} x - A b c x}{2 \, {\left (c x^{2} + b\right )} c^{3}} + \frac {B c^{4} x^{3} - 6 \, B b c^{3} x + 3 \, A c^{4} x}{3 \, c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 104, normalized size = 1.17 \begin {gather*} x\,\left (\frac {A}{c^2}-\frac {2\,B\,b}{c^3}\right )-\frac {x\,\left (\frac {B\,b^2}{2}-\frac {A\,b\,c}{2}\right )}{c^4\,x^2+b\,c^3}+\frac {B\,x^3}{3\,c^2}+\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c}\,x\,\left (3\,A\,c-5\,B\,b\right )}{5\,B\,b^2-3\,A\,b\,c}\right )\,\left (3\,A\,c-5\,B\,b\right )}{2\,c^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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