Optimal. Leaf size=95 \[ \frac {B x}{c^3}-\frac {b (b B-A c) x}{4 c^3 \left (b+c x^2\right )^2}+\frac {(9 b B-5 A c) x}{8 c^3 \left (b+c x^2\right )}-\frac {3 (5 b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 \sqrt {b} c^{7/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1598, 466,
1171, 396, 211} \begin {gather*} -\frac {3 (5 b B-A c) \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 \sqrt {b} c^{7/2}}+\frac {x (9 b B-5 A c)}{8 c^3 \left (b+c x^2\right )}-\frac {b x (b B-A c)}{4 c^3 \left (b+c x^2\right )^2}+\frac {B x}{c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 396
Rule 466
Rule 1171
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^{10} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {x^4 \left (A+B x^2\right )}{\left (b+c x^2\right )^3} \, dx\\ &=-\frac {b (b B-A c) x}{4 c^3 \left (b+c x^2\right )^2}-\frac {\int \frac {-b (b B-A c)+4 c (b B-A c) x^2-4 B c^2 x^4}{\left (b+c x^2\right )^2} \, dx}{4 c^3}\\ &=-\frac {b (b B-A c) x}{4 c^3 \left (b+c x^2\right )^2}+\frac {(9 b B-5 A c) x}{8 c^3 \left (b+c x^2\right )}+\frac {\int \frac {-b (7 b B-3 A c)+8 b B c x^2}{b+c x^2} \, dx}{8 b c^3}\\ &=\frac {B x}{c^3}-\frac {b (b B-A c) x}{4 c^3 \left (b+c x^2\right )^2}+\frac {(9 b B-5 A c) x}{8 c^3 \left (b+c x^2\right )}-\frac {(3 (5 b B-A c)) \int \frac {1}{b+c x^2} \, dx}{8 c^3}\\ &=\frac {B x}{c^3}-\frac {b (b B-A c) x}{4 c^3 \left (b+c x^2\right )^2}+\frac {(9 b B-5 A c) x}{8 c^3 \left (b+c x^2\right )}-\frac {3 (5 b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 \sqrt {b} c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 92, normalized size = 0.97 \begin {gather*} \frac {x \left (15 b^2 B+c^2 x^2 \left (-5 A+8 B x^2\right )+b \left (-3 A c+25 B c x^2\right )\right )}{8 c^3 \left (b+c x^2\right )^2}-\frac {3 (5 b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 \sqrt {b} c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 77, normalized size = 0.81
method | result | size |
default | \(\frac {B x}{c^{3}}+\frac {\frac {\left (-\frac {5}{8} A \,c^{2}+\frac {9}{8} b B c \right ) x^{3}-\frac {b \left (3 A c -7 B b \right ) x}{8}}{\left (c \,x^{2}+b \right )^{2}}+\frac {3 \left (A c -5 B b \right ) \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}}}{c^{3}}\) | \(77\) |
risch | \(\frac {B x}{c^{3}}+\frac {\left (-\frac {5}{8} A \,c^{2}+\frac {9}{8} b B c \right ) x^{3}-\frac {b \left (3 A c -7 B b \right ) x}{8}}{c^{3} \left (c \,x^{2}+b \right )^{2}}-\frac {3 \ln \left (c x +\sqrt {-b c}\right ) A}{16 c^{2} \sqrt {-b c}}+\frac {15 \ln \left (c x +\sqrt {-b c}\right ) B b}{16 c^{3} \sqrt {-b c}}+\frac {3 \ln \left (-c x +\sqrt {-b c}\right ) A}{16 c^{2} \sqrt {-b c}}-\frac {15 \ln \left (-c x +\sqrt {-b c}\right ) B b}{16 c^{3} \sqrt {-b c}}\) | \(147\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 94, normalized size = 0.99 \begin {gather*} \frac {{\left (9 \, B b c - 5 \, A c^{2}\right )} x^{3} + {\left (7 \, B b^{2} - 3 \, A b c\right )} x}{8 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}} + \frac {B x}{c^{3}} - \frac {3 \, {\left (5 \, B b - A c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.07, size = 328, normalized size = 3.45 \begin {gather*} \left [\frac {16 \, B b c^{3} x^{5} + 10 \, {\left (5 \, B b^{2} c^{2} - A b c^{3}\right )} x^{3} + 3 \, {\left ({\left (5 \, B b c^{2} - A c^{3}\right )} x^{4} + 5 \, B b^{3} - A b^{2} c + 2 \, {\left (5 \, B b^{2} c - A b c^{2}\right )} x^{2}\right )} \sqrt {-b c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-b c} x - b}{c x^{2} + b}\right ) + 6 \, {\left (5 \, B b^{3} c - A b^{2} c^{2}\right )} x}{16 \, {\left (b c^{6} x^{4} + 2 \, b^{2} c^{5} x^{2} + b^{3} c^{4}\right )}}, \frac {8 \, B b c^{3} x^{5} + 5 \, {\left (5 \, B b^{2} c^{2} - A b c^{3}\right )} x^{3} - 3 \, {\left ({\left (5 \, B b c^{2} - A c^{3}\right )} x^{4} + 5 \, B b^{3} - A b^{2} c + 2 \, {\left (5 \, B b^{2} c - A b c^{2}\right )} x^{2}\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c} x}{b}\right ) + 3 \, {\left (5 \, B b^{3} c - A b^{2} c^{2}\right )} x}{8 \, {\left (b c^{6} x^{4} + 2 \, b^{2} c^{5} x^{2} + b^{3} c^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 194 vs.
\(2 (92) = 184\).
time = 0.59, size = 194, normalized size = 2.04 \begin {gather*} \frac {B x}{c^{3}} + \frac {3 \sqrt {- \frac {1}{b c^{7}}} \left (- A c + 5 B b\right ) \log {\left (- \frac {3 b c^{3} \sqrt {- \frac {1}{b c^{7}}} \left (- A c + 5 B b\right )}{- 3 A c + 15 B b} + x \right )}}{16} - \frac {3 \sqrt {- \frac {1}{b c^{7}}} \left (- A c + 5 B b\right ) \log {\left (\frac {3 b c^{3} \sqrt {- \frac {1}{b c^{7}}} \left (- A c + 5 B b\right )}{- 3 A c + 15 B b} + x \right )}}{16} + \frac {x^{3} \left (- 5 A c^{2} + 9 B b c\right ) + x \left (- 3 A b c + 7 B b^{2}\right )}{8 b^{2} c^{3} + 16 b c^{4} x^{2} + 8 c^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.99, size = 80, normalized size = 0.84 \begin {gather*} \frac {B x}{c^{3}} - \frac {3 \, {\left (5 \, B b - A c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} c^{3}} + \frac {9 \, B b c x^{3} - 5 \, A c^{2} x^{3} + 7 \, B b^{2} x - 3 \, A b c x}{8 \, {\left (c x^{2} + b\right )}^{2} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 92, normalized size = 0.97 \begin {gather*} \frac {B\,x}{c^3}-\frac {x^3\,\left (\frac {5\,A\,c^2}{8}-\frac {9\,B\,b\,c}{8}\right )-x\,\left (\frac {7\,B\,b^2}{8}-\frac {3\,A\,b\,c}{8}\right )}{b^2\,c^3+2\,b\,c^4\,x^2+c^5\,x^4}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )\,\left (A\,c-5\,B\,b\right )}{8\,\sqrt {b}\,c^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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