Optimal. Leaf size=118 \[ -\frac {(3 b B-A c) x}{c^4}+\frac {B x^3}{3 c^3}+\frac {b^2 (b B-A c) x}{4 c^4 \left (b+c x^2\right )^2}-\frac {b (13 b B-9 A c) x}{8 c^4 \left (b+c x^2\right )}+\frac {5 \sqrt {b} (7 b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 c^{9/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1598, 466,
1828, 1167, 211} \begin {gather*} \frac {5 \sqrt {b} (7 b B-3 A c) \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 c^{9/2}}+\frac {b^2 x (b B-A c)}{4 c^4 \left (b+c x^2\right )^2}-\frac {b x (13 b B-9 A c)}{8 c^4 \left (b+c x^2\right )}-\frac {x (3 b B-A c)}{c^4}+\frac {B x^3}{3 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 466
Rule 1167
Rule 1598
Rule 1828
Rubi steps
\begin {align*} \int \frac {x^{12} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {x^6 \left (A+B x^2\right )}{\left (b+c x^2\right )^3} \, dx\\ &=\frac {b^2 (b B-A c) x}{4 c^4 \left (b+c x^2\right )^2}-\frac {\int \frac {b^2 (b B-A c)-4 b c (b B-A c) x^2+4 c^2 (b B-A c) x^4-4 B c^3 x^6}{\left (b+c x^2\right )^2} \, dx}{4 c^4}\\ &=\frac {b^2 (b B-A c) x}{4 c^4 \left (b+c x^2\right )^2}-\frac {b (13 b B-9 A c) x}{8 c^4 \left (b+c x^2\right )}+\frac {\int \frac {b^2 (11 b B-7 A c)-8 b c (2 b B-A c) x^2+8 b B c^2 x^4}{b+c x^2} \, dx}{8 b c^4}\\ &=\frac {b^2 (b B-A c) x}{4 c^4 \left (b+c x^2\right )^2}-\frac {b (13 b B-9 A c) x}{8 c^4 \left (b+c x^2\right )}+\frac {\int \left (-8 b (3 b B-A c)+8 b B c x^2+\frac {5 \left (7 b^3 B-3 A b^2 c\right )}{b+c x^2}\right ) \, dx}{8 b c^4}\\ &=-\frac {(3 b B-A c) x}{c^4}+\frac {B x^3}{3 c^3}+\frac {b^2 (b B-A c) x}{4 c^4 \left (b+c x^2\right )^2}-\frac {b (13 b B-9 A c) x}{8 c^4 \left (b+c x^2\right )}+\frac {(5 b (7 b B-3 A c)) \int \frac {1}{b+c x^2} \, dx}{8 c^4}\\ &=-\frac {(3 b B-A c) x}{c^4}+\frac {B x^3}{3 c^3}+\frac {b^2 (b B-A c) x}{4 c^4 \left (b+c x^2\right )^2}-\frac {b (13 b B-9 A c) x}{8 c^4 \left (b+c x^2\right )}+\frac {5 \sqrt {b} (7 b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 c^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 113, normalized size = 0.96 \begin {gather*} \frac {-105 b^3 B x+b c^2 x^3 \left (75 A-56 B x^2\right )+5 b^2 c x \left (9 A-35 B x^2\right )+8 c^3 x^5 \left (3 A+B x^2\right )}{24 c^4 \left (b+c x^2\right )^2}+\frac {5 \sqrt {b} (7 b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 95, normalized size = 0.81
method | result | size |
default | \(\frac {\frac {1}{3} B c \,x^{3}+A c x -3 b B x}{c^{4}}-\frac {b \left (\frac {\left (-\frac {9}{8} A \,c^{2}+\frac {13}{8} b B c \right ) x^{3}-\frac {b \left (7 A c -11 B b \right ) x}{8}}{\left (c \,x^{2}+b \right )^{2}}+\frac {5 \left (3 A c -7 B b \right ) \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}}\right )}{c^{4}}\) | \(95\) |
risch | \(\frac {B \,x^{3}}{3 c^{3}}+\frac {A x}{c^{3}}-\frac {3 b B x}{c^{4}}+\frac {\left (\frac {9}{8} A b \,c^{2}-\frac {13}{8} B \,b^{2} c \right ) x^{3}+\frac {b^{2} \left (7 A c -11 B b \right ) x}{8}}{c^{4} \left (c \,x^{2}+b \right )^{2}}+\frac {15 \sqrt {-b c}\, \ln \left (-\sqrt {-b c}\, x -b \right ) A}{16 c^{4}}-\frac {35 \sqrt {-b c}\, \ln \left (-\sqrt {-b c}\, x -b \right ) B b}{16 c^{5}}-\frac {15 \sqrt {-b c}\, \ln \left (\sqrt {-b c}\, x -b \right ) A}{16 c^{4}}+\frac {35 \sqrt {-b c}\, \ln \left (\sqrt {-b c}\, x -b \right ) B b}{16 c^{5}}\) | \(177\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 120, normalized size = 1.02 \begin {gather*} -\frac {{\left (13 \, B b^{2} c - 9 \, A b c^{2}\right )} x^{3} + {\left (11 \, B b^{3} - 7 \, A b^{2} c\right )} x}{8 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}} + \frac {5 \, {\left (7 \, B b^{2} - 3 \, A b c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} c^{4}} + \frac {B c x^{3} - 3 \, {\left (3 \, B b - A c\right )} x}{3 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.38, size = 358, normalized size = 3.03 \begin {gather*} \left [\frac {16 \, B c^{3} x^{7} - 16 \, {\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{5} - 50 \, {\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x^{3} - 15 \, {\left ({\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{4} + 7 \, B b^{3} - 3 \, A b^{2} c + 2 \, {\left (7 \, B b^{2} c - 3 \, A b 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 ) - 30 \, {\left (7 \, B b^{3} - 3 \, A b^{2} c\right )} x}{48 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}}, \frac {8 \, B c^{3} x^{7} - 8 \, {\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{5} - 25 \, {\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x^{3} + 15 \, {\left ({\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{4} + 7 \, B b^{3} - 3 \, A b^{2} c + 2 \, {\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x^{2}\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right ) - 15 \, {\left (7 \, B b^{3} - 3 \, A b^{2} c\right )} x}{24 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.71, size = 214, normalized size = 1.81 \begin {gather*} \frac {B x^{3}}{3 c^{3}} + x \left (\frac {A}{c^{3}} - \frac {3 B b}{c^{4}}\right ) - \frac {5 \sqrt {- \frac {b}{c^{9}}} \left (- 3 A c + 7 B b\right ) \log {\left (- \frac {5 c^{4} \sqrt {- \frac {b}{c^{9}}} \left (- 3 A c + 7 B b\right )}{- 15 A c + 35 B b} + x \right )}}{16} + \frac {5 \sqrt {- \frac {b}{c^{9}}} \left (- 3 A c + 7 B b\right ) \log {\left (\frac {5 c^{4} \sqrt {- \frac {b}{c^{9}}} \left (- 3 A c + 7 B b\right )}{- 15 A c + 35 B b} + x \right )}}{16} + \frac {x^{3} \cdot \left (9 A b c^{2} - 13 B b^{2} c\right ) + x \left (7 A b^{2} c - 11 B b^{3}\right )}{8 b^{2} c^{4} + 16 b c^{5} x^{2} + 8 c^{6} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.70, size = 111, normalized size = 0.94 \begin {gather*} \frac {5 \, {\left (7 \, B b^{2} - 3 \, A b c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} c^{4}} - \frac {13 \, B b^{2} c x^{3} - 9 \, A b c^{2} x^{3} + 11 \, B b^{3} x - 7 \, A b^{2} c x}{8 \, {\left (c x^{2} + b\right )}^{2} c^{4}} + \frac {B c^{6} x^{3} - 9 \, B b c^{5} x + 3 \, A c^{6} x}{3 \, c^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 138, normalized size = 1.17 \begin {gather*} \frac {x^3\,\left (\frac {9\,A\,b\,c^2}{8}-\frac {13\,B\,b^2\,c}{8}\right )-x\,\left (\frac {11\,B\,b^3}{8}-\frac {7\,A\,b^2\,c}{8}\right )}{b^2\,c^4+2\,b\,c^5\,x^2+c^6\,x^4}+x\,\left (\frac {A}{c^3}-\frac {3\,B\,b}{c^4}\right )+\frac {B\,x^3}{3\,c^3}+\frac {5\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c}\,x\,\left (3\,A\,c-7\,B\,b\right )}{7\,B\,b^2-3\,A\,b\,c}\right )\,\left (3\,A\,c-7\,B\,b\right )}{8\,c^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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