Optimal. Leaf size=96 \[ \frac {3 (b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{7/2} \sqrt {c}}+\frac {x (3 b B-7 A c)}{8 b^3 \left (b+c x^2\right )}-\frac {A}{b^3 x}+\frac {x (b B-A c)}{4 b^2 \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.12, antiderivative size = 96, 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, 456, 453, 205} \[ \frac {x (3 b B-7 A c)}{8 b^3 \left (b+c x^2\right )}+\frac {x (b B-A c)}{4 b^2 \left (b+c x^2\right )^2}+\frac {3 (b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{7/2} \sqrt {c}}-\frac {A}{b^3 x} \]
Antiderivative was successfully verified.
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Rule 205
Rule 453
Rule 456
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^4 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {A+B x^2}{x^2 \left (b+c x^2\right )^3} \, dx\\ &=\frac {(b B-A c) x}{4 b^2 \left (b+c x^2\right )^2}-\frac {1}{4} \int \frac {-\frac {4 A}{b}-\frac {3 (b B-A c) x^2}{b^2}}{x^2 \left (b+c x^2\right )^2} \, dx\\ &=\frac {(b B-A c) x}{4 b^2 \left (b+c x^2\right )^2}+\frac {(3 b B-7 A c) x}{8 b^3 \left (b+c x^2\right )}+\frac {1}{8} \int \frac {\frac {8 A}{b^2}+\frac {(3 b B-7 A c) x^2}{b^3}}{x^2 \left (b+c x^2\right )} \, dx\\ &=-\frac {A}{b^3 x}+\frac {(b B-A c) x}{4 b^2 \left (b+c x^2\right )^2}+\frac {(3 b B-7 A c) x}{8 b^3 \left (b+c x^2\right )}+\frac {(3 (b B-5 A c)) \int \frac {1}{b+c x^2} \, dx}{8 b^3}\\ &=-\frac {A}{b^3 x}+\frac {(b B-A c) x}{4 b^2 \left (b+c x^2\right )^2}+\frac {(3 b B-7 A c) x}{8 b^3 \left (b+c x^2\right )}+\frac {3 (b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{7/2} \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 96, normalized size = 1.00 \[ \frac {3 (b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{7/2} \sqrt {c}}+\frac {x (3 b B-7 A c)}{8 b^3 \left (b+c x^2\right )}-\frac {A}{b^3 x}+\frac {x (b B-A c)}{4 b^2 \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 324, normalized size = 3.38 \[ \left [-\frac {16 \, A b^{3} c - 6 \, {\left (B b^{2} c^{2} - 5 \, A b c^{3}\right )} x^{4} - 10 \, {\left (B b^{3} c - 5 \, A b^{2} c^{2}\right )} x^{2} - 3 \, {\left ({\left (B b c^{2} - 5 \, A c^{3}\right )} x^{5} + 2 \, {\left (B b^{2} c - 5 \, A b c^{2}\right )} x^{3} + {\left (B b^{3} - 5 \, A b^{2} c\right )} x\right )} \sqrt {-b c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-b c} x - b}{c x^{2} + b}\right )}{16 \, {\left (b^{4} c^{3} x^{5} + 2 \, b^{5} c^{2} x^{3} + b^{6} c x\right )}}, -\frac {8 \, A b^{3} c - 3 \, {\left (B b^{2} c^{2} - 5 \, A b c^{3}\right )} x^{4} - 5 \, {\left (B b^{3} c - 5 \, A b^{2} c^{2}\right )} x^{2} - 3 \, {\left ({\left (B b c^{2} - 5 \, A c^{3}\right )} x^{5} + 2 \, {\left (B b^{2} c - 5 \, A b c^{2}\right )} x^{3} + {\left (B b^{3} - 5 \, A b^{2} c\right )} x\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c} x}{b}\right )}{8 \, {\left (b^{4} c^{3} x^{5} + 2 \, b^{5} c^{2} x^{3} + b^{6} c x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 82, normalized size = 0.85 \[ \frac {3 \, {\left (B b - 5 \, A c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b^{3}} - \frac {A}{b^{3} x} + \frac {3 \, B b c x^{3} - 7 \, A c^{2} x^{3} + 5 \, B b^{2} x - 9 \, A b c x}{8 \, {\left (c x^{2} + b\right )}^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 125, normalized size = 1.30 \[ -\frac {7 A \,c^{2} x^{3}}{8 \left (c \,x^{2}+b \right )^{2} b^{3}}+\frac {3 B c \,x^{3}}{8 \left (c \,x^{2}+b \right )^{2} b^{2}}-\frac {9 A c x}{8 \left (c \,x^{2}+b \right )^{2} b^{2}}+\frac {5 B x}{8 \left (c \,x^{2}+b \right )^{2} b}-\frac {15 A c \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}\, b^{3}}+\frac {3 B \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}\, b^{2}}-\frac {A}{b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 96, normalized size = 1.00 \[ \frac {3 \, {\left (B b c - 5 \, A c^{2}\right )} x^{4} - 8 \, A b^{2} + 5 \, {\left (B b^{2} - 5 \, A b c\right )} x^{2}}{8 \, {\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )}} + \frac {3 \, {\left (B b - 5 \, A c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 113, normalized size = 1.18 \[ -\frac {\frac {A}{b}+\frac {5\,x^2\,\left (5\,A\,c-B\,b\right )}{8\,b^2}+\frac {3\,c\,x^4\,\left (5\,A\,c-B\,b\right )}{8\,b^3}}{b^2\,x+2\,b\,c\,x^3+c^2\,x^5}-\frac {3\,\mathrm {atan}\left (\frac {3\,\sqrt {c}\,x\,\left (5\,A\,c-B\,b\right )}{\sqrt {b}\,\left (15\,A\,c-3\,B\,b\right )}\right )\,\left (5\,A\,c-B\,b\right )}{8\,b^{7/2}\,\sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.73, size = 194, normalized size = 2.02 \[ - \frac {3 \sqrt {- \frac {1}{b^{7} c}} \left (- 5 A c + B b\right ) \log {\left (- \frac {3 b^{4} \sqrt {- \frac {1}{b^{7} c}} \left (- 5 A c + B b\right )}{- 15 A c + 3 B b} + x \right )}}{16} + \frac {3 \sqrt {- \frac {1}{b^{7} c}} \left (- 5 A c + B b\right ) \log {\left (\frac {3 b^{4} \sqrt {- \frac {1}{b^{7} c}} \left (- 5 A c + B b\right )}{- 15 A c + 3 B b} + x \right )}}{16} + \frac {- 8 A b^{2} + x^{4} \left (- 15 A c^{2} + 3 B b c\right ) + x^{2} \left (- 25 A b c + 5 B b^{2}\right )}{8 b^{5} x + 16 b^{4} c x^{3} + 8 b^{3} c^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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