Optimal. Leaf size=92 \[ \frac {(3 A c+b B) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{5/2} c^{3/2}}+\frac {x (3 A c+b B)}{8 b^2 c \left (b+c x^2\right )}-\frac {x (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.05, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1584, 385, 199, 205} \[ \frac {(3 A c+b B) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{5/2} c^{3/2}}+\frac {x (3 A c+b B)}{8 b^2 c \left (b+c x^2\right )}-\frac {x (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 385
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^6 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {A+B x^2}{\left (b+c x^2\right )^3} \, dx\\ &=-\frac {(b B-A c) x}{4 b c \left (b+c x^2\right )^2}+\frac {(b B+3 A c) \int \frac {1}{\left (b+c x^2\right )^2} \, dx}{4 b c}\\ &=-\frac {(b B-A c) x}{4 b c \left (b+c x^2\right )^2}+\frac {(b B+3 A c) x}{8 b^2 c \left (b+c x^2\right )}+\frac {(b B+3 A c) \int \frac {1}{b+c x^2} \, dx}{8 b^2 c}\\ &=-\frac {(b B-A c) x}{4 b c \left (b+c x^2\right )^2}+\frac {(b B+3 A c) x}{8 b^2 c \left (b+c x^2\right )}+\frac {(b B+3 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{5/2} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 84, normalized size = 0.91 \[ \frac {(3 A c+b B) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{5/2} c^{3/2}}+\frac {x \left (b c \left (5 A+B x^2\right )+3 A c^2 x^2+b^2 (-B)\right )}{8 b^2 c \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 300, normalized size = 3.26 \[ \left [\frac {2 \, {\left (B b^{2} c^{2} + 3 \, A b c^{3}\right )} x^{3} - {\left ({\left (B b c^{2} + 3 \, A c^{3}\right )} x^{4} + B b^{3} + 3 \, A b^{2} c + 2 \, {\left (B b^{2} c + 3 \, A b c^{2}\right )} x^{2}\right )} \sqrt {-b c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-b c} x - b}{c x^{2} + b}\right ) - 2 \, {\left (B b^{3} c - 5 \, A b^{2} c^{2}\right )} x}{16 \, {\left (b^{3} c^{4} x^{4} + 2 \, b^{4} c^{3} x^{2} + b^{5} c^{2}\right )}}, \frac {{\left (B b^{2} c^{2} + 3 \, A b c^{3}\right )} x^{3} + {\left ({\left (B b c^{2} + 3 \, A c^{3}\right )} x^{4} + B b^{3} + 3 \, A b^{2} c + 2 \, {\left (B b^{2} c + 3 \, A b c^{2}\right )} x^{2}\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c} x}{b}\right ) - {\left (B b^{3} c - 5 \, A b^{2} c^{2}\right )} x}{8 \, {\left (b^{3} c^{4} x^{4} + 2 \, b^{4} c^{3} x^{2} + b^{5} c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 78, normalized size = 0.85 \[ \frac {{\left (B b + 3 \, A c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b^{2} c} + \frac {B b c x^{3} + 3 \, A c^{2} x^{3} - B b^{2} x + 5 \, A b c x}{8 \, {\left (c x^{2} + b\right )}^{2} b^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 90, normalized size = 0.98 \[ \frac {3 A \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}\, b^{2}}+\frac {B \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}\, b c}+\frac {\frac {\left (3 A c +b B \right ) x^{3}}{8 b^{2}}+\frac {\left (5 A c -b B \right ) x}{8 b c}}{\left (c \,x^{2}+b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.95, size = 92, normalized size = 1.00 \[ \frac {{\left (B b c + 3 \, A c^{2}\right )} x^{3} - {\left (B b^{2} - 5 \, A b c\right )} x}{8 \, {\left (b^{2} c^{3} x^{4} + 2 \, b^{3} c^{2} x^{2} + b^{4} c\right )}} + \frac {{\left (B b + 3 \, A c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 82, normalized size = 0.89 \[ \frac {\frac {x^3\,\left (3\,A\,c+B\,b\right )}{8\,b^2}+\frac {x\,\left (5\,A\,c-B\,b\right )}{8\,b\,c}}{b^2+2\,b\,c\,x^2+c^2\,x^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )\,\left (3\,A\,c+B\,b\right )}{8\,b^{5/2}\,c^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.60, size = 150, normalized size = 1.63 \[ - \frac {\sqrt {- \frac {1}{b^{5} c^{3}}} \left (3 A c + B b\right ) \log {\left (- b^{3} c \sqrt {- \frac {1}{b^{5} c^{3}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{b^{5} c^{3}}} \left (3 A c + B b\right ) \log {\left (b^{3} c \sqrt {- \frac {1}{b^{5} c^{3}}} + x \right )}}{16} + \frac {x^{3} \left (3 A c^{2} + B b c\right ) + x \left (5 A b c - B b^{2}\right )}{8 b^{4} c + 16 b^{3} c^{2} x^{2} + 8 b^{2} c^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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