Optimal. Leaf size=63 \[ \frac {(A c+b B) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{3/2} c^{3/2}}-\frac {x (b B-A c)}{2 b c \left (b+c x^2\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1584, 385, 205} \begin {gather*} \frac {(A c+b B) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{3/2} c^{3/2}}-\frac {x (b B-A c)}{2 b c \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 385
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^4 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {A+B x^2}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac {(b B-A c) x}{2 b c \left (b+c x^2\right )}+\frac {(b B+A c) \int \frac {1}{b+c x^2} \, dx}{2 b c}\\ &=-\frac {(b B-A c) x}{2 b c \left (b+c x^2\right )}+\frac {(b B+A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{3/2} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 63, normalized size = 1.00 \begin {gather*} \frac {(A c+b B) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{3/2} c^{3/2}}-\frac {x (b B-A c)}{2 b c \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 182, normalized size = 2.89 \begin {gather*} \left [-\frac {{\left (B b^{2} + A b c + {\left (B b c + A 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^{2} c - A b c^{2}\right )} x}{4 \, {\left (b^{2} c^{3} x^{2} + b^{3} c^{2}\right )}}, \frac {{\left (B b^{2} + A b c + {\left (B b c + A c^{2}\right )} x^{2}\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c} x}{b}\right ) - {\left (B b^{2} c - A b c^{2}\right )} x}{2 \, {\left (b^{2} c^{3} x^{2} + b^{3} c^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 57, normalized size = 0.90 \begin {gather*} \frac {{\left (B b + A c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} b c} - \frac {B b x - A c x}{2 \, {\left (c x^{2} + b\right )} b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 68, normalized size = 1.08 \begin {gather*} \frac {A \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}\, b}+\frac {B \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}\, c}+\frac {\left (A c -b B \right ) x}{2 \left (c \,x^{2}+b \right ) b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 57, normalized size = 0.90 \begin {gather*} -\frac {{\left (B b - A c\right )} x}{2 \, {\left (b c^{2} x^{2} + b^{2} c\right )}} + \frac {{\left (B b + A c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 51, normalized size = 0.81 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )\,\left (A\,c+B\,b\right )}{2\,b^{3/2}\,c^{3/2}}+\frac {x\,\left (A\,c-B\,b\right )}{2\,b\,c\,\left (c\,x^2+b\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.42, size = 112, normalized size = 1.78 \begin {gather*} \frac {x \left (A c - B b\right )}{2 b^{2} c + 2 b c^{2} x^{2}} - \frac {\sqrt {- \frac {1}{b^{3} c^{3}}} \left (A c + B b\right ) \log {\left (- b^{2} c \sqrt {- \frac {1}{b^{3} c^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{b^{3} c^{3}}} \left (A c + B b\right ) \log {\left (b^{2} c \sqrt {- \frac {1}{b^{3} c^{3}}} + x \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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