Optimal. Leaf size=45 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 \sqrt {b} c^{3/2}}-\frac {x}{2 c \left (b+c x^2\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1584, 288, 205} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 \sqrt {b} c^{3/2}}-\frac {x}{2 c \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 288
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^6}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {x^2}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac {x}{2 c \left (b+c x^2\right )}+\frac {\int \frac {1}{b+c x^2} \, dx}{2 c}\\ &=-\frac {x}{2 c \left (b+c x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 \sqrt {b} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 \sqrt {b} c^{3/2}}-\frac {x}{2 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^6}{\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 = 0.51, size = 120, normalized size = 2.67 \begin {gather*} \left [-\frac {2 \, b c x + {\left (c x^{2} + b\right )} \sqrt {-b c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-b c} x - b}{c x^{2} + b}\right )}{4 \, {\left (b c^{3} x^{2} + b^{2} c^{2}\right )}}, -\frac {b c x - {\left (c x^{2} + b\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c} x}{b}\right )}{2 \, {\left (b c^{3} x^{2} + b^{2} c^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 35, normalized size = 0.78 \begin {gather*} \frac {\arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} c} - \frac {x}{2 \, {\left (c x^{2} + b\right )} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 0.80 \begin {gather*} -\frac {x}{2 \left (c \,x^{2}+b \right ) c}+\frac {\arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}\, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 36, normalized size = 0.80 \begin {gather*} -\frac {x}{2 \, {\left (c^{2} x^{2} + b c\right )}} + \frac {\arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.15, size = 33, normalized size = 0.73 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{2\,\sqrt {b}\,c^{3/2}}-\frac {x}{2\,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 = 0.23, size = 78, normalized size = 1.73 \begin {gather*} - \frac {x}{2 b c + 2 c^{2} x^{2}} - \frac {\sqrt {- \frac {1}{b c^{3}}} \log {\left (- b c \sqrt {- \frac {1}{b c^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{b c^{3}}} \log {\left (b c \sqrt {- \frac {1}{b c^{3}}} + x \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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