Optimal. Leaf size=55 \[ -\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{5/2}}-\frac {x^3}{2 c \left (b+c x^2\right )}+\frac {3 x}{2 c^2} \]
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Rubi [A] time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1584, 288, 321, 205} \[ -\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{5/2}}-\frac {x^3}{2 c \left (b+c x^2\right )}+\frac {3 x}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 288
Rule 321
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^8}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {x^4}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac {x^3}{2 c \left (b+c x^2\right )}+\frac {3 \int \frac {x^2}{b+c x^2} \, dx}{2 c}\\ &=\frac {3 x}{2 c^2}-\frac {x^3}{2 c \left (b+c x^2\right )}-\frac {(3 b) \int \frac {1}{b+c x^2} \, dx}{2 c^2}\\ &=\frac {3 x}{2 c^2}-\frac {x^3}{2 c \left (b+c x^2\right )}-\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 0.93 \[ -\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{5/2}}+\frac {b x}{2 c^2 \left (b+c x^2\right )}+\frac {x}{c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 136, normalized size = 2.47 \[ \left [\frac {4 \, c x^{3} + 3 \, {\left (c x^{2} + b\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} - 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right ) + 6 \, b x}{4 \, {\left (c^{3} x^{2} + b c^{2}\right )}}, \frac {2 \, c x^{3} - 3 \, {\left (c x^{2} + b\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right ) + 3 \, b x}{2 \, {\left (c^{3} x^{2} + b c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 42, normalized size = 0.76 \[ -\frac {3 \, b \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} c^{2}} + \frac {b x}{2 \, {\left (c x^{2} + b\right )} c^{2}} + \frac {x}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 43, normalized size = 0.78 \[ \frac {b x}{2 \left (c \,x^{2}+b \right ) c^{2}}-\frac {3 b \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}\, c^{2}}+\frac {x}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 45, normalized size = 0.82 \[ \frac {b x}{2 \, {\left (c^{3} x^{2} + b c^{2}\right )}} - \frac {3 \, b \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} c^{2}} + \frac {x}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.17, size = 43, normalized size = 0.78 \[ \frac {x}{c^2}+\frac {b\,x}{2\,\left (c^3\,x^2+b\,c^2\right )}-\frac {3\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{2\,c^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 83, normalized size = 1.51 \[ \frac {b x}{2 b c^{2} + 2 c^{3} x^{2}} + \frac {3 \sqrt {- \frac {b}{c^{5}}} \log {\left (- c^{2} \sqrt {- \frac {b}{c^{5}}} + x \right )}}{4} - \frac {3 \sqrt {- \frac {b}{c^{5}}} \log {\left (c^{2} \sqrt {- \frac {b}{c^{5}}} + x \right )}}{4} + \frac {x}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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