Optimal. Leaf size=64 \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 \sqrt {b} c^{5/2}}-\frac {3 x}{8 c^2 \left (b+c x^2\right )}-\frac {x^3}{4 c \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1584, 288, 205} \[ -\frac {3 x}{8 c^2 \left (b+c x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 \sqrt {b} c^{5/2}}-\frac {x^3}{4 c \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 288
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{10}}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {x^4}{\left (b+c x^2\right )^3} \, dx\\ &=-\frac {x^3}{4 c \left (b+c x^2\right )^2}+\frac {3 \int \frac {x^2}{\left (b+c x^2\right )^2} \, dx}{4 c}\\ &=-\frac {x^3}{4 c \left (b+c x^2\right )^2}-\frac {3 x}{8 c^2 \left (b+c x^2\right )}+\frac {3 \int \frac {1}{b+c x^2} \, dx}{8 c^2}\\ &=-\frac {x^3}{4 c \left (b+c x^2\right )^2}-\frac {3 x}{8 c^2 \left (b+c x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 \sqrt {b} c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 55, normalized size = 0.86 \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 \sqrt {b} c^{5/2}}-\frac {3 b x+5 c x^3}{8 c^2 \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 188, normalized size = 2.94 \[ \left [-\frac {10 \, b c^{2} x^{3} + 6 \, b^{2} c x + 3 \, {\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt {-b c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-b c} x - b}{c x^{2} + b}\right )}{16 \, {\left (b c^{5} x^{4} + 2 \, b^{2} c^{4} x^{2} + b^{3} c^{3}\right )}}, -\frac {5 \, b c^{2} x^{3} + 3 \, b^{2} c x - 3 \, {\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c} x}{b}\right )}{8 \, {\left (b c^{5} x^{4} + 2 \, b^{2} c^{4} x^{2} + b^{3} c^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 45, normalized size = 0.70 \[ \frac {3 \, \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} c^{2}} - \frac {5 \, c x^{3} + 3 \, b x}{8 \, {\left (c x^{2} + b\right )}^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 0.73 \[ \frac {3 \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}\, c^{2}}+\frac {-\frac {5 x^{3}}{8 c}-\frac {3 b x}{8 c^{2}}}{\left (c \,x^{2}+b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.91, size = 59, normalized size = 0.92 \[ -\frac {5 \, c x^{3} + 3 \, b x}{8 \, {\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )}} + \frac {3 \, \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.23, size = 56, normalized size = 0.88 \[ \frac {3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{8\,\sqrt {b}\,c^{5/2}}-\frac {\frac {5\,x^3}{8\,c}+\frac {3\,b\,x}{8\,c^2}}{b^2+2\,b\,c\,x^2+c^2\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 110, normalized size = 1.72 \[ - \frac {3 \sqrt {- \frac {1}{b c^{5}}} \log {\left (- b c^{2} \sqrt {- \frac {1}{b c^{5}}} + x \right )}}{16} + \frac {3 \sqrt {- \frac {1}{b c^{5}}} \log {\left (b c^{2} \sqrt {- \frac {1}{b c^{5}}} + x \right )}}{16} + \frac {- 3 b x - 5 c x^{3}}{8 b^{2} c^{2} + 16 b c^{3} x^{2} + 8 c^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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