Optimal. Leaf size=23 \[ -\frac {b^2}{3 x^3}-\frac {2 b c}{x}+c^2 x \]
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Rubi [A] time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1584, 270} \[ -\frac {b^2}{3 x^3}-\frac {2 b c}{x}+c^2 x \]
Antiderivative was successfully verified.
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Rule 270
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^2}{x^8} \, dx &=\int \frac {\left (b+c x^2\right )^2}{x^4} \, dx\\ &=\int \left (c^2+\frac {b^2}{x^4}+\frac {2 b c}{x^2}\right ) \, dx\\ &=-\frac {b^2}{3 x^3}-\frac {2 b c}{x}+c^2 x\\ \end {align*}
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Mathematica [A] time = 0.00, size = 23, normalized size = 1.00 \[ -\frac {b^2}{3 x^3}-\frac {2 b c}{x}+c^2 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 26, normalized size = 1.13 \[ \frac {3 \, c^{2} x^{4} - 6 \, b c x^{2} - b^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 22, normalized size = 0.96 \[ c^{2} x - \frac {6 \, b c x^{2} + b^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 22, normalized size = 0.96 \[ c^{2} x -\frac {2 b c}{x}-\frac {b^{2}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 22, normalized size = 0.96 \[ c^{2} x - \frac {6 \, b c x^{2} + b^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 24, normalized size = 1.04 \[ c^2\,x-\frac {\frac {b^2}{3}+2\,c\,b\,x^2}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 22, normalized size = 0.96 \[ c^{2} x + \frac {- b^{2} - 6 b c x^{2}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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