Optimal. Leaf size=34 \[ -\frac {b^3}{x}+3 b^2 c x+b c^2 x^3+\frac {c^3 x^5}{5} \]
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Rubi [A] time = 0.02, antiderivative size = 34, 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} \[ 3 b^2 c x-\frac {b^3}{x}+b c^2 x^3+\frac {c^3 x^5}{5} \]
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 )^3}{x^8} \, dx &=\int \frac {\left (b+c x^2\right )^3}{x^2} \, dx\\ &=\int \left (3 b^2 c+\frac {b^3}{x^2}+3 b c^2 x^2+c^3 x^4\right ) \, dx\\ &=-\frac {b^3}{x}+3 b^2 c x+b c^2 x^3+\frac {c^3 x^5}{5}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 34, normalized size = 1.00 \[ -\frac {b^3}{x}+3 b^2 c x+b c^2 x^3+\frac {c^3 x^5}{5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 36, normalized size = 1.06 \[ \frac {c^{3} x^{6} + 5 \, b c^{2} x^{4} + 15 \, b^{2} c x^{2} - 5 \, b^{3}}{5 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 32, normalized size = 0.94 \[ \frac {1}{5} \, c^{3} x^{5} + b c^{2} x^{3} + 3 \, b^{2} c x - \frac {b^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 33, normalized size = 0.97 \[ \frac {c^{3} x^{5}}{5}+b \,c^{2} x^{3}+3 b^{2} c x -\frac {b^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 32, normalized size = 0.94 \[ \frac {1}{5} \, c^{3} x^{5} + b c^{2} x^{3} + 3 \, b^{2} c x - \frac {b^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 32, normalized size = 0.94 \[ \frac {c^3\,x^5}{5}-\frac {b^3}{x}+b\,c^2\,x^3+3\,b^2\,c\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 29, normalized size = 0.85 \[ - \frac {b^{3}}{x} + 3 b^{2} c x + b c^{2} x^{3} + \frac {c^{3} x^{5}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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