Optimal. Leaf size=30 \[ \frac {b^2 x^4}{4}+\frac {1}{3} b c x^6+\frac {c^2 x^8}{8} \]
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Rubi [A] time = 0.03, antiderivative size = 30, 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, 266, 43} \[ \frac {b^2 x^4}{4}+\frac {1}{3} b c x^6+\frac {c^2 x^8}{8} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^2}{x} \, dx &=\int x^3 \left (b+c x^2\right )^2 \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int x (b+c x)^2 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (b^2 x+2 b c x^2+c^2 x^3\right ) \, dx,x,x^2\right )\\ &=\frac {b^2 x^4}{4}+\frac {1}{3} b c x^6+\frac {c^2 x^8}{8}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 30, normalized size = 1.00 \[ \frac {b^2 x^4}{4}+\frac {1}{3} b c x^6+\frac {c^2 x^8}{8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 24, normalized size = 0.80 \[ \frac {1}{8} \, c^{2} x^{8} + \frac {1}{3} \, b c x^{6} + \frac {1}{4} \, b^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 24, normalized size = 0.80 \[ \frac {1}{8} \, c^{2} x^{8} + \frac {1}{3} \, b c x^{6} + \frac {1}{4} \, b^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.83 \[ \frac {1}{8} c^{2} x^{8}+\frac {1}{3} b c \,x^{6}+\frac {1}{4} b^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 24, normalized size = 0.80 \[ \frac {1}{8} \, c^{2} x^{8} + \frac {1}{3} \, b c x^{6} + \frac {1}{4} \, b^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 24, normalized size = 0.80 \[ \frac {b^2\,x^4}{4}+\frac {b\,c\,x^6}{3}+\frac {c^2\,x^8}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 24, normalized size = 0.80 \[ \frac {b^{2} x^{4}}{4} + \frac {b c x^{6}}{3} + \frac {c^{2} x^{8}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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