Optimal. Leaf size=30 \[ \frac {a^2 x^4}{4}+\frac {1}{3} a b x^6+\frac {b^2 x^8}{8} \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \begin {gather*} \frac {a^2 x^4}{4}+\frac {1}{3} a b x^6+\frac {b^2 x^8}{8} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^3 \left (a+b x^2\right )^2 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (a+b x)^2 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 x^4}{4}+\frac {1}{3} a b x^6+\frac {b^2 x^8}{8}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {a^2 x^4}{4}+\frac {1}{3} a b x^6+\frac {b^2 x^8}{8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^3 \left (a+b x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.84, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{8} x^{8} b^{2} + \frac {1}{3} x^{6} b a + \frac {1}{4} x^{4} a^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{8} \, b^{2} x^{8} + \frac {1}{3} \, a b x^{6} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}
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 \begin {gather*} \frac {1}{8} b^{2} x^{8}+\frac {1}{3} a b \,x^{6}+\frac {1}{4} a^{2} x^{4} \end {gather*}
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 \begin {gather*} \frac {1}{8} \, b^{2} x^{8} + \frac {1}{3} \, a b x^{6} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 24, normalized size = 0.80 \begin {gather*} \frac {a^2\,x^4}{4}+\frac {a\,b\,x^6}{3}+\frac {b^2\,x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.06, size = 24, normalized size = 0.80 \begin {gather*} \frac {a^{2} x^{4}}{4} + \frac {a b x^{6}}{3} + \frac {b^{2} x^{8}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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