Optimal. Leaf size=43 \[ \frac {a^3 x^6}{6}+\frac {3}{8} a^2 b x^8+\frac {3}{10} a b^2 x^{10}+\frac {b^3 x^{12}}{12} \]
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Rubi [A] time = 0.03, antiderivative size = 43, 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 {3}{8} a^2 b x^8+\frac {a^3 x^6}{6}+\frac {3}{10} a b^2 x^{10}+\frac {b^3 x^{12}}{12} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^5 \left (a+b x^2\right )^3 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 (a+b x)^3 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a^3 x^2+3 a^2 b x^3+3 a b^2 x^4+b^3 x^5\right ) \, dx,x,x^2\right )\\ &=\frac {a^3 x^6}{6}+\frac {3}{8} a^2 b x^8+\frac {3}{10} a b^2 x^{10}+\frac {b^3 x^{12}}{12}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 43, normalized size = 1.00 \begin {gather*} \frac {a^3 x^6}{6}+\frac {3}{8} a^2 b x^8+\frac {3}{10} a b^2 x^{10}+\frac {b^3 x^{12}}{12} \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^5 \left (a+b x^2\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.47, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{12} x^{12} b^{3} + \frac {3}{10} x^{10} b^{2} a + \frac {3}{8} x^{8} b a^{2} + \frac {1}{6} x^{6} a^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{12} \, b^{3} x^{12} + \frac {3}{10} \, a b^{2} x^{10} + \frac {3}{8} \, a^{2} b x^{8} + \frac {1}{6} \, a^{3} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 0.84 \begin {gather*} \frac {1}{12} b^{3} x^{12}+\frac {3}{10} a \,b^{2} x^{10}+\frac {3}{8} a^{2} b \,x^{8}+\frac {1}{6} a^{3} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{12} \, b^{3} x^{12} + \frac {3}{10} \, a b^{2} x^{10} + \frac {3}{8} \, a^{2} b x^{8} + \frac {1}{6} \, a^{3} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 35, normalized size = 0.81 \begin {gather*} \frac {a^3\,x^6}{6}+\frac {3\,a^2\,b\,x^8}{8}+\frac {3\,a\,b^2\,x^{10}}{10}+\frac {b^3\,x^{12}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 39, normalized size = 0.91 \begin {gather*} \frac {a^{3} x^{6}}{6} + \frac {3 a^{2} b x^{8}}{8} + \frac {3 a b^{2} x^{10}}{10} + \frac {b^{3} x^{12}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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