Optimal. Leaf size=53 \[ \frac {a^2 \left (a+b x^3\right )^9}{27 b^3}-\frac {a \left (a+b x^3\right )^{10}}{15 b^3}+\frac {\left (a+b x^3\right )^{11}}{33 b^3} \]
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Rubi [A]
time = 0.06, antiderivative size = 53, 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 = {272, 45}
\begin {gather*} \frac {a^2 \left (a+b x^3\right )^9}{27 b^3}+\frac {\left (a+b x^3\right )^{11}}{33 b^3}-\frac {a \left (a+b x^3\right )^{10}}{15 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^8 \left (a+b x^3\right )^8 \, dx &=\frac {1}{3} \text {Subst}\left (\int x^2 (a+b x)^8 \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a^2 (a+b x)^8}{b^2}-\frac {2 a (a+b x)^9}{b^2}+\frac {(a+b x)^{10}}{b^2}\right ) \, dx,x,x^3\right )\\ &=\frac {a^2 \left (a+b x^3\right )^9}{27 b^3}-\frac {a \left (a+b x^3\right )^{10}}{15 b^3}+\frac {\left (a+b x^3\right )^{11}}{33 b^3}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(53)=106\).
time = 0.00, size = 108, normalized size = 2.04 \begin {gather*} \frac {a^8 x^9}{9}+\frac {2}{3} a^7 b x^{12}+\frac {28}{15} a^6 b^2 x^{15}+\frac {28}{9} a^5 b^3 x^{18}+\frac {10}{3} a^4 b^4 x^{21}+\frac {7}{3} a^3 b^5 x^{24}+\frac {28}{27} a^2 b^6 x^{27}+\frac {4}{15} a b^7 x^{30}+\frac {b^8 x^{33}}{33} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 91, normalized size = 1.72
method | result | size |
gosper | \(\frac {4}{15} a \,b^{7} x^{30}+\frac {1}{33} b^{8} x^{33}+\frac {28}{27} a^{2} b^{6} x^{27}+\frac {7}{3} a^{3} b^{5} x^{24}+\frac {28}{9} a^{5} b^{3} x^{18}+\frac {10}{3} a^{4} b^{4} x^{21}+\frac {1}{9} a^{8} x^{9}+\frac {2}{3} a^{7} b \,x^{12}+\frac {28}{15} a^{6} b^{2} x^{15}\) | \(91\) |
default | \(\frac {4}{15} a \,b^{7} x^{30}+\frac {1}{33} b^{8} x^{33}+\frac {28}{27} a^{2} b^{6} x^{27}+\frac {7}{3} a^{3} b^{5} x^{24}+\frac {28}{9} a^{5} b^{3} x^{18}+\frac {10}{3} a^{4} b^{4} x^{21}+\frac {1}{9} a^{8} x^{9}+\frac {2}{3} a^{7} b \,x^{12}+\frac {28}{15} a^{6} b^{2} x^{15}\) | \(91\) |
norman | \(\frac {4}{15} a \,b^{7} x^{30}+\frac {1}{33} b^{8} x^{33}+\frac {28}{27} a^{2} b^{6} x^{27}+\frac {7}{3} a^{3} b^{5} x^{24}+\frac {28}{9} a^{5} b^{3} x^{18}+\frac {10}{3} a^{4} b^{4} x^{21}+\frac {1}{9} a^{8} x^{9}+\frac {2}{3} a^{7} b \,x^{12}+\frac {28}{15} a^{6} b^{2} x^{15}\) | \(91\) |
risch | \(\frac {4}{15} a \,b^{7} x^{30}+\frac {1}{33} b^{8} x^{33}+\frac {28}{27} a^{2} b^{6} x^{27}+\frac {7}{3} a^{3} b^{5} x^{24}+\frac {28}{9} a^{5} b^{3} x^{18}+\frac {10}{3} a^{4} b^{4} x^{21}+\frac {1}{9} a^{8} x^{9}+\frac {2}{3} a^{7} b \,x^{12}+\frac {28}{15} a^{6} b^{2} x^{15}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 90, normalized size = 1.70 \begin {gather*} \frac {1}{33} \, b^{8} x^{33} + \frac {4}{15} \, a b^{7} x^{30} + \frac {28}{27} \, a^{2} b^{6} x^{27} + \frac {7}{3} \, a^{3} b^{5} x^{24} + \frac {10}{3} \, a^{4} b^{4} x^{21} + \frac {28}{9} \, a^{5} b^{3} x^{18} + \frac {28}{15} \, a^{6} b^{2} x^{15} + \frac {2}{3} \, a^{7} b x^{12} + \frac {1}{9} \, a^{8} x^{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 90, normalized size = 1.70 \begin {gather*} \frac {1}{33} \, b^{8} x^{33} + \frac {4}{15} \, a b^{7} x^{30} + \frac {28}{27} \, a^{2} b^{6} x^{27} + \frac {7}{3} \, a^{3} b^{5} x^{24} + \frac {10}{3} \, a^{4} b^{4} x^{21} + \frac {28}{9} \, a^{5} b^{3} x^{18} + \frac {28}{15} \, a^{6} b^{2} x^{15} + \frac {2}{3} \, a^{7} b x^{12} + \frac {1}{9} \, a^{8} x^{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 107 vs.
\(2 (44) = 88\).
time = 0.02, size = 107, normalized size = 2.02 \begin {gather*} \frac {a^{8} x^{9}}{9} + \frac {2 a^{7} b x^{12}}{3} + \frac {28 a^{6} b^{2} x^{15}}{15} + \frac {28 a^{5} b^{3} x^{18}}{9} + \frac {10 a^{4} b^{4} x^{21}}{3} + \frac {7 a^{3} b^{5} x^{24}}{3} + \frac {28 a^{2} b^{6} x^{27}}{27} + \frac {4 a b^{7} x^{30}}{15} + \frac {b^{8} x^{33}}{33} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.10, size = 90, normalized size = 1.70 \begin {gather*} \frac {1}{33} \, b^{8} x^{33} + \frac {4}{15} \, a b^{7} x^{30} + \frac {28}{27} \, a^{2} b^{6} x^{27} + \frac {7}{3} \, a^{3} b^{5} x^{24} + \frac {10}{3} \, a^{4} b^{4} x^{21} + \frac {28}{9} \, a^{5} b^{3} x^{18} + \frac {28}{15} \, a^{6} b^{2} x^{15} + \frac {2}{3} \, a^{7} b x^{12} + \frac {1}{9} \, a^{8} x^{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 90, normalized size = 1.70 \begin {gather*} \frac {a^8\,x^9}{9}+\frac {2\,a^7\,b\,x^{12}}{3}+\frac {28\,a^6\,b^2\,x^{15}}{15}+\frac {28\,a^5\,b^3\,x^{18}}{9}+\frac {10\,a^4\,b^4\,x^{21}}{3}+\frac {7\,a^3\,b^5\,x^{24}}{3}+\frac {28\,a^2\,b^6\,x^{27}}{27}+\frac {4\,a\,b^7\,x^{30}}{15}+\frac {b^8\,x^{33}}{33} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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