Optimal. Leaf size=91 \[ \frac {a^4 \left (a+b x^3\right )^9}{27 b^5}-\frac {2 a^3 \left (a+b x^3\right )^{10}}{15 b^5}+\frac {2 a^2 \left (a+b x^3\right )^{11}}{11 b^5}-\frac {a \left (a+b x^3\right )^{12}}{9 b^5}+\frac {\left (a+b x^3\right )^{13}}{39 b^5} \]
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Rubi [A]
time = 0.10, antiderivative size = 91, 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^4 \left (a+b x^3\right )^9}{27 b^5}-\frac {2 a^3 \left (a+b x^3\right )^{10}}{15 b^5}+\frac {2 a^2 \left (a+b x^3\right )^{11}}{11 b^5}+\frac {\left (a+b x^3\right )^{13}}{39 b^5}-\frac {a \left (a+b x^3\right )^{12}}{9 b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^{14} \left (a+b x^3\right )^8 \, dx &=\frac {1}{3} \text {Subst}\left (\int x^4 (a+b x)^8 \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a^4 (a+b x)^8}{b^4}-\frac {4 a^3 (a+b x)^9}{b^4}+\frac {6 a^2 (a+b x)^{10}}{b^4}-\frac {4 a (a+b x)^{11}}{b^4}+\frac {(a+b x)^{12}}{b^4}\right ) \, dx,x,x^3\right )\\ &=\frac {a^4 \left (a+b x^3\right )^9}{27 b^5}-\frac {2 a^3 \left (a+b x^3\right )^{10}}{15 b^5}+\frac {2 a^2 \left (a+b x^3\right )^{11}}{11 b^5}-\frac {a \left (a+b x^3\right )^{12}}{9 b^5}+\frac {\left (a+b x^3\right )^{13}}{39 b^5}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 108, normalized size = 1.19 \begin {gather*} \frac {a^8 x^{15}}{15}+\frac {4}{9} a^7 b x^{18}+\frac {4}{3} a^6 b^2 x^{21}+\frac {7}{3} a^5 b^3 x^{24}+\frac {70}{27} a^4 b^4 x^{27}+\frac {28}{15} a^3 b^5 x^{30}+\frac {28}{33} a^2 b^6 x^{33}+\frac {2}{9} a b^7 x^{36}+\frac {b^8 x^{39}}{39} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 91, normalized size = 1.00
method | result | size |
gosper | \(\frac {1}{39} b^{8} x^{39}+\frac {28}{33} a^{2} b^{6} x^{33}+\frac {2}{9} a \,b^{7} x^{36}+\frac {4}{3} a^{6} b^{2} x^{21}+\frac {7}{3} a^{5} b^{3} x^{24}+\frac {70}{27} a^{4} b^{4} x^{27}+\frac {28}{15} a^{3} b^{5} x^{30}+\frac {1}{15} a^{8} x^{15}+\frac {4}{9} b \,a^{7} x^{18}\) | \(91\) |
default | \(\frac {1}{39} b^{8} x^{39}+\frac {28}{33} a^{2} b^{6} x^{33}+\frac {2}{9} a \,b^{7} x^{36}+\frac {4}{3} a^{6} b^{2} x^{21}+\frac {7}{3} a^{5} b^{3} x^{24}+\frac {70}{27} a^{4} b^{4} x^{27}+\frac {28}{15} a^{3} b^{5} x^{30}+\frac {1}{15} a^{8} x^{15}+\frac {4}{9} b \,a^{7} x^{18}\) | \(91\) |
risch | \(\frac {1}{39} b^{8} x^{39}+\frac {28}{33} a^{2} b^{6} x^{33}+\frac {2}{9} a \,b^{7} x^{36}+\frac {4}{3} a^{6} b^{2} x^{21}+\frac {7}{3} a^{5} b^{3} x^{24}+\frac {70}{27} a^{4} b^{4} x^{27}+\frac {28}{15} a^{3} b^{5} x^{30}+\frac {1}{15} a^{8} x^{15}+\frac {4}{9} b \,a^{7} x^{18}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 90, normalized size = 0.99 \begin {gather*} \frac {1}{39} \, b^{8} x^{39} + \frac {2}{9} \, a b^{7} x^{36} + \frac {28}{33} \, a^{2} b^{6} x^{33} + \frac {28}{15} \, a^{3} b^{5} x^{30} + \frac {70}{27} \, a^{4} b^{4} x^{27} + \frac {7}{3} \, a^{5} b^{3} x^{24} + \frac {4}{3} \, a^{6} b^{2} x^{21} + \frac {4}{9} \, a^{7} b x^{18} + \frac {1}{15} \, a^{8} x^{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 90, normalized size = 0.99 \begin {gather*} \frac {1}{39} \, b^{8} x^{39} + \frac {2}{9} \, a b^{7} x^{36} + \frac {28}{33} \, a^{2} b^{6} x^{33} + \frac {28}{15} \, a^{3} b^{5} x^{30} + \frac {70}{27} \, a^{4} b^{4} x^{27} + \frac {7}{3} \, a^{5} b^{3} x^{24} + \frac {4}{3} \, a^{6} b^{2} x^{21} + \frac {4}{9} \, a^{7} b x^{18} + \frac {1}{15} \, a^{8} x^{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 107, normalized size = 1.18 \begin {gather*} \frac {a^{8} x^{15}}{15} + \frac {4 a^{7} b x^{18}}{9} + \frac {4 a^{6} b^{2} x^{21}}{3} + \frac {7 a^{5} b^{3} x^{24}}{3} + \frac {70 a^{4} b^{4} x^{27}}{27} + \frac {28 a^{3} b^{5} x^{30}}{15} + \frac {28 a^{2} b^{6} x^{33}}{33} + \frac {2 a b^{7} x^{36}}{9} + \frac {b^{8} x^{39}}{39} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.61, size = 90, normalized size = 0.99 \begin {gather*} \frac {1}{39} \, b^{8} x^{39} + \frac {2}{9} \, a b^{7} x^{36} + \frac {28}{33} \, a^{2} b^{6} x^{33} + \frac {28}{15} \, a^{3} b^{5} x^{30} + \frac {70}{27} \, a^{4} b^{4} x^{27} + \frac {7}{3} \, a^{5} b^{3} x^{24} + \frac {4}{3} \, a^{6} b^{2} x^{21} + \frac {4}{9} \, a^{7} b x^{18} + \frac {1}{15} \, a^{8} x^{15} \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 = 0.99 \begin {gather*} \frac {a^8\,x^{15}}{15}+\frac {4\,a^7\,b\,x^{18}}{9}+\frac {4\,a^6\,b^2\,x^{21}}{3}+\frac {7\,a^5\,b^3\,x^{24}}{3}+\frac {70\,a^4\,b^4\,x^{27}}{27}+\frac {28\,a^3\,b^5\,x^{30}}{15}+\frac {28\,a^2\,b^6\,x^{33}}{33}+\frac {2\,a\,b^7\,x^{36}}{9}+\frac {b^8\,x^{39}}{39} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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