Optimal. Leaf size=129 \[ \frac{5 a^2 \left (a+b x^3\right )^{13}}{13 b^7}-\frac{5 a^3 \left (a+b x^3\right )^{12}}{9 b^7}+\frac{5 a^4 \left (a+b x^3\right )^{11}}{11 b^7}-\frac{a^5 \left (a+b x^3\right )^{10}}{5 b^7}+\frac{a^6 \left (a+b x^3\right )^9}{27 b^7}+\frac{\left (a+b x^3\right )^{15}}{45 b^7}-\frac{a \left (a+b x^3\right )^{14}}{7 b^7} \]
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Rubi [A] time = 0.216476, antiderivative size = 129, normalized size of antiderivative = 1., 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} \[ \frac{5 a^2 \left (a+b x^3\right )^{13}}{13 b^7}-\frac{5 a^3 \left (a+b x^3\right )^{12}}{9 b^7}+\frac{5 a^4 \left (a+b x^3\right )^{11}}{11 b^7}-\frac{a^5 \left (a+b x^3\right )^{10}}{5 b^7}+\frac{a^6 \left (a+b x^3\right )^9}{27 b^7}+\frac{\left (a+b x^3\right )^{15}}{45 b^7}-\frac{a \left (a+b x^3\right )^{14}}{7 b^7} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{20} \left (a+b x^3\right )^8 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^6 (a+b x)^8 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^6 (a+b x)^8}{b^6}-\frac{6 a^5 (a+b x)^9}{b^6}+\frac{15 a^4 (a+b x)^{10}}{b^6}-\frac{20 a^3 (a+b x)^{11}}{b^6}+\frac{15 a^2 (a+b x)^{12}}{b^6}-\frac{6 a (a+b x)^{13}}{b^6}+\frac{(a+b x)^{14}}{b^6}\right ) \, dx,x,x^3\right )\\ &=\frac{a^6 \left (a+b x^3\right )^9}{27 b^7}-\frac{a^5 \left (a+b x^3\right )^{10}}{5 b^7}+\frac{5 a^4 \left (a+b x^3\right )^{11}}{11 b^7}-\frac{5 a^3 \left (a+b x^3\right )^{12}}{9 b^7}+\frac{5 a^2 \left (a+b x^3\right )^{13}}{13 b^7}-\frac{a \left (a+b x^3\right )^{14}}{7 b^7}+\frac{\left (a+b x^3\right )^{15}}{45 b^7}\\ \end{align*}
Mathematica [A] time = 0.002804, size = 108, normalized size = 0.84 \[ \frac{28}{39} a^2 b^6 x^{39}+\frac{14}{9} a^3 b^5 x^{36}+\frac{70}{33} a^4 b^4 x^{33}+\frac{28}{15} a^5 b^3 x^{30}+\frac{28}{27} a^6 b^2 x^{27}+\frac{1}{3} a^7 b x^{24}+\frac{a^8 x^{21}}{21}+\frac{4}{21} a b^7 x^{42}+\frac{b^8 x^{45}}{45} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 91, normalized size = 0.7 \begin{align*}{\frac{{b}^{8}{x}^{45}}{45}}+{\frac{4\,a{b}^{7}{x}^{42}}{21}}+{\frac{28\,{b}^{6}{a}^{2}{x}^{39}}{39}}+{\frac{14\,{a}^{3}{b}^{5}{x}^{36}}{9}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{33}}{33}}+{\frac{28\,{a}^{5}{b}^{3}{x}^{30}}{15}}+{\frac{28\,{a}^{6}{b}^{2}{x}^{27}}{27}}+{\frac{{a}^{7}b{x}^{24}}{3}}+{\frac{{a}^{8}{x}^{21}}{21}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.95288, size = 122, normalized size = 0.95 \begin{align*} \frac{1}{45} \, b^{8} x^{45} + \frac{4}{21} \, a b^{7} x^{42} + \frac{28}{39} \, a^{2} b^{6} x^{39} + \frac{14}{9} \, a^{3} b^{5} x^{36} + \frac{70}{33} \, a^{4} b^{4} x^{33} + \frac{28}{15} \, a^{5} b^{3} x^{30} + \frac{28}{27} \, a^{6} b^{2} x^{27} + \frac{1}{3} \, a^{7} b x^{24} + \frac{1}{21} \, a^{8} x^{21} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50925, size = 230, normalized size = 1.78 \begin{align*} \frac{1}{45} x^{45} b^{8} + \frac{4}{21} x^{42} b^{7} a + \frac{28}{39} x^{39} b^{6} a^{2} + \frac{14}{9} x^{36} b^{5} a^{3} + \frac{70}{33} x^{33} b^{4} a^{4} + \frac{28}{15} x^{30} b^{3} a^{5} + \frac{28}{27} x^{27} b^{2} a^{6} + \frac{1}{3} x^{24} b a^{7} + \frac{1}{21} x^{21} a^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.113934, size = 105, normalized size = 0.81 \begin{align*} \frac{a^{8} x^{21}}{21} + \frac{a^{7} b x^{24}}{3} + \frac{28 a^{6} b^{2} x^{27}}{27} + \frac{28 a^{5} b^{3} x^{30}}{15} + \frac{70 a^{4} b^{4} x^{33}}{33} + \frac{14 a^{3} b^{5} x^{36}}{9} + \frac{28 a^{2} b^{6} x^{39}}{39} + \frac{4 a b^{7} x^{42}}{21} + \frac{b^{8} x^{45}}{45} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12356, size = 122, normalized size = 0.95 \begin{align*} \frac{1}{45} \, b^{8} x^{45} + \frac{4}{21} \, a b^{7} x^{42} + \frac{28}{39} \, a^{2} b^{6} x^{39} + \frac{14}{9} \, a^{3} b^{5} x^{36} + \frac{70}{33} \, a^{4} b^{4} x^{33} + \frac{28}{15} \, a^{5} b^{3} x^{30} + \frac{28}{27} \, a^{6} b^{2} x^{27} + \frac{1}{3} \, a^{7} b x^{24} + \frac{1}{21} \, a^{8} x^{21} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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