Optimal. Leaf size=95 \[ \frac{a^2 \left (a+b x^3\right )^6 (A b-a B)}{18 b^4}+\frac{\left (a+b x^3\right )^8 (A b-3 a B)}{24 b^4}-\frac{a \left (a+b x^3\right )^7 (2 A b-3 a B)}{21 b^4}+\frac{B \left (a+b x^3\right )^9}{27 b^4} \]
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Rubi [A] time = 0.236616, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 76} \[ \frac{a^2 \left (a+b x^3\right )^6 (A b-a B)}{18 b^4}+\frac{\left (a+b x^3\right )^8 (A b-3 a B)}{24 b^4}-\frac{a \left (a+b x^3\right )^7 (2 A b-3 a B)}{21 b^4}+\frac{B \left (a+b x^3\right )^9}{27 b^4} \]
Antiderivative was successfully verified.
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Rule 446
Rule 76
Rubi steps
\begin{align*} \int x^8 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^2 (a+b x)^5 (A+B x) \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^2 (-A b+a B) (a+b x)^5}{b^3}+\frac{a (-2 A b+3 a B) (a+b x)^6}{b^3}+\frac{(A b-3 a B) (a+b x)^7}{b^3}+\frac{B (a+b x)^8}{b^3}\right ) \, dx,x,x^3\right )\\ &=\frac{a^2 (A b-a B) \left (a+b x^3\right )^6}{18 b^4}-\frac{a (2 A b-3 a B) \left (a+b x^3\right )^7}{21 b^4}+\frac{(A b-3 a B) \left (a+b x^3\right )^8}{24 b^4}+\frac{B \left (a+b x^3\right )^9}{27 b^4}\\ \end{align*}
Mathematica [A] time = 0.0250756, size = 107, normalized size = 1.13 \[ \frac{x^9 \left (840 a^2 b^2 x^9 (a B+A b)+504 a^3 b x^6 (a B+2 A b)+126 a^4 x^3 (a B+5 A b)+168 a^5 A+63 b^4 x^{15} (5 a B+A b)+360 a b^3 x^{12} (2 a B+A b)+56 b^5 B x^{18}\right )}{1512} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 124, normalized size = 1.3 \begin{align*}{\frac{{b}^{5}B{x}^{27}}{27}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{24}}{24}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{21}}{21}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{18}}{18}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{15}}{15}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{12}}{12}}+{\frac{{a}^{5}A{x}^{9}}{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40945, size = 161, normalized size = 1.69 \begin{align*} \frac{1}{27} \, B b^{5} x^{27} + \frac{1}{24} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{24} + \frac{5}{21} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{21} + \frac{5}{9} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{18} + \frac{1}{3} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{15} + \frac{1}{9} \, A a^{5} x^{9} + \frac{1}{12} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26366, size = 316, normalized size = 3.33 \begin{align*} \frac{1}{27} x^{27} b^{5} B + \frac{5}{24} x^{24} b^{4} a B + \frac{1}{24} x^{24} b^{5} A + \frac{10}{21} x^{21} b^{3} a^{2} B + \frac{5}{21} x^{21} b^{4} a A + \frac{5}{9} x^{18} b^{2} a^{3} B + \frac{5}{9} x^{18} b^{3} a^{2} A + \frac{1}{3} x^{15} b a^{4} B + \frac{2}{3} x^{15} b^{2} a^{3} A + \frac{1}{12} x^{12} a^{5} B + \frac{5}{12} x^{12} b a^{4} A + \frac{1}{9} x^{9} a^{5} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.084674, size = 136, normalized size = 1.43 \begin{align*} \frac{A a^{5} x^{9}}{9} + \frac{B b^{5} x^{27}}{27} + x^{24} \left (\frac{A b^{5}}{24} + \frac{5 B a b^{4}}{24}\right ) + x^{21} \left (\frac{5 A a b^{4}}{21} + \frac{10 B a^{2} b^{3}}{21}\right ) + x^{18} \left (\frac{5 A a^{2} b^{3}}{9} + \frac{5 B a^{3} b^{2}}{9}\right ) + x^{15} \left (\frac{2 A a^{3} b^{2}}{3} + \frac{B a^{4} b}{3}\right ) + x^{12} \left (\frac{5 A a^{4} b}{12} + \frac{B a^{5}}{12}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13727, size = 169, normalized size = 1.78 \begin{align*} \frac{1}{27} \, B b^{5} x^{27} + \frac{5}{24} \, B a b^{4} x^{24} + \frac{1}{24} \, A b^{5} x^{24} + \frac{10}{21} \, B a^{2} b^{3} x^{21} + \frac{5}{21} \, A a b^{4} x^{21} + \frac{5}{9} \, B a^{3} b^{2} x^{18} + \frac{5}{9} \, A a^{2} b^{3} x^{18} + \frac{1}{3} \, B a^{4} b x^{15} + \frac{2}{3} \, A a^{3} b^{2} x^{15} + \frac{1}{12} \, B a^{5} x^{12} + \frac{5}{12} \, A a^{4} b x^{12} + \frac{1}{9} \, A a^{5} x^{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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