Optimal. Leaf size=42 \[ \frac{\left (a+b x^3\right )^6 (A b-a B)}{18 b^2}+\frac{B \left (a+b x^3\right )^7}{21 b^2} \]
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Rubi [A] time = 0.068837, antiderivative size = 42, 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 = {444, 43} \[ \frac{\left (a+b x^3\right )^6 (A b-a B)}{18 b^2}+\frac{B \left (a+b x^3\right )^7}{21 b^2} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int x^2 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int (a+b x)^5 (A+B x) \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{(A b-a B) (a+b x)^5}{b}+\frac{B (a+b x)^6}{b}\right ) \, dx,x,x^3\right )\\ &=\frac{(A b-a B) \left (a+b x^3\right )^6}{18 b^2}+\frac{B \left (a+b x^3\right )^7}{21 b^2}\\ \end{align*}
Mathematica [B] time = 0.0250203, size = 107, normalized size = 2.55 \[ \frac{1}{126} x^3 \left (105 a^2 b^2 x^9 (a B+A b)+70 a^3 b x^6 (a B+2 A b)+21 a^4 x^3 (a B+5 A b)+42 a^5 A+7 b^4 x^{15} (5 a B+A b)+42 a b^3 x^{12} (2 a B+A b)+6 b^5 B x^{18}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.001, size = 124, normalized size = 3. \begin{align*}{\frac{{b}^{5}B{x}^{21}}{21}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{18}}{18}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{15}}{15}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{12}}{12}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{9}}{9}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{6}}{6}}+{\frac{{a}^{5}A{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.32201, size = 161, normalized size = 3.83 \begin{align*} \frac{1}{21} \, B b^{5} x^{21} + \frac{1}{18} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{18} + \frac{1}{3} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{15} + \frac{5}{6} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{12} + \frac{5}{9} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{9} + \frac{1}{3} \, A a^{5} x^{3} + \frac{1}{6} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.2861, size = 305, normalized size = 7.26 \begin{align*} \frac{1}{21} x^{21} b^{5} B + \frac{5}{18} x^{18} b^{4} a B + \frac{1}{18} x^{18} b^{5} A + \frac{2}{3} x^{15} b^{3} a^{2} B + \frac{1}{3} x^{15} b^{4} a A + \frac{5}{6} x^{12} b^{2} a^{3} B + \frac{5}{6} x^{12} b^{3} a^{2} A + \frac{5}{9} x^{9} b a^{4} B + \frac{10}{9} x^{9} b^{2} a^{3} A + \frac{1}{6} x^{6} a^{5} B + \frac{5}{6} x^{6} b a^{4} A + \frac{1}{3} x^{3} a^{5} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.084238, size = 136, normalized size = 3.24 \begin{align*} \frac{A a^{5} x^{3}}{3} + \frac{B b^{5} x^{21}}{21} + x^{18} \left (\frac{A b^{5}}{18} + \frac{5 B a b^{4}}{18}\right ) + x^{15} \left (\frac{A a b^{4}}{3} + \frac{2 B a^{2} b^{3}}{3}\right ) + x^{12} \left (\frac{5 A a^{2} b^{3}}{6} + \frac{5 B a^{3} b^{2}}{6}\right ) + x^{9} \left (\frac{10 A a^{3} b^{2}}{9} + \frac{5 B a^{4} b}{9}\right ) + x^{6} \left (\frac{5 A a^{4} b}{6} + \frac{B a^{5}}{6}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17044, size = 169, normalized size = 4.02 \begin{align*} \frac{1}{21} \, B b^{5} x^{21} + \frac{5}{18} \, B a b^{4} x^{18} + \frac{1}{18} \, A b^{5} x^{18} + \frac{2}{3} \, B a^{2} b^{3} x^{15} + \frac{1}{3} \, A a b^{4} x^{15} + \frac{5}{6} \, B a^{3} b^{2} x^{12} + \frac{5}{6} \, A a^{2} b^{3} x^{12} + \frac{5}{9} \, B a^{4} b x^{9} + \frac{10}{9} \, A a^{3} b^{2} x^{9} + \frac{1}{6} \, B a^{5} x^{6} + \frac{5}{6} \, A a^{4} b x^{6} + \frac{1}{3} \, A a^{5} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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