Optimal. Leaf size=67 \[ \frac {\left (a+b x^3\right )^7 (A b-2 a B)}{21 b^3}-\frac {a \left (a+b x^3\right )^6 (A b-a B)}{18 b^3}+\frac {B \left (a+b x^3\right )^8}{24 b^3} \]
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Rubi [A] time = 0.15, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 76} \begin {gather*} \frac {\left (a+b x^3\right )^7 (A b-2 a B)}{21 b^3}-\frac {a \left (a+b x^3\right )^6 (A b-a B)}{18 b^3}+\frac {B \left (a+b x^3\right )^8}{24 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 446
Rubi steps
\begin {align*} \int x^5 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x (a+b x)^5 (A+B x) \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {a (-A b+a B) (a+b x)^5}{b^2}+\frac {(A b-2 a B) (a+b x)^6}{b^2}+\frac {B (a+b x)^7}{b^2}\right ) \, dx,x,x^3\right )\\ &=-\frac {a (A b-a B) \left (a+b x^3\right )^6}{18 b^3}+\frac {(A b-2 a B) \left (a+b x^3\right )^7}{21 b^3}+\frac {B \left (a+b x^3\right )^8}{24 b^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 107, normalized size = 1.60 \begin {gather*} \frac {1}{504} x^6 \left (84 a^5 A+56 a^4 x^3 (a B+5 A b)+210 a^3 b x^6 (a B+2 A b)+336 a^2 b^2 x^9 (a B+A b)+24 b^4 x^{15} (5 a B+A b)+140 a b^3 x^{12} (2 a B+A b)+21 b^5 B x^{18}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^5 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.28, size = 125, normalized size = 1.87 \begin {gather*} \frac {1}{24} x^{24} b^{5} B + \frac {5}{21} x^{21} b^{4} a B + \frac {1}{21} x^{21} b^{5} A + \frac {5}{9} x^{18} b^{3} a^{2} B + \frac {5}{18} x^{18} b^{4} a A + \frac {2}{3} x^{15} b^{2} a^{3} B + \frac {2}{3} x^{15} b^{3} a^{2} A + \frac {5}{12} x^{12} b a^{4} B + \frac {5}{6} x^{12} b^{2} a^{3} A + \frac {1}{9} x^{9} a^{5} B + \frac {5}{9} x^{9} b a^{4} A + \frac {1}{6} x^{6} a^{5} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 125, normalized size = 1.87 \begin {gather*} \frac {1}{24} \, B b^{5} x^{24} + \frac {5}{21} \, B a b^{4} x^{21} + \frac {1}{21} \, A b^{5} x^{21} + \frac {5}{9} \, B a^{2} b^{3} x^{18} + \frac {5}{18} \, A a b^{4} x^{18} + \frac {2}{3} \, B a^{3} b^{2} x^{15} + \frac {2}{3} \, A a^{2} b^{3} x^{15} + \frac {5}{12} \, B a^{4} b x^{12} + \frac {5}{6} \, A a^{3} b^{2} x^{12} + \frac {1}{9} \, B a^{5} x^{9} + \frac {5}{9} \, A a^{4} b x^{9} + \frac {1}{6} \, A a^{5} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 124, normalized size = 1.85 \begin {gather*} \frac {B \,b^{5} x^{24}}{24}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{21}}{21}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{18}}{18}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{15}}{15}+\frac {A \,a^{5} x^{6}}{6}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{12}}{12}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{9}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 119, normalized size = 1.78 \begin {gather*} \frac {1}{24} \, B b^{5} x^{24} + \frac {1}{21} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{21} + \frac {5}{18} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{18} + \frac {2}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{15} + \frac {5}{12} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{12} + \frac {1}{6} \, A a^{5} x^{6} + \frac {1}{9} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 107, normalized size = 1.60 \begin {gather*} x^9\,\left (\frac {B\,a^5}{9}+\frac {5\,A\,b\,a^4}{9}\right )+x^{21}\,\left (\frac {A\,b^5}{21}+\frac {5\,B\,a\,b^4}{21}\right )+\frac {A\,a^5\,x^6}{6}+\frac {B\,b^5\,x^{24}}{24}+\frac {2\,a^2\,b^2\,x^{15}\,\left (A\,b+B\,a\right )}{3}+\frac {5\,a^3\,b\,x^{12}\,\left (2\,A\,b+B\,a\right )}{12}+\frac {5\,a\,b^3\,x^{18}\,\left (A\,b+2\,B\,a\right )}{18} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 138, normalized size = 2.06 \begin {gather*} \frac {A a^{5} x^{6}}{6} + \frac {B b^{5} x^{24}}{24} + x^{21} \left (\frac {A b^{5}}{21} + \frac {5 B a b^{4}}{21}\right ) + x^{18} \left (\frac {5 A a b^{4}}{18} + \frac {5 B a^{2} b^{3}}{9}\right ) + x^{15} \left (\frac {2 A a^{2} b^{3}}{3} + \frac {2 B a^{3} b^{2}}{3}\right ) + x^{12} \left (\frac {5 A a^{3} b^{2}}{6} + \frac {5 B a^{4} b}{12}\right ) + x^{9} \left (\frac {5 A a^{4} b}{9} + \frac {B a^{5}}{9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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