Optimal. Leaf size=88 \[ a^5 A \log (x)+\frac {5}{3} a^4 A b x^3+\frac {5}{3} a^3 A b^2 x^6+\frac {10}{9} a^2 A b^3 x^9+\frac {5}{12} a A b^4 x^{12}+\frac {B \left (a+b x^3\right )^6}{18 b}+\frac {1}{15} A b^5 x^{15} \]
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Rubi [A] time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {446, 80, 43} \begin {gather*} \frac {10}{9} a^2 A b^3 x^9+\frac {5}{3} a^3 A b^2 x^6+\frac {5}{3} a^4 A b x^3+a^5 A \log (x)+\frac {5}{12} a A b^4 x^{12}+\frac {B \left (a+b x^3\right )^6}{18 b}+\frac {1}{15} A b^5 x^{15} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 80
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(a+b x)^5 (A+B x)}{x} \, dx,x,x^3\right )\\ &=\frac {B \left (a+b x^3\right )^6}{18 b}+\frac {1}{3} A \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x} \, dx,x,x^3\right )\\ &=\frac {B \left (a+b x^3\right )^6}{18 b}+\frac {1}{3} A \operatorname {Subst}\left (\int \left (5 a^4 b+\frac {a^5}{x}+10 a^3 b^2 x+10 a^2 b^3 x^2+5 a b^4 x^3+b^5 x^4\right ) \, dx,x,x^3\right )\\ &=\frac {5}{3} a^4 A b x^3+\frac {5}{3} a^3 A b^2 x^6+\frac {10}{9} a^2 A b^3 x^9+\frac {5}{12} a A b^4 x^{12}+\frac {1}{15} A b^5 x^{15}+\frac {B \left (a+b x^3\right )^6}{18 b}+a^5 A \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 113, normalized size = 1.28 \begin {gather*} a^5 A \log (x)+\frac {1}{3} a^4 x^3 (a B+5 A b)+\frac {5}{6} a^3 b x^6 (a B+2 A b)+\frac {10}{9} a^2 b^2 x^9 (a B+A b)+\frac {1}{15} b^4 x^{15} (5 a B+A b)+\frac {5}{12} a b^3 x^{12} (2 a B+A b)+\frac {1}{18} b^5 B x^{18} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.82, size = 117, normalized size = 1.33 \begin {gather*} \frac {1}{18} \, B b^{5} x^{18} + \frac {1}{15} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + \frac {5}{12} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + \frac {10}{9} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac {5}{6} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + A a^{5} \log \relax (x) + \frac {1}{3} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 124, normalized size = 1.41 \begin {gather*} \frac {1}{18} \, B b^{5} x^{18} + \frac {1}{3} \, B a b^{4} x^{15} + \frac {1}{15} \, A b^{5} x^{15} + \frac {5}{6} \, B a^{2} b^{3} x^{12} + \frac {5}{12} \, A a b^{4} x^{12} + \frac {10}{9} \, B a^{3} b^{2} x^{9} + \frac {10}{9} \, A a^{2} b^{3} x^{9} + \frac {5}{6} \, B a^{4} b x^{6} + \frac {5}{3} \, A a^{3} b^{2} x^{6} + \frac {1}{3} \, B a^{5} x^{3} + \frac {5}{3} \, A a^{4} b x^{3} + A a^{5} \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 124, normalized size = 1.41 \begin {gather*} \frac {B \,b^{5} x^{18}}{18}+\frac {A \,b^{5} x^{15}}{15}+\frac {B a \,b^{4} x^{15}}{3}+\frac {5 A a \,b^{4} x^{12}}{12}+\frac {5 B \,a^{2} b^{3} x^{12}}{6}+\frac {10 A \,a^{2} b^{3} x^{9}}{9}+\frac {10 B \,a^{3} b^{2} x^{9}}{9}+\frac {5 A \,a^{3} b^{2} x^{6}}{3}+\frac {5 B \,a^{4} b \,x^{6}}{6}+\frac {5 A \,a^{4} b \,x^{3}}{3}+\frac {B \,a^{5} x^{3}}{3}+A \,a^{5} \ln \relax (x ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 120, normalized size = 1.36 \begin {gather*} \frac {1}{18} \, B b^{5} x^{18} + \frac {1}{15} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + \frac {5}{12} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + \frac {10}{9} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac {5}{6} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + \frac {1}{3} \, A a^{5} \log \left (x^{3}\right ) + \frac {1}{3} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 105, normalized size = 1.19 \begin {gather*} x^3\,\left (\frac {B\,a^5}{3}+\frac {5\,A\,b\,a^4}{3}\right )+x^{15}\,\left (\frac {A\,b^5}{15}+\frac {B\,a\,b^4}{3}\right )+\frac {B\,b^5\,x^{18}}{18}+A\,a^5\,\ln \relax (x)+\frac {10\,a^2\,b^2\,x^9\,\left (A\,b+B\,a\right )}{9}+\frac {5\,a^3\,b\,x^6\,\left (2\,A\,b+B\,a\right )}{6}+\frac {5\,a\,b^3\,x^{12}\,\left (A\,b+2\,B\,a\right )}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 134, normalized size = 1.52 \begin {gather*} A a^{5} \log {\relax (x )} + \frac {B b^{5} x^{18}}{18} + x^{15} \left (\frac {A b^{5}}{15} + \frac {B a b^{4}}{3}\right ) + x^{12} \left (\frac {5 A a b^{4}}{12} + \frac {5 B a^{2} b^{3}}{6}\right ) + x^{9} \left (\frac {10 A a^{2} b^{3}}{9} + \frac {10 B a^{3} b^{2}}{9}\right ) + x^{6} \left (\frac {5 A a^{3} b^{2}}{3} + \frac {5 B a^{4} b}{6}\right ) + x^{3} \left (\frac {5 A a^{4} b}{3} + \frac {B a^{5}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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