Optimal. Leaf size=80 \[ a^5 A \log (x)+5 a^4 A b x+5 a^3 A b^2 x^2+\frac {10}{3} a^2 A b^3 x^3+\frac {5}{4} a A b^4 x^4+\frac {B (a+b x)^6}{6 b}+\frac {1}{5} A b^5 x^5 \]
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Rubi [A] time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {80, 43} \[ \frac {10}{3} a^2 A b^3 x^3+5 a^3 A b^2 x^2+5 a^4 A b x+a^5 A \log (x)+\frac {5}{4} a A b^4 x^4+\frac {B (a+b x)^6}{6 b}+\frac {1}{5} A b^5 x^5 \]
Antiderivative was successfully verified.
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Rule 43
Rule 80
Rubi steps
\begin {align*} \int \frac {(a+b x)^5 (A+B x)}{x} \, dx &=\frac {B (a+b x)^6}{6 b}+A \int \frac {(a+b x)^5}{x} \, dx\\ &=\frac {B (a+b x)^6}{6 b}+A \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\\ &=5 a^4 A b x+5 a^3 A b^2 x^2+\frac {10}{3} a^2 A b^3 x^3+\frac {5}{4} a A b^4 x^4+\frac {1}{5} A b^5 x^5+\frac {B (a+b x)^6}{6 b}+a^5 A \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 108, normalized size = 1.35 \[ a^5 A \log (x)+a^4 x (a B+5 A b)+\frac {5}{2} a^3 b x^2 (a B+2 A b)+\frac {10}{3} a^2 b^2 x^3 (a B+A b)+\frac {1}{5} b^4 x^5 (5 a B+A b)+\frac {5}{4} a b^3 x^4 (2 a B+A b)+\frac {1}{6} b^5 B x^6 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 114, normalized size = 1.42 \[ \frac {1}{6} \, B b^{5} x^{6} + A a^{5} \log \relax (x) + \frac {1}{5} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + \frac {10}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + \frac {5}{2} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + {\left (B a^{5} + 5 \, A a^{4} b\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 118, normalized size = 1.48 \[ \frac {1}{6} \, B b^{5} x^{6} + B a b^{4} x^{5} + \frac {1}{5} \, A b^{5} x^{5} + \frac {5}{2} \, B a^{2} b^{3} x^{4} + \frac {5}{4} \, A a b^{4} x^{4} + \frac {10}{3} \, B a^{3} b^{2} x^{3} + \frac {10}{3} \, A a^{2} b^{3} x^{3} + \frac {5}{2} \, B a^{4} b x^{2} + 5 \, A a^{3} b^{2} x^{2} + B a^{5} x + 5 \, A a^{4} b x + A a^{5} \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 118, normalized size = 1.48 \[ \frac {B \,b^{5} x^{6}}{6}+\frac {A \,b^{5} x^{5}}{5}+B a \,b^{4} x^{5}+\frac {5 A a \,b^{4} x^{4}}{4}+\frac {5 B \,a^{2} b^{3} x^{4}}{2}+\frac {10 A \,a^{2} b^{3} x^{3}}{3}+\frac {10 B \,a^{3} b^{2} x^{3}}{3}+5 A \,a^{3} b^{2} x^{2}+\frac {5 B \,a^{4} b \,x^{2}}{2}+A \,a^{5} \ln \relax (x )+5 A \,a^{4} b x +B \,a^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 114, normalized size = 1.42 \[ \frac {1}{6} \, B b^{5} x^{6} + A a^{5} \log \relax (x) + \frac {1}{5} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + \frac {10}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + \frac {5}{2} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + {\left (B a^{5} + 5 \, A a^{4} b\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 101, normalized size = 1.26 \[ x\,\left (B\,a^5+5\,A\,b\,a^4\right )+x^5\,\left (\frac {A\,b^5}{5}+B\,a\,b^4\right )+\frac {B\,b^5\,x^6}{6}+A\,a^5\,\ln \relax (x)+\frac {10\,a^2\,b^2\,x^3\,\left (A\,b+B\,a\right )}{3}+\frac {5\,a^3\,b\,x^2\,\left (2\,A\,b+B\,a\right )}{2}+\frac {5\,a\,b^3\,x^4\,\left (A\,b+2\,B\,a\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 126, normalized size = 1.58 \[ A a^{5} \log {\relax (x )} + \frac {B b^{5} x^{6}}{6} + x^{5} \left (\frac {A b^{5}}{5} + B a b^{4}\right ) + x^{4} \left (\frac {5 A a b^{4}}{4} + \frac {5 B a^{2} b^{3}}{2}\right ) + x^{3} \left (\frac {10 A a^{2} b^{3}}{3} + \frac {10 B a^{3} b^{2}}{3}\right ) + x^{2} \left (5 A a^{3} b^{2} + \frac {5 B a^{4} b}{2}\right ) + x \left (5 A a^{4} b + B a^{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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