Optimal. Leaf size=105 \[ -\frac {a^5 A}{x}+a^4 \log (x) (a B+5 A b)+5 a^3 b x (a B+2 A b)+5 a^2 b^2 x^2 (a B+A b)+\frac {1}{4} b^4 x^4 (5 a B+A b)+\frac {5}{3} a b^3 x^3 (2 a B+A b)+\frac {1}{5} b^5 B x^5 \]
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Rubi [A] time = 0.06, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {76} \[ 5 a^2 b^2 x^2 (a B+A b)+5 a^3 b x (a B+2 A b)+a^4 \log (x) (a B+5 A b)-\frac {a^5 A}{x}+\frac {5}{3} a b^3 x^3 (2 a B+A b)+\frac {1}{4} b^4 x^4 (5 a B+A b)+\frac {1}{5} b^5 B x^5 \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x)^5 (A+B x)}{x^2} \, dx &=\int \left (5 a^3 b (2 A b+a B)+\frac {a^5 A}{x^2}+\frac {a^4 (5 A b+a B)}{x}+10 a^2 b^2 (A b+a B) x+5 a b^3 (A b+2 a B) x^2+b^4 (A b+5 a B) x^3+b^5 B x^4\right ) \, dx\\ &=-\frac {a^5 A}{x}+5 a^3 b (2 A b+a B) x+5 a^2 b^2 (A b+a B) x^2+\frac {5}{3} a b^3 (A b+2 a B) x^3+\frac {1}{4} b^4 (A b+5 a B) x^4+\frac {1}{5} b^5 B x^5+a^4 (5 A b+a B) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.06, size = 107, normalized size = 1.02 \[ -\frac {a^5 A}{x}+5 a^3 b x (a B+2 A b)+5 a^2 b^2 x^2 (a B+A b)+\log (x) \left (a^5 B+5 a^4 A b\right )+\frac {1}{4} b^4 x^4 (5 a B+A b)+\frac {5}{3} a b^3 x^3 (2 a B+A b)+\frac {1}{5} b^5 B x^5 \]
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 121, normalized size = 1.15 \[ \frac {12 \, B b^{5} x^{6} - 60 \, A a^{5} + 15 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 100 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 300 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 300 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 60 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x \log \relax (x)}{60 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.20, size = 119, normalized size = 1.13 \[ \frac {1}{5} \, B b^{5} x^{5} + \frac {5}{4} \, B a b^{4} x^{4} + \frac {1}{4} \, A b^{5} x^{4} + \frac {10}{3} \, B a^{2} b^{3} x^{3} + \frac {5}{3} \, A a b^{4} x^{3} + 5 \, B a^{3} b^{2} x^{2} + 5 \, A a^{2} b^{3} x^{2} + 5 \, B a^{4} b x + 10 \, A a^{3} b^{2} x - \frac {A a^{5}}{x} + {\left (B a^{5} + 5 \, A a^{4} b\right )} \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 119, normalized size = 1.13 \[ \frac {B \,b^{5} x^{5}}{5}+\frac {A \,b^{5} x^{4}}{4}+\frac {5 B a \,b^{4} x^{4}}{4}+\frac {5 A a \,b^{4} x^{3}}{3}+\frac {10 B \,a^{2} b^{3} x^{3}}{3}+5 A \,a^{2} b^{3} x^{2}+5 B \,a^{3} b^{2} x^{2}+5 A \,a^{4} b \ln \relax (x )+10 A \,a^{3} b^{2} x +B \,a^{5} \ln \relax (x )+5 B \,a^{4} b x -\frac {A \,a^{5}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 115, normalized size = 1.10 \[ \frac {1}{5} \, B b^{5} x^{5} - \frac {A a^{5}}{x} + \frac {1}{4} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{3} + 5 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{2} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x + {\left (B a^{5} + 5 \, A a^{4} b\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 103, normalized size = 0.98 \[ x^4\,\left (\frac {A\,b^5}{4}+\frac {5\,B\,a\,b^4}{4}\right )+\ln \relax (x)\,\left (B\,a^5+5\,A\,b\,a^4\right )-\frac {A\,a^5}{x}+\frac {B\,b^5\,x^5}{5}+5\,a^2\,b^2\,x^2\,\left (A\,b+B\,a\right )+5\,a^3\,b\,x\,\left (2\,A\,b+B\,a\right )+\frac {5\,a\,b^3\,x^3\,\left (A\,b+2\,B\,a\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 121, normalized size = 1.15 \[ - \frac {A a^{5}}{x} + \frac {B b^{5} x^{5}}{5} + a^{4} \left (5 A b + B a\right ) \log {\relax (x )} + x^{4} \left (\frac {A b^{5}}{4} + \frac {5 B a b^{4}}{4}\right ) + x^{3} \left (\frac {5 A a b^{4}}{3} + \frac {10 B a^{2} b^{3}}{3}\right ) + x^{2} \left (5 A a^{2} b^{3} + 5 B a^{3} b^{2}\right ) + x \left (10 A a^{3} b^{2} + 5 B a^{4} b\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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