Optimal. Leaf size=105 \[ -\frac {a^5 A}{x}+5 a^3 b (2 A b+a B) x+5 a^2 b^2 (A b+a B) x^2+\frac {5}{3} a b^3 (A b+2 a B) x^3+\frac {1}{4} b^4 (A b+5 a B) x^4+\frac {1}{5} b^5 B x^5+a^4 (5 A b+a B) \log (x) \]
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Rubi [A]
time = 0.04, 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 = {77}
\begin {gather*} -\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 \end {gather*}
Antiderivative was successfully verified.
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Rule 77
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 = 0.04, size = 107, normalized size = 1.02 \begin {gather*} -\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+\left (5 a^4 A b+a^5 B\right ) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 117, normalized size = 1.11
method | result | size |
default | \(\frac {b^{5} B \,x^{5}}{5}+\frac {A \,b^{5} x^{4}}{4}+\frac {5 B a \,b^{4} x^{4}}{4}+\frac {5 A a \,b^{4} x^{3}}{3}+\frac {10 B \,a^{2} b^{3} x^{3}}{3}+5 A \,a^{2} b^{3} x^{2}+5 B \,a^{3} b^{2} x^{2}+10 a^{3} b^{2} A x +5 a^{4} b B x -\frac {a^{5} A}{x}+a^{4} \left (5 A b +B a \right ) \ln \left (x \right )\) | \(117\) |
risch | \(\frac {b^{5} B \,x^{5}}{5}+\frac {A \,b^{5} x^{4}}{4}+\frac {5 B a \,b^{4} x^{4}}{4}+\frac {5 A a \,b^{4} x^{3}}{3}+\frac {10 B \,a^{2} b^{3} x^{3}}{3}+5 A \,a^{2} b^{3} x^{2}+5 B \,a^{3} b^{2} x^{2}+10 a^{3} b^{2} A x +5 a^{4} b B x -\frac {a^{5} A}{x}+5 A \ln \left (x \right ) a^{4} b +B \ln \left (x \right ) a^{5}\) | \(119\) |
norman | \(\frac {\left (\frac {1}{4} b^{5} A +\frac {5}{4} a \,b^{4} B \right ) x^{5}+\left (\frac {5}{3} a \,b^{4} A +\frac {10}{3} a^{2} b^{3} B \right ) x^{4}+\left (5 a^{2} b^{3} A +5 a^{3} b^{2} B \right ) x^{3}+\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{2}-a^{5} A +\frac {b^{5} B \,x^{6}}{5}}{x}+\left (5 a^{4} b A +a^{5} B \right ) \ln \left (x \right )\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 115, normalized size = 1.10 \begin {gather*} \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 \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.77, size = 121, normalized size = 1.15 \begin {gather*} \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 \left (x\right )}{60 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 121, normalized size = 1.15 \begin {gather*} - \frac {A a^{5}}{x} + \frac {B b^{5} x^{5}}{5} + a^{4} \cdot \left (5 A b + B a\right ) \log {\left (x \right )} + x^{4} \left (\frac {A b^{5}}{4} + \frac {5 B a b^{4}}{4}\right ) + x^{3} \cdot \left (\frac {5 A a b^{4}}{3} + \frac {10 B a^{2} b^{3}}{3}\right ) + x^{2} \cdot \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.31, size = 119, normalized size = 1.13 \begin {gather*} \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 ) \end {gather*}
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 \begin {gather*} x^4\,\left (\frac {A\,b^5}{4}+\frac {5\,B\,a\,b^4}{4}\right )+\ln \left (x\right )\,\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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