Optimal. Leaf size=131 \[ -\frac {a^6 A}{5 x^5}-\frac {a^5 (6 A b+a B)}{4 x^4}-\frac {a^4 b (5 A b+2 a B)}{x^3}-\frac {5 a^3 b^2 (4 A b+3 a B)}{2 x^2}-\frac {5 a^2 b^3 (3 A b+4 a B)}{x}+b^5 (A b+6 a B) x+\frac {1}{2} b^6 B x^2+3 a b^4 (2 A b+5 a B) \log (x) \]
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Rubi [A]
time = 0.06, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {27, 77}
\begin {gather*} -\frac {a^6 A}{5 x^5}-\frac {a^5 (a B+6 A b)}{4 x^4}-\frac {a^4 b (2 a B+5 A b)}{x^3}-\frac {5 a^3 b^2 (3 a B+4 A b)}{2 x^2}-\frac {5 a^2 b^3 (4 a B+3 A b)}{x}+b^5 x (6 a B+A b)+3 a b^4 \log (x) (5 a B+2 A b)+\frac {1}{2} b^6 B x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^6} \, dx &=\int \frac {(a+b x)^6 (A+B x)}{x^6} \, dx\\ &=\int \left (b^5 (A b+6 a B)+\frac {a^6 A}{x^6}+\frac {a^5 (6 A b+a B)}{x^5}+\frac {3 a^4 b (5 A b+2 a B)}{x^4}+\frac {5 a^3 b^2 (4 A b+3 a B)}{x^3}+\frac {5 a^2 b^3 (3 A b+4 a B)}{x^2}+\frac {3 a b^4 (2 A b+5 a B)}{x}+b^6 B x\right ) \, dx\\ &=-\frac {a^6 A}{5 x^5}-\frac {a^5 (6 A b+a B)}{4 x^4}-\frac {a^4 b (5 A b+2 a B)}{x^3}-\frac {5 a^3 b^2 (4 A b+3 a B)}{2 x^2}-\frac {5 a^2 b^3 (3 A b+4 a B)}{x}+b^5 (A b+6 a B) x+\frac {1}{2} b^6 B x^2+3 a b^4 (2 A b+5 a B) \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 128, normalized size = 0.98 \begin {gather*} -\frac {15 a^2 A b^4}{x}+6 a b^5 B x+\frac {1}{2} b^6 x (2 A+B x)-\frac {10 a^3 b^3 (A+2 B x)}{x^2}-\frac {5 a^4 b^2 (2 A+3 B x)}{2 x^3}-\frac {a^5 b (3 A+4 B x)}{2 x^4}-\frac {a^6 (4 A+5 B x)}{20 x^5}+3 a b^4 (2 A b+5 a B) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, size = 125, normalized size = 0.95
method | result | size |
default | \(\frac {b^{6} B \,x^{2}}{2}+A \,b^{6} x +6 B a \,b^{5} x -\frac {a^{6} A}{5 x^{5}}-\frac {a^{5} \left (6 A b +B a \right )}{4 x^{4}}-\frac {5 a^{3} b^{2} \left (4 A b +3 B a \right )}{2 x^{2}}-\frac {a^{4} b \left (5 A b +2 B a \right )}{x^{3}}+3 a \,b^{4} \left (2 A b +5 B a \right ) \ln \left (x \right )-\frac {5 a^{2} b^{3} \left (3 A b +4 B a \right )}{x}\) | \(125\) |
risch | \(\frac {b^{6} B \,x^{2}}{2}+A \,b^{6} x +6 B a \,b^{5} x +\frac {\left (-15 A \,a^{2} b^{4}-20 B \,a^{3} b^{3}\right ) x^{4}+\left (-10 A \,a^{3} b^{3}-\frac {15}{2} B \,a^{4} b^{2}\right ) x^{3}+\left (-5 A \,a^{4} b^{2}-2 B \,a^{5} b \right ) x^{2}+\left (-\frac {3}{2} A \,a^{5} b -\frac {1}{4} B \,a^{6}\right ) x -\frac {A \,a^{6}}{5}}{x^{5}}+6 A \ln \left (x \right ) a \,b^{5}+15 B \ln \left (x \right ) a^{2} b^{4}\) | \(139\) |
norman | \(\frac {\left (-10 A \,a^{3} b^{3}-\frac {15}{2} B \,a^{4} b^{2}\right ) x^{3}+\left (-\frac {3}{2} A \,a^{5} b -\frac {1}{4} B \,a^{6}\right ) x +\left (A \,b^{6}+6 B a \,b^{5}\right ) x^{6}+\left (-15 A \,a^{2} b^{4}-20 B \,a^{3} b^{3}\right ) x^{4}+\left (-5 A \,a^{4} b^{2}-2 B \,a^{5} b \right ) x^{2}-\frac {A \,a^{6}}{5}+\frac {b^{6} B \,x^{7}}{2}}{x^{5}}+\left (6 A a \,b^{5}+15 B \,a^{2} b^{4}\right ) \ln \left (x \right )\) | \(142\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 144, normalized size = 1.10 \begin {gather*} \frac {1}{2} \, B b^{6} x^{2} + {\left (6 \, B a b^{5} + A b^{6}\right )} x + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} \log \left (x\right ) - \frac {4 \, A a^{6} + 100 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 50 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 20 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 5 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{20 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.04, size = 149, normalized size = 1.14 \begin {gather*} \frac {10 \, B b^{6} x^{7} - 4 \, A a^{6} + 20 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 60 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} \log \left (x\right ) - 100 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 50 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 20 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 5 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{20 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.52, size = 150, normalized size = 1.15 \begin {gather*} \frac {B b^{6} x^{2}}{2} + 3 a b^{4} \cdot \left (2 A b + 5 B a\right ) \log {\left (x \right )} + x \left (A b^{6} + 6 B a b^{5}\right ) + \frac {- 4 A a^{6} + x^{4} \left (- 300 A a^{2} b^{4} - 400 B a^{3} b^{3}\right ) + x^{3} \left (- 200 A a^{3} b^{3} - 150 B a^{4} b^{2}\right ) + x^{2} \left (- 100 A a^{4} b^{2} - 40 B a^{5} b\right ) + x \left (- 30 A a^{5} b - 5 B a^{6}\right )}{20 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, size = 144, normalized size = 1.10 \begin {gather*} \frac {1}{2} \, B b^{6} x^{2} + 6 \, B a b^{5} x + A b^{6} x + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} \log \left ({\left | x \right |}\right ) - \frac {4 \, A a^{6} + 100 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 50 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 20 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 5 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{20 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.09, size = 140, normalized size = 1.07 \begin {gather*} x\,\left (A\,b^6+6\,B\,a\,b^5\right )-\frac {x\,\left (\frac {B\,a^6}{4}+\frac {3\,A\,b\,a^5}{2}\right )+\frac {A\,a^6}{5}+x^2\,\left (2\,B\,a^5\,b+5\,A\,a^4\,b^2\right )+x^3\,\left (\frac {15\,B\,a^4\,b^2}{2}+10\,A\,a^3\,b^3\right )+x^4\,\left (20\,B\,a^3\,b^3+15\,A\,a^2\,b^4\right )}{x^5}+\ln \left (x\right )\,\left (15\,B\,a^2\,b^4+6\,A\,a\,b^5\right )+\frac {B\,b^6\,x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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