Optimal. Leaf size=134 \[ -\frac {a^6 A}{4 x^4}-\frac {a^5 (a B+6 A b)}{3 x^3}-\frac {3 a^4 b (2 a B+5 A b)}{2 x^2}-\frac {5 a^3 b^2 (3 a B+4 A b)}{x}+5 a^2 b^3 \log (x) (4 a B+3 A b)+\frac {1}{2} b^5 x^2 (6 a B+A b)+3 a b^4 x (5 a B+2 A b)+\frac {1}{3} b^6 B x^3 \]
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Rubi [A] time = 0.08, antiderivative size = 134, 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, 76} \begin {gather*} -\frac {5 a^3 b^2 (3 a B+4 A b)}{x}+5 a^2 b^3 \log (x) (4 a B+3 A b)-\frac {a^5 (a B+6 A b)}{3 x^3}-\frac {3 a^4 b (2 a B+5 A b)}{2 x^2}-\frac {a^6 A}{4 x^4}+\frac {1}{2} b^5 x^2 (6 a B+A b)+3 a b^4 x (5 a B+2 A b)+\frac {1}{3} b^6 B x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^5} \, dx &=\int \frac {(a+b x)^6 (A+B x)}{x^5} \, dx\\ &=\int \left (3 a b^4 (2 A b+5 a B)+\frac {a^6 A}{x^5}+\frac {a^5 (6 A b+a B)}{x^4}+\frac {3 a^4 b (5 A b+2 a B)}{x^3}+\frac {5 a^3 b^2 (4 A b+3 a B)}{x^2}+\frac {5 a^2 b^3 (3 A b+4 a B)}{x}+b^5 (A b+6 a B) x+b^6 B x^2\right ) \, dx\\ &=-\frac {a^6 A}{4 x^4}-\frac {a^5 (6 A b+a B)}{3 x^3}-\frac {3 a^4 b (5 A b+2 a B)}{2 x^2}-\frac {5 a^3 b^2 (4 A b+3 a B)}{x}+3 a b^4 (2 A b+5 a B) x+\frac {1}{2} b^5 (A b+6 a B) x^2+\frac {1}{3} b^6 B x^3+5 a^2 b^3 (3 A b+4 a B) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 128, normalized size = 0.96 \begin {gather*} -\frac {a^6 (3 A+4 B x)}{12 x^4}-\frac {a^5 b (2 A+3 B x)}{x^3}-\frac {15 a^4 b^2 (A+2 B x)}{2 x^2}-\frac {20 a^3 A b^3}{x}+5 a^2 b^3 \log (x) (4 a B+3 A b)+15 a^2 b^4 B x+3 a b^5 x (2 A+B x)+\frac {1}{6} b^6 x^2 (3 A+2 B x) \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 {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 149, normalized size = 1.11 \begin {gather*} \frac {4 \, B b^{6} x^{7} - 3 \, A a^{6} + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 36 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 60 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} \log \relax (x) - 60 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 18 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 4 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 145, normalized size = 1.08 \begin {gather*} \frac {1}{3} \, B b^{6} x^{3} + 3 \, B a b^{5} x^{2} + \frac {1}{2} \, A b^{6} x^{2} + 15 \, B a^{2} b^{4} x + 6 \, A a b^{5} x + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} \log \left ({\left | x \right |}\right ) - \frac {3 \, A a^{6} + 60 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 18 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 4 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 144, normalized size = 1.07 \begin {gather*} \frac {B \,b^{6} x^{3}}{3}+\frac {A \,b^{6} x^{2}}{2}+3 B a \,b^{5} x^{2}+15 A \,a^{2} b^{4} \ln \relax (x )+6 A a \,b^{5} x +20 B \,a^{3} b^{3} \ln \relax (x )+15 B \,a^{2} b^{4} x -\frac {20 A \,a^{3} b^{3}}{x}-\frac {15 B \,a^{4} b^{2}}{x}-\frac {15 A \,a^{4} b^{2}}{2 x^{2}}-\frac {3 B \,a^{5} b}{x^{2}}-\frac {2 A \,a^{5} b}{x^{3}}-\frac {B \,a^{6}}{3 x^{3}}-\frac {A \,a^{6}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 145, normalized size = 1.08 \begin {gather*} \frac {1}{3} \, B b^{6} x^{3} + \frac {1}{2} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{2} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} \log \relax (x) - \frac {3 \, A a^{6} + 60 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 18 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 4 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 138, normalized size = 1.03 \begin {gather*} x^2\,\left (\frac {A\,b^6}{2}+3\,B\,a\,b^5\right )-\frac {x\,\left (\frac {B\,a^6}{3}+2\,A\,b\,a^5\right )+\frac {A\,a^6}{4}+x^2\,\left (3\,B\,a^5\,b+\frac {15\,A\,a^4\,b^2}{2}\right )+x^3\,\left (15\,B\,a^4\,b^2+20\,A\,a^3\,b^3\right )}{x^4}+\ln \relax (x)\,\left (20\,B\,a^3\,b^3+15\,A\,a^2\,b^4\right )+\frac {B\,b^6\,x^3}{3}+3\,a\,b^4\,x\,\left (2\,A\,b+5\,B\,a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.35, size = 150, normalized size = 1.12 \begin {gather*} \frac {B b^{6} x^{3}}{3} + 5 a^{2} b^{3} \left (3 A b + 4 B a\right ) \log {\relax (x )} + x^{2} \left (\frac {A b^{6}}{2} + 3 B a b^{5}\right ) + x \left (6 A a b^{5} + 15 B a^{2} b^{4}\right ) + \frac {- 3 A a^{6} + x^{3} \left (- 240 A a^{3} b^{3} - 180 B a^{4} b^{2}\right ) + x^{2} \left (- 90 A a^{4} b^{2} - 36 B a^{5} b\right ) + x \left (- 24 A a^{5} b - 4 B a^{6}\right )}{12 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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