Optimal. Leaf size=131 \[ -\frac {a^6 A}{2 x^2}-\frac {a^5 (a B+6 A b)}{x}+3 a^4 b \log (x) (2 a B+5 A b)+5 a^3 b^2 x (3 a B+4 A b)+\frac {5}{2} a^2 b^3 x^2 (4 a B+3 A b)+\frac {1}{4} b^5 x^4 (6 a B+A b)+a b^4 x^3 (5 a B+2 A b)+\frac {1}{5} b^6 B x^5 \]
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Rubi [A] time = 0.08, 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, 76} \begin {gather*} \frac {5}{2} a^2 b^3 x^2 (4 a B+3 A b)+5 a^3 b^2 x (3 a B+4 A b)-\frac {a^5 (a B+6 A b)}{x}+3 a^4 b \log (x) (2 a B+5 A b)-\frac {a^6 A}{2 x^2}+a b^4 x^3 (5 a B+2 A b)+\frac {1}{4} b^5 x^4 (6 a B+A b)+\frac {1}{5} b^6 B x^5 \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^3} \, dx &=\int \frac {(a+b x)^6 (A+B x)}{x^3} \, dx\\ &=\int \left (5 a^3 b^2 (4 A b+3 a B)+\frac {a^6 A}{x^3}+\frac {a^5 (6 A b+a B)}{x^2}+\frac {3 a^4 b (5 A b+2 a B)}{x}+5 a^2 b^3 (3 A b+4 a B) x+3 a b^4 (2 A b+5 a B) x^2+b^5 (A b+6 a B) x^3+b^6 B x^4\right ) \, dx\\ &=-\frac {a^6 A}{2 x^2}-\frac {a^5 (6 A b+a B)}{x}+5 a^3 b^2 (4 A b+3 a B) x+\frac {5}{2} a^2 b^3 (3 A b+4 a B) x^2+a b^4 (2 A b+5 a B) x^3+\frac {1}{4} b^5 (A b+6 a B) x^4+\frac {1}{5} b^6 B x^5+3 a^4 b (5 A b+2 a B) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.06, size = 128, normalized size = 0.98 \begin {gather*} -\frac {a^6 (A+2 B x)}{2 x^2}-\frac {6 a^5 A b}{x}+3 a^4 b \log (x) (2 a B+5 A b)+15 a^4 b^2 B x+10 a^3 b^3 x (2 A+B x)+\frac {5}{2} a^2 b^4 x^2 (3 A+2 B x)+\frac {1}{2} a b^5 x^3 (4 A+3 B x)+\frac {1}{20} b^6 x^4 (5 A+4 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^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 149, normalized size = 1.14 \begin {gather*} \frac {4 \, B b^{6} x^{7} - 10 \, A a^{6} + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 20 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 50 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 100 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 60 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} \log \relax (x) - 20 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{20 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 144, normalized size = 1.10 \begin {gather*} \frac {1}{5} \, B b^{6} x^{5} + \frac {3}{2} \, B a b^{5} x^{4} + \frac {1}{4} \, A b^{6} x^{4} + 5 \, B a^{2} b^{4} x^{3} + 2 \, A a b^{5} x^{3} + 10 \, B a^{3} b^{3} x^{2} + \frac {15}{2} \, A a^{2} b^{4} x^{2} + 15 \, B a^{4} b^{2} x + 20 \, A a^{3} b^{3} x + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac {A a^{6} + 2 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{2 \, x^{2}} \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 = 1.10 \begin {gather*} \frac {B \,b^{6} x^{5}}{5}+\frac {A \,b^{6} x^{4}}{4}+\frac {3 B a \,b^{5} x^{4}}{2}+2 A a \,b^{5} x^{3}+5 B \,a^{2} b^{4} x^{3}+\frac {15 A \,a^{2} b^{4} x^{2}}{2}+10 B \,a^{3} b^{3} x^{2}+15 A \,a^{4} b^{2} \ln \relax (x )+20 A \,a^{3} b^{3} x +6 B \,a^{5} b \ln \relax (x )+15 B \,a^{4} b^{2} x -\frac {6 A \,a^{5} b}{x}-\frac {B \,a^{6}}{x}-\frac {A \,a^{6}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 143, normalized size = 1.09 \begin {gather*} \frac {1}{5} \, B b^{6} x^{5} + \frac {1}{4} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{4} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{3} + \frac {5}{2} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{2} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} \log \relax (x) - \frac {A a^{6} + 2 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 130, normalized size = 0.99 \begin {gather*} \ln \relax (x)\,\left (6\,B\,a^5\,b+15\,A\,a^4\,b^2\right )-\frac {x\,\left (B\,a^6+6\,A\,b\,a^5\right )+\frac {A\,a^6}{2}}{x^2}+x^4\,\left (\frac {A\,b^6}{4}+\frac {3\,B\,a\,b^5}{2}\right )+\frac {B\,b^6\,x^5}{5}+\frac {5\,a^2\,b^3\,x^2\,\left (3\,A\,b+4\,B\,a\right )}{2}+5\,a^3\,b^2\,x\,\left (4\,A\,b+3\,B\,a\right )+a\,b^4\,x^3\,\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 = 0.51, size = 148, normalized size = 1.13 \begin {gather*} \frac {B b^{6} x^{5}}{5} + 3 a^{4} b \left (5 A b + 2 B a\right ) \log {\relax (x )} + x^{4} \left (\frac {A b^{6}}{4} + \frac {3 B a b^{5}}{2}\right ) + x^{3} \left (2 A a b^{5} + 5 B a^{2} b^{4}\right ) + x^{2} \left (\frac {15 A a^{2} b^{4}}{2} + 10 B a^{3} b^{3}\right ) + x \left (20 A a^{3} b^{3} + 15 B a^{4} b^{2}\right ) + \frac {- A a^{6} + x \left (- 12 A a^{5} b - 2 B a^{6}\right )}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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