Optimal. Leaf size=86 \[ -\frac {a^4 A}{4 x^4}-\frac {a^3 (a B+4 A b)}{3 x^3}-\frac {a^2 b (2 a B+3 A b)}{x^2}+b^3 \log (x) (4 a B+A b)-\frac {2 a b^2 (3 a B+2 A b)}{x}+b^4 B x \]
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Rubi [A] time = 0.05, antiderivative size = 86, 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 {a^3 (a B+4 A b)}{3 x^3}-\frac {a^2 b (2 a B+3 A b)}{x^2}-\frac {a^4 A}{4 x^4}-\frac {2 a b^2 (3 a B+2 A b)}{x}+b^3 \log (x) (4 a B+A b)+b^4 B x \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 )^2}{x^5} \, dx &=\int \frac {(a+b x)^4 (A+B x)}{x^5} \, dx\\ &=\int \left (b^4 B+\frac {a^4 A}{x^5}+\frac {a^3 (4 A b+a B)}{x^4}+\frac {2 a^2 b (3 A b+2 a B)}{x^3}+\frac {2 a b^2 (2 A b+3 a B)}{x^2}+\frac {b^3 (A b+4 a B)}{x}\right ) \, dx\\ &=-\frac {a^4 A}{4 x^4}-\frac {a^3 (4 A b+a B)}{3 x^3}-\frac {a^2 b (3 A b+2 a B)}{x^2}-\frac {2 a b^2 (2 A b+3 a B)}{x}+b^4 B x+b^3 (A b+4 a B) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 85, normalized size = 0.99 \begin {gather*} -\frac {a^4 (3 A+4 B x)}{12 x^4}-\frac {2 a^3 b (2 A+3 B x)}{3 x^3}-\frac {3 a^2 b^2 (A+2 B x)}{x^2}+b^3 \log (x) (4 a B+A b)-\frac {4 a A b^3}{x}+b^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 )^2}{x^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 101, normalized size = 1.17 \begin {gather*} \frac {12 \, B b^{4} x^{5} + 12 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} \log \relax (x) - 3 \, A a^{4} - 24 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} - 12 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} - 4 \, {\left (B a^{4} + 4 \, A a^{3} 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 = 0.17, size = 96, normalized size = 1.12 \begin {gather*} B b^{4} x + {\left (4 \, B a b^{3} + A b^{4}\right )} \log \left ({\left | x \right |}\right ) - \frac {3 \, A a^{4} + 24 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 12 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 4 \, {\left (B a^{4} + 4 \, A a^{3} 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 = 0.05, size = 96, normalized size = 1.12 \begin {gather*} A \,b^{4} \ln \relax (x )+4 B a \,b^{3} \ln \relax (x )+B \,b^{4} x -\frac {4 A a \,b^{3}}{x}-\frac {6 B \,a^{2} b^{2}}{x}-\frac {3 A \,a^{2} b^{2}}{x^{2}}-\frac {2 B \,a^{3} b}{x^{2}}-\frac {4 A \,a^{3} b}{3 x^{3}}-\frac {B \,a^{4}}{3 x^{3}}-\frac {A \,a^{4}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 95, normalized size = 1.10 \begin {gather*} B b^{4} x + {\left (4 \, B a b^{3} + A b^{4}\right )} \log \relax (x) - \frac {3 \, A a^{4} + 24 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 12 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 4 \, {\left (B a^{4} + 4 \, A a^{3} 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 = 1.09, size = 93, normalized size = 1.08 \begin {gather*} \ln \relax (x)\,\left (A\,b^4+4\,B\,a\,b^3\right )-\frac {x\,\left (\frac {B\,a^4}{3}+\frac {4\,A\,b\,a^3}{3}\right )+\frac {A\,a^4}{4}+x^2\,\left (2\,B\,a^3\,b+3\,A\,a^2\,b^2\right )+x^3\,\left (6\,B\,a^2\,b^2+4\,A\,a\,b^3\right )}{x^4}+B\,b^4\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.22, size = 99, normalized size = 1.15 \begin {gather*} B b^{4} x + b^{3} \left (A b + 4 B a\right ) \log {\relax (x )} + \frac {- 3 A a^{4} + x^{3} \left (- 48 A a b^{3} - 72 B a^{2} b^{2}\right ) + x^{2} \left (- 36 A a^{2} b^{2} - 24 B a^{3} b\right ) + x \left (- 16 A a^{3} b - 4 B a^{4}\right )}{12 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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