Optimal. Leaf size=133 \[ \frac {b^3 \log (x) (5 A b-4 a B)}{a^6}-\frac {b^3 (5 A b-4 a B) \log (a+b x)}{a^6}+\frac {b^3 (A b-a B)}{a^5 (a+b x)}+\frac {b^2 (4 A b-3 a B)}{a^5 x}-\frac {b (3 A b-2 a B)}{2 a^4 x^2}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {A}{4 a^2 x^4} \]
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Rubi [A] time = 0.11, antiderivative size = 133, 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 {b^3 (A b-a B)}{a^5 (a+b x)}+\frac {b^2 (4 A b-3 a B)}{a^5 x}+\frac {b^3 \log (x) (5 A b-4 a B)}{a^6}-\frac {b^3 (5 A b-4 a B) \log (a+b x)}{a^6}-\frac {b (3 A b-2 a B)}{2 a^4 x^2}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {A}{4 a^2 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{x^5 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac {A+B x}{x^5 (a+b x)^2} \, dx\\ &=\int \left (\frac {A}{a^2 x^5}+\frac {-2 A b+a B}{a^3 x^4}-\frac {b (-3 A b+2 a B)}{a^4 x^3}+\frac {b^2 (-4 A b+3 a B)}{a^5 x^2}-\frac {b^3 (-5 A b+4 a B)}{a^6 x}+\frac {b^4 (-A b+a B)}{a^5 (a+b x)^2}+\frac {b^4 (-5 A b+4 a B)}{a^6 (a+b x)}\right ) \, dx\\ &=-\frac {A}{4 a^2 x^4}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {b (3 A b-2 a B)}{2 a^4 x^2}+\frac {b^2 (4 A b-3 a B)}{a^5 x}+\frac {b^3 (A b-a B)}{a^5 (a+b x)}+\frac {b^3 (5 A b-4 a B) \log (x)}{a^6}-\frac {b^3 (5 A b-4 a B) \log (a+b x)}{a^6}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 129, normalized size = 0.97 \begin {gather*} \frac {\frac {a \left (-\left (a^4 (3 A+4 B x)\right )+a^3 b x (5 A+8 B x)-2 a^2 b^2 x^2 (5 A+12 B x)+6 a b^3 x^3 (5 A-8 B x)+60 A b^4 x^4\right )}{x^4 (a+b x)}+12 b^3 \log (x) (5 A b-4 a B)+12 b^3 (4 a B-5 A b) \log (a+b x)}{12 a^6} \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}{x^5 \left (a^2+2 a b x+b^2 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 203, normalized size = 1.53 \begin {gather*} -\frac {3 \, A a^{5} + 12 \, {\left (4 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + 6 \, {\left (4 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 2 \, {\left (4 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} + {\left (4 \, B a^{5} - 5 \, A a^{4} b\right )} x - 12 \, {\left ({\left (4 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + {\left (4 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4}\right )} \log \left (b x + a\right ) + 12 \, {\left ({\left (4 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + {\left (4 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4}\right )} \log \relax (x)}{12 \, {\left (a^{6} b x^{5} + a^{7} x^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 157, normalized size = 1.18 \begin {gather*} -\frac {{\left (4 \, B a b^{3} - 5 \, A b^{4}\right )} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac {{\left (4 \, B a b^{4} - 5 \, A b^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{6} b} - \frac {3 \, A a^{5} + 12 \, {\left (4 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + 6 \, {\left (4 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 2 \, {\left (4 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} + {\left (4 \, B a^{5} - 5 \, A a^{4} b\right )} x}{12 \, {\left (b x + a\right )} a^{6} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 158, normalized size = 1.19 \begin {gather*} \frac {A \,b^{4}}{\left (b x +a \right ) a^{5}}+\frac {5 A \,b^{4} \ln \relax (x )}{a^{6}}-\frac {5 A \,b^{4} \ln \left (b x +a \right )}{a^{6}}-\frac {B \,b^{3}}{\left (b x +a \right ) a^{4}}-\frac {4 B \,b^{3} \ln \relax (x )}{a^{5}}+\frac {4 B \,b^{3} \ln \left (b x +a \right )}{a^{5}}+\frac {4 A \,b^{3}}{a^{5} x}-\frac {3 B \,b^{2}}{a^{4} x}-\frac {3 A \,b^{2}}{2 a^{4} x^{2}}+\frac {B b}{a^{3} x^{2}}+\frac {2 A b}{3 a^{3} x^{3}}-\frac {B}{3 a^{2} x^{3}}-\frac {A}{4 a^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 152, normalized size = 1.14 \begin {gather*} -\frac {3 \, A a^{4} + 12 \, {\left (4 \, B a b^{3} - 5 \, A b^{4}\right )} x^{4} + 6 \, {\left (4 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{3} - 2 \, {\left (4 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{2} + {\left (4 \, B a^{4} - 5 \, A a^{3} b\right )} x}{12 \, {\left (a^{5} b x^{5} + a^{6} x^{4}\right )}} + \frac {{\left (4 \, B a b^{3} - 5 \, A b^{4}\right )} \log \left (b x + a\right )}{a^{6}} - \frac {{\left (4 \, B a b^{3} - 5 \, A b^{4}\right )} \log \relax (x)}{a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 150, normalized size = 1.13 \begin {gather*} \frac {\frac {x\,\left (5\,A\,b-4\,B\,a\right )}{12\,a^2}-\frac {A}{4\,a}+\frac {b^2\,x^3\,\left (5\,A\,b-4\,B\,a\right )}{2\,a^4}+\frac {b^3\,x^4\,\left (5\,A\,b-4\,B\,a\right )}{a^5}-\frac {b\,x^2\,\left (5\,A\,b-4\,B\,a\right )}{6\,a^3}}{b\,x^5+a\,x^4}-\frac {2\,b^3\,\mathrm {atanh}\left (\frac {b^3\,\left (5\,A\,b-4\,B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (5\,A\,b^4-4\,B\,a\,b^3\right )}\right )\,\left (5\,A\,b-4\,B\,a\right )}{a^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.73, size = 243, normalized size = 1.83 \begin {gather*} \frac {- 3 A a^{4} + x^{4} \left (60 A b^{4} - 48 B a b^{3}\right ) + x^{3} \left (30 A a b^{3} - 24 B a^{2} b^{2}\right ) + x^{2} \left (- 10 A a^{2} b^{2} + 8 B a^{3} b\right ) + x \left (5 A a^{3} b - 4 B a^{4}\right )}{12 a^{6} x^{4} + 12 a^{5} b x^{5}} - \frac {b^{3} \left (- 5 A b + 4 B a\right ) \log {\left (x + \frac {- 5 A a b^{4} + 4 B a^{2} b^{3} - a b^{3} \left (- 5 A b + 4 B a\right )}{- 10 A b^{5} + 8 B a b^{4}} \right )}}{a^{6}} + \frac {b^{3} \left (- 5 A b + 4 B a\right ) \log {\left (x + \frac {- 5 A a b^{4} + 4 B a^{2} b^{3} + a b^{3} \left (- 5 A b + 4 B a\right )}{- 10 A b^{5} + 8 B a b^{4}} \right )}}{a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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