Optimal. Leaf size=102 \[ -\frac {A \log (a+b x)}{a^6}+\frac {A \log (x)}{a^6}+\frac {A}{a^5 (a+b x)}+\frac {A}{2 a^4 (a+b x)^2}+\frac {A}{3 a^3 (a+b x)^3}+\frac {A}{4 a^2 (a+b x)^4}+\frac {A b-a B}{5 a b (a+b x)^5} \]
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Rubi [A] time = 0.07, antiderivative size = 102, 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} \[ \frac {A}{a^5 (a+b x)}+\frac {A}{2 a^4 (a+b x)^2}+\frac {A}{3 a^3 (a+b x)^3}+\frac {A}{4 a^2 (a+b x)^4}-\frac {A \log (a+b x)}{a^6}+\frac {A \log (x)}{a^6}+\frac {A b-a B}{5 a b (a+b x)^5} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{x \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {A+B x}{x (a+b x)^6} \, dx\\ &=\int \left (\frac {A}{a^6 x}+\frac {-A b+a B}{a (a+b x)^6}-\frac {A b}{a^2 (a+b x)^5}-\frac {A b}{a^3 (a+b x)^4}-\frac {A b}{a^4 (a+b x)^3}-\frac {A b}{a^5 (a+b x)^2}-\frac {A b}{a^6 (a+b x)}\right ) \, dx\\ &=\frac {A b-a B}{5 a b (a+b x)^5}+\frac {A}{4 a^2 (a+b x)^4}+\frac {A}{3 a^3 (a+b x)^3}+\frac {A}{2 a^4 (a+b x)^2}+\frac {A}{a^5 (a+b x)}+\frac {A \log (x)}{a^6}-\frac {A \log (a+b x)}{a^6}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 89, normalized size = 0.87 \[ \frac {\frac {a \left (-12 a^5 B+137 a^4 A b+385 a^3 A b^2 x+470 a^2 A b^3 x^2+270 a A b^4 x^3+60 A b^5 x^4\right )}{b (a+b x)^5}-60 A \log (a+b x)+60 A \log (x)}{60 a^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 250, normalized size = 2.45 \[ \frac {60 \, A a b^{5} x^{4} + 270 \, A a^{2} b^{4} x^{3} + 470 \, A a^{3} b^{3} x^{2} + 385 \, A a^{4} b^{2} x - 12 \, B a^{6} + 137 \, A a^{5} b - 60 \, {\left (A b^{6} x^{5} + 5 \, A a b^{5} x^{4} + 10 \, A a^{2} b^{4} x^{3} + 10 \, A a^{3} b^{3} x^{2} + 5 \, A a^{4} b^{2} x + A a^{5} b\right )} \log \left (b x + a\right ) + 60 \, {\left (A b^{6} x^{5} + 5 \, A a b^{5} x^{4} + 10 \, A a^{2} b^{4} x^{3} + 10 \, A a^{3} b^{3} x^{2} + 5 \, A a^{4} b^{2} x + A a^{5} b\right )} \log \relax (x)}{60 \, {\left (a^{6} b^{6} x^{5} + 5 \, a^{7} b^{5} x^{4} + 10 \, a^{8} b^{4} x^{3} + 10 \, a^{9} b^{3} x^{2} + 5 \, a^{10} b^{2} x + a^{11} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 95, normalized size = 0.93 \[ -\frac {A \log \left ({\left | b x + a \right |}\right )}{a^{6}} + \frac {A \log \left ({\left | x \right |}\right )}{a^{6}} + \frac {60 \, A a b^{5} x^{4} + 270 \, A a^{2} b^{4} x^{3} + 470 \, A a^{3} b^{3} x^{2} + 385 \, A a^{4} b^{2} x - 12 \, B a^{6} + 137 \, A a^{5} b}{60 \, {\left (b x + a\right )}^{5} a^{6} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 98, normalized size = 0.96 \[ \frac {A}{5 \left (b x +a \right )^{5} a}-\frac {B}{5 \left (b x +a \right )^{5} b}+\frac {A}{4 \left (b x +a \right )^{4} a^{2}}+\frac {A}{3 \left (b x +a \right )^{3} a^{3}}+\frac {A}{2 \left (b x +a \right )^{2} a^{4}}+\frac {A}{\left (b x +a \right ) a^{5}}+\frac {A \ln \relax (x )}{a^{6}}-\frac {A \ln \left (b x +a \right )}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 137, normalized size = 1.34 \[ \frac {60 \, A b^{5} x^{4} + 270 \, A a b^{4} x^{3} + 470 \, A a^{2} b^{3} x^{2} + 385 \, A a^{3} b^{2} x - 12 \, B a^{5} + 137 \, A a^{4} b}{60 \, {\left (a^{5} b^{6} x^{5} + 5 \, a^{6} b^{5} x^{4} + 10 \, a^{7} b^{4} x^{3} + 10 \, a^{8} b^{3} x^{2} + 5 \, a^{9} b^{2} x + a^{10} b\right )}} - \frac {A \log \left (b x + a\right )}{a^{6}} + \frac {A \log \relax (x)}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 130, normalized size = 1.27 \[ \frac {\frac {137\,A\,b-12\,B\,a}{60\,a\,b}+\frac {77\,A\,b\,x}{12\,a^2}+\frac {47\,A\,b^2\,x^2}{6\,a^3}+\frac {9\,A\,b^3\,x^3}{2\,a^4}+\frac {A\,b^4\,x^4}{a^5}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5}-\frac {2\,A\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.77, size = 141, normalized size = 1.38 \[ \frac {A \left (\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{6}} + \frac {137 A a^{4} b + 385 A a^{3} b^{2} x + 470 A a^{2} b^{3} x^{2} + 270 A a b^{4} x^{3} + 60 A b^{5} x^{4} - 12 B a^{5}}{60 a^{10} b + 300 a^{9} b^{2} x + 600 a^{8} b^{3} x^{2} + 600 a^{7} b^{4} x^{3} + 300 a^{6} b^{5} x^{4} + 60 a^{5} b^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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