Optimal. Leaf size=135 \[ \frac {2 b \log (x) (5 A b-2 a B)}{a^6}-\frac {2 b (5 A b-2 a B) \log (a+b x)}{a^6}+\frac {4 A b-a B}{a^5 x}+\frac {3 b (2 A b-a B)}{a^5 (a+b x)}+\frac {b (3 A b-2 a B)}{2 a^4 (a+b x)^2}-\frac {A}{2 a^4 x^2}+\frac {b (A b-a B)}{3 a^3 (a+b x)^3} \]
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Rubi [A] time = 0.12, antiderivative size = 135, 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} \[ \frac {4 A b-a B}{a^5 x}+\frac {3 b (2 A b-a B)}{a^5 (a+b x)}+\frac {b (3 A b-2 a B)}{2 a^4 (a+b x)^2}+\frac {b (A b-a B)}{3 a^3 (a+b x)^3}+\frac {2 b \log (x) (5 A b-2 a B)}{a^6}-\frac {2 b (5 A b-2 a B) \log (a+b x)}{a^6}-\frac {A}{2 a^4 x^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {A+B x}{x^3 (a+b x)^4} \, dx\\ &=\int \left (\frac {A}{a^4 x^3}+\frac {-4 A b+a B}{a^5 x^2}-\frac {2 b (-5 A b+2 a B)}{a^6 x}+\frac {b^2 (-A b+a B)}{a^3 (a+b x)^4}+\frac {b^2 (-3 A b+2 a B)}{a^4 (a+b x)^3}+\frac {3 b^2 (-2 A b+a B)}{a^5 (a+b x)^2}+\frac {2 b^2 (-5 A b+2 a B)}{a^6 (a+b x)}\right ) \, dx\\ &=-\frac {A}{2 a^4 x^2}+\frac {4 A b-a B}{a^5 x}+\frac {b (A b-a B)}{3 a^3 (a+b x)^3}+\frac {b (3 A b-2 a B)}{2 a^4 (a+b x)^2}+\frac {3 b (2 A b-a B)}{a^5 (a+b x)}+\frac {2 b (5 A b-2 a B) \log (x)}{a^6}-\frac {2 b (5 A b-2 a B) \log (a+b x)}{a^6}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 123, normalized size = 0.91 \[ \frac {\frac {a \left (-3 a^4 (A+2 B x)+a^3 b x (15 A-44 B x)+10 a^2 b^2 x^2 (11 A-6 B x)+6 a b^3 x^3 (25 A-4 B x)+60 A b^4 x^4\right )}{x^2 (a+b x)^3}+12 b \log (x) (5 A b-2 a B)+12 b (2 a B-5 A b) \log (a+b x)}{6 a^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 318, normalized size = 2.36 \[ -\frac {3 \, A a^{5} + 12 \, {\left (2 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + 30 \, {\left (2 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} + 22 \, {\left (2 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} + 3 \, {\left (2 \, B a^{5} - 5 \, A a^{4} b\right )} x - 12 \, {\left ({\left (2 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 3 \, {\left (2 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + 3 \, {\left (2 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} + {\left (2 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 12 \, {\left ({\left (2 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 3 \, {\left (2 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + 3 \, {\left (2 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} + {\left (2 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2}\right )} \log \relax (x)}{6 \, {\left (a^{6} b^{3} x^{5} + 3 \, a^{7} b^{2} x^{4} + 3 \, a^{8} b x^{3} + a^{9} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 157, normalized size = 1.16 \[ -\frac {2 \, {\left (2 \, B a b - 5 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac {2 \, {\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{6} b} - \frac {3 \, A a^{5} + 12 \, {\left (2 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + 30 \, {\left (2 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} + 22 \, {\left (2 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} + 3 \, {\left (2 \, B a^{5} - 5 \, A a^{4} b\right )} x}{6 \, {\left (b x + a\right )}^{3} a^{6} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 168, normalized size = 1.24 \[ \frac {A \,b^{2}}{3 \left (b x +a \right )^{3} a^{3}}-\frac {B b}{3 \left (b x +a \right )^{3} a^{2}}+\frac {3 A \,b^{2}}{2 \left (b x +a \right )^{2} a^{4}}-\frac {B b}{\left (b x +a \right )^{2} a^{3}}+\frac {6 A \,b^{2}}{\left (b x +a \right ) a^{5}}+\frac {10 A \,b^{2} \ln \relax (x )}{a^{6}}-\frac {10 A \,b^{2} \ln \left (b x +a \right )}{a^{6}}-\frac {3 B b}{\left (b x +a \right ) a^{4}}-\frac {4 B b \ln \relax (x )}{a^{5}}+\frac {4 B b \ln \left (b x +a \right )}{a^{5}}+\frac {4 A b}{a^{5} x}-\frac {B}{a^{4} x}-\frac {A}{2 a^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 172, normalized size = 1.27 \[ -\frac {3 \, A a^{4} + 12 \, {\left (2 \, B a b^{3} - 5 \, A b^{4}\right )} x^{4} + 30 \, {\left (2 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{3} + 22 \, {\left (2 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{2} + 3 \, {\left (2 \, B a^{4} - 5 \, A a^{3} b\right )} x}{6 \, {\left (a^{5} b^{3} x^{5} + 3 \, a^{6} b^{2} x^{4} + 3 \, a^{7} b x^{3} + a^{8} x^{2}\right )}} + \frac {2 \, {\left (2 \, B a b - 5 \, A b^{2}\right )} \log \left (b x + a\right )}{a^{6}} - \frac {2 \, {\left (2 \, B a b - 5 \, A b^{2}\right )} \log \relax (x)}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 168, normalized size = 1.24 \[ \frac {\frac {x\,\left (5\,A\,b-2\,B\,a\right )}{2\,a^2}-\frac {A}{2\,a}+\frac {5\,b^2\,x^3\,\left (5\,A\,b-2\,B\,a\right )}{a^4}+\frac {2\,b^3\,x^4\,\left (5\,A\,b-2\,B\,a\right )}{a^5}+\frac {11\,b\,x^2\,\left (5\,A\,b-2\,B\,a\right )}{3\,a^3}}{a^3\,x^2+3\,a^2\,b\,x^3+3\,a\,b^2\,x^4+b^3\,x^5}-\frac {4\,b\,\mathrm {atanh}\left (\frac {2\,b\,\left (5\,A\,b-2\,B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (10\,A\,b^2-4\,B\,a\,b\right )}\right )\,\left (5\,A\,b-2\,B\,a\right )}{a^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.88, size = 264, normalized size = 1.96 \[ \frac {- 3 A a^{4} + x^{4} \left (60 A b^{4} - 24 B a b^{3}\right ) + x^{3} \left (150 A a b^{3} - 60 B a^{2} b^{2}\right ) + x^{2} \left (110 A a^{2} b^{2} - 44 B a^{3} b\right ) + x \left (15 A a^{3} b - 6 B a^{4}\right )}{6 a^{8} x^{2} + 18 a^{7} b x^{3} + 18 a^{6} b^{2} x^{4} + 6 a^{5} b^{3} x^{5}} - \frac {2 b \left (- 5 A b + 2 B a\right ) \log {\left (x + \frac {- 10 A a b^{2} + 4 B a^{2} b - 2 a b \left (- 5 A b + 2 B a\right )}{- 20 A b^{3} + 8 B a b^{2}} \right )}}{a^{6}} + \frac {2 b \left (- 5 A b + 2 B a\right ) \log {\left (x + \frac {- 10 A a b^{2} + 4 B a^{2} b + 2 a b \left (- 5 A b + 2 B a\right )}{- 20 A b^{3} + 8 B a b^{2}} \right )}}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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