Optimal. Leaf size=157 \[ -\frac {\log (x) (6 A b-a B)}{a^7}+\frac {(6 A b-a B) \log (a+b x)}{a^7}-\frac {5 A b-a B}{a^6 (a+b x)}-\frac {A}{a^6 x}-\frac {4 A b-a B}{2 a^5 (a+b x)^2}-\frac {3 A b-a B}{3 a^4 (a+b x)^3}-\frac {2 A b-a B}{4 a^3 (a+b x)^4}-\frac {A b-a B}{5 a^2 (a+b x)^5} \]
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Rubi [A] time = 0.16, antiderivative size = 157, 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 {5 A b-a B}{a^6 (a+b x)}-\frac {4 A b-a B}{2 a^5 (a+b x)^2}-\frac {3 A b-a B}{3 a^4 (a+b x)^3}-\frac {2 A b-a B}{4 a^3 (a+b x)^4}-\frac {A b-a B}{5 a^2 (a+b x)^5}-\frac {\log (x) (6 A b-a B)}{a^7}+\frac {(6 A b-a B) \log (a+b x)}{a^7}-\frac {A}{a^6 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {A+B x}{x^2 (a+b x)^6} \, dx\\ &=\int \left (\frac {A}{a^6 x^2}+\frac {-6 A b+a B}{a^7 x}-\frac {b (-A b+a B)}{a^2 (a+b x)^6}-\frac {b (-2 A b+a B)}{a^3 (a+b x)^5}-\frac {b (-3 A b+a B)}{a^4 (a+b x)^4}-\frac {b (-4 A b+a B)}{a^5 (a+b x)^3}-\frac {b (-5 A b+a B)}{a^6 (a+b x)^2}-\frac {b (-6 A b+a B)}{a^7 (a+b x)}\right ) \, dx\\ &=-\frac {A}{a^6 x}-\frac {A b-a B}{5 a^2 (a+b x)^5}-\frac {2 A b-a B}{4 a^3 (a+b x)^4}-\frac {3 A b-a B}{3 a^4 (a+b x)^3}-\frac {4 A b-a B}{2 a^5 (a+b x)^2}-\frac {5 A b-a B}{a^6 (a+b x)}-\frac {(6 A b-a B) \log (x)}{a^7}+\frac {(6 A b-a B) \log (a+b x)}{a^7}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 142, normalized size = 0.90 \begin {gather*} \frac {\frac {12 a^5 (a B-A b)}{(a+b x)^5}+\frac {15 a^4 (a B-2 A b)}{(a+b x)^4}+\frac {20 a^3 (a B-3 A b)}{(a+b x)^3}+\frac {30 a^2 (a B-4 A b)}{(a+b x)^2}+\frac {60 a (a B-5 A b)}{a+b x}+60 \log (x) (a B-6 A b)+60 (6 A b-a B) \log (a+b x)-\frac {60 a A}{x}}{60 a^7} \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^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.43, size = 427, normalized size = 2.72 \begin {gather*} -\frac {60 \, A a^{6} - 60 \, {\left (B a^{2} b^{4} - 6 \, A a b^{5}\right )} x^{5} - 270 \, {\left (B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} x^{4} - 470 \, {\left (B a^{4} b^{2} - 6 \, A a^{3} b^{3}\right )} x^{3} - 385 \, {\left (B a^{5} b - 6 \, A a^{4} b^{2}\right )} x^{2} - 137 \, {\left (B a^{6} - 6 \, A a^{5} b\right )} x + 60 \, {\left ({\left (B a b^{5} - 6 \, A b^{6}\right )} x^{6} + 5 \, {\left (B a^{2} b^{4} - 6 \, A a b^{5}\right )} x^{5} + 10 \, {\left (B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} x^{4} + 10 \, {\left (B a^{4} b^{2} - 6 \, A a^{3} b^{3}\right )} x^{3} + 5 \, {\left (B a^{5} b - 6 \, A a^{4} b^{2}\right )} x^{2} + {\left (B a^{6} - 6 \, A a^{5} b\right )} x\right )} \log \left (b x + a\right ) - 60 \, {\left ({\left (B a b^{5} - 6 \, A b^{6}\right )} x^{6} + 5 \, {\left (B a^{2} b^{4} - 6 \, A a b^{5}\right )} x^{5} + 10 \, {\left (B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} x^{4} + 10 \, {\left (B a^{4} b^{2} - 6 \, A a^{3} b^{3}\right )} x^{3} + 5 \, {\left (B a^{5} b - 6 \, A a^{4} b^{2}\right )} x^{2} + {\left (B a^{6} - 6 \, A a^{5} b\right )} x\right )} \log \relax (x)}{60 \, {\left (a^{7} b^{5} x^{6} + 5 \, a^{8} b^{4} x^{5} + 10 \, a^{9} b^{3} x^{4} + 10 \, a^{10} b^{2} x^{3} + 5 \, a^{11} b x^{2} + a^{12} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 168, normalized size = 1.07 \begin {gather*} \frac {{\left (B a - 6 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{7}} - \frac {{\left (B a b - 6 \, A b^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{7} b} - \frac {60 \, A a^{6} - 60 \, {\left (B a^{2} b^{4} - 6 \, A a b^{5}\right )} x^{5} - 270 \, {\left (B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} x^{4} - 470 \, {\left (B a^{4} b^{2} - 6 \, A a^{3} b^{3}\right )} x^{3} - 385 \, {\left (B a^{5} b - 6 \, A a^{4} b^{2}\right )} x^{2} - 137 \, {\left (B a^{6} - 6 \, A a^{5} b\right )} x}{60 \, {\left (b x + a\right )}^{5} a^{7} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 186, normalized size = 1.18 \begin {gather*} -\frac {A b}{5 \left (b x +a \right )^{5} a^{2}}+\frac {B}{5 \left (b x +a \right )^{5} a}-\frac {A b}{2 \left (b x +a \right )^{4} a^{3}}+\frac {B}{4 \left (b x +a \right )^{4} a^{2}}-\frac {A b}{\left (b x +a \right )^{3} a^{4}}+\frac {B}{3 \left (b x +a \right )^{3} a^{3}}-\frac {2 A b}{\left (b x +a \right )^{2} a^{5}}+\frac {B}{2 \left (b x +a \right )^{2} a^{4}}-\frac {5 A b}{\left (b x +a \right ) a^{6}}-\frac {6 A b \ln \relax (x )}{a^{7}}+\frac {6 A b \ln \left (b x +a \right )}{a^{7}}+\frac {B}{\left (b x +a \right ) a^{5}}+\frac {B \ln \relax (x )}{a^{6}}-\frac {B \ln \left (b x +a \right )}{a^{6}}-\frac {A}{a^{6} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 202, normalized size = 1.29 \begin {gather*} -\frac {60 \, A a^{5} - 60 \, {\left (B a b^{4} - 6 \, A b^{5}\right )} x^{5} - 270 \, {\left (B a^{2} b^{3} - 6 \, A a b^{4}\right )} x^{4} - 470 \, {\left (B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{3} - 385 \, {\left (B a^{4} b - 6 \, A a^{3} b^{2}\right )} x^{2} - 137 \, {\left (B a^{5} - 6 \, A a^{4} b\right )} x}{60 \, {\left (a^{6} b^{5} x^{6} + 5 \, a^{7} b^{4} x^{5} + 10 \, a^{8} b^{3} x^{4} + 10 \, a^{9} b^{2} x^{3} + 5 \, a^{10} b x^{2} + a^{11} x\right )}} - \frac {{\left (B a - 6 \, A b\right )} \log \left (b x + a\right )}{a^{7}} + \frac {{\left (B a - 6 \, A b\right )} \log \relax (x)}{a^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 180, normalized size = 1.15 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )\,\left (6\,A\,b-B\,a\right )}{a^7}-\frac {\frac {A}{a}+\frac {137\,x\,\left (6\,A\,b-B\,a\right )}{60\,a^2}+\frac {47\,b^2\,x^3\,\left (6\,A\,b-B\,a\right )}{6\,a^4}+\frac {9\,b^3\,x^4\,\left (6\,A\,b-B\,a\right )}{2\,a^5}+\frac {b^4\,x^5\,\left (6\,A\,b-B\,a\right )}{a^6}+\frac {77\,b\,x^2\,\left (6\,A\,b-B\,a\right )}{12\,a^3}}{a^5\,x+5\,a^4\,b\,x^2+10\,a^3\,b^2\,x^3+10\,a^2\,b^3\,x^4+5\,a\,b^4\,x^5+b^5\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.06, size = 275, normalized size = 1.75 \begin {gather*} \frac {- 60 A a^{5} + x^{5} \left (- 360 A b^{5} + 60 B a b^{4}\right ) + x^{4} \left (- 1620 A a b^{4} + 270 B a^{2} b^{3}\right ) + x^{3} \left (- 2820 A a^{2} b^{3} + 470 B a^{3} b^{2}\right ) + x^{2} \left (- 2310 A a^{3} b^{2} + 385 B a^{4} b\right ) + x \left (- 822 A a^{4} b + 137 B a^{5}\right )}{60 a^{11} x + 300 a^{10} b x^{2} + 600 a^{9} b^{2} x^{3} + 600 a^{8} b^{3} x^{4} + 300 a^{7} b^{4} x^{5} + 60 a^{6} b^{5} x^{6}} + \frac {\left (- 6 A b + B a\right ) \log {\left (x + \frac {- 6 A a b + B a^{2} - a \left (- 6 A b + B a\right )}{- 12 A b^{2} + 2 B a b} \right )}}{a^{7}} - \frac {\left (- 6 A b + B a\right ) \log {\left (x + \frac {- 6 A a b + B a^{2} + a \left (- 6 A b + B a\right )}{- 12 A b^{2} + 2 B a b} \right )}}{a^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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