Optimal. Leaf size=166 \[ -\frac {10 b^2 \log (x) (2 A b-a B)}{a^7}+\frac {10 b^2 (2 A b-a B) \log (a+b x)}{a^7}-\frac {2 b^2 (5 A b-3 a B)}{a^6 (a+b x)}-\frac {2 b (5 A b-2 a B)}{a^6 x}-\frac {b^2 (4 A b-3 a B)}{2 a^5 (a+b x)^2}+\frac {4 A b-a B}{2 a^5 x^2}-\frac {b^2 (A b-a B)}{3 a^4 (a+b x)^3}-\frac {A}{3 a^4 x^3} \]
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Rubi [A] time = 0.17, antiderivative size = 166, 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 {2 b^2 (5 A b-3 a B)}{a^6 (a+b x)}-\frac {b^2 (4 A b-3 a B)}{2 a^5 (a+b x)^2}-\frac {b^2 (A b-a B)}{3 a^4 (a+b x)^3}-\frac {10 b^2 \log (x) (2 A b-a B)}{a^7}+\frac {10 b^2 (2 A b-a B) \log (a+b x)}{a^7}+\frac {4 A b-a B}{2 a^5 x^2}-\frac {2 b (5 A b-2 a B)}{a^6 x}-\frac {A}{3 a^4 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{x^4 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {A+B x}{x^4 (a+b x)^4} \, dx\\ &=\int \left (\frac {A}{a^4 x^4}+\frac {-4 A b+a B}{a^5 x^3}-\frac {2 b (-5 A b+2 a B)}{a^6 x^2}+\frac {10 b^2 (-2 A b+a B)}{a^7 x}-\frac {b^3 (-A b+a B)}{a^4 (a+b x)^4}-\frac {b^3 (-4 A b+3 a B)}{a^5 (a+b x)^3}-\frac {2 b^3 (-5 A b+3 a B)}{a^6 (a+b x)^2}-\frac {10 b^3 (-2 A b+a B)}{a^7 (a+b x)}\right ) \, dx\\ &=-\frac {A}{3 a^4 x^3}+\frac {4 A b-a B}{2 a^5 x^2}-\frac {2 b (5 A b-2 a B)}{a^6 x}-\frac {b^2 (A b-a B)}{3 a^4 (a+b x)^3}-\frac {b^2 (4 A b-3 a B)}{2 a^5 (a+b x)^2}-\frac {2 b^2 (5 A b-3 a B)}{a^6 (a+b x)}-\frac {10 b^2 (2 A b-a B) \log (x)}{a^7}+\frac {10 b^2 (2 A b-a B) \log (a+b x)}{a^7}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 148, normalized size = 0.89 \begin {gather*} \frac {\frac {a \left (-\left (a^5 (2 A+3 B x)\right )+3 a^4 b x (2 A+5 B x)+10 a^3 b^2 x^2 (11 B x-3 A)+10 a^2 b^3 x^3 (15 B x-22 A)+60 a b^4 x^4 (B x-5 A)-120 A b^5 x^5\right )}{x^3 (a+b x)^3}-60 b^2 \log (x) (2 A b-a B)+60 b^2 (2 A b-a B) \log (a+b x)}{6 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^4 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 333, normalized size = 2.01 \begin {gather*} -\frac {2 \, A a^{6} - 60 \, {\left (B a^{2} b^{4} - 2 \, A a b^{5}\right )} x^{5} - 150 \, {\left (B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} x^{4} - 110 \, {\left (B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{3} - 15 \, {\left (B a^{5} b - 2 \, A a^{4} b^{2}\right )} x^{2} + 3 \, {\left (B a^{6} - 2 \, A a^{5} b\right )} x + 60 \, {\left ({\left (B a b^{5} - 2 \, A b^{6}\right )} x^{6} + 3 \, {\left (B a^{2} b^{4} - 2 \, A a b^{5}\right )} x^{5} + 3 \, {\left (B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} x^{4} + {\left (B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) - 60 \, {\left ({\left (B a b^{5} - 2 \, A b^{6}\right )} x^{6} + 3 \, {\left (B a^{2} b^{4} - 2 \, A a b^{5}\right )} x^{5} + 3 \, {\left (B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} x^{4} + {\left (B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{3}\right )} \log \relax (x)}{6 \, {\left (a^{7} b^{3} x^{6} + 3 \, a^{8} b^{2} x^{5} + 3 \, a^{9} b x^{4} + a^{10} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 175, normalized size = 1.05 \begin {gather*} \frac {10 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{7}} - \frac {10 \, {\left (B a b^{3} - 2 \, A b^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{7} b} + \frac {60 \, B a b^{4} x^{5} - 120 \, A b^{5} x^{5} + 150 \, B a^{2} b^{3} x^{4} - 300 \, A a b^{4} x^{4} + 110 \, B a^{3} b^{2} x^{3} - 220 \, A a^{2} b^{3} x^{3} + 15 \, B a^{4} b x^{2} - 30 \, A a^{3} b^{2} x^{2} - 3 \, B a^{5} x + 6 \, A a^{4} b x - 2 \, A a^{5}}{6 \, {\left (b x^{2} + a x\right )}^{3} a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 200, normalized size = 1.20 \begin {gather*} -\frac {A \,b^{3}}{3 \left (b x +a \right )^{3} a^{4}}+\frac {B \,b^{2}}{3 \left (b x +a \right )^{3} a^{3}}-\frac {2 A \,b^{3}}{\left (b x +a \right )^{2} a^{5}}+\frac {3 B \,b^{2}}{2 \left (b x +a \right )^{2} a^{4}}-\frac {10 A \,b^{3}}{\left (b x +a \right ) a^{6}}-\frac {20 A \,b^{3} \ln \relax (x )}{a^{7}}+\frac {20 A \,b^{3} \ln \left (b x +a \right )}{a^{7}}+\frac {6 B \,b^{2}}{\left (b x +a \right ) a^{5}}+\frac {10 B \,b^{2} \ln \relax (x )}{a^{6}}-\frac {10 B \,b^{2} \ln \left (b x +a \right )}{a^{6}}-\frac {10 A \,b^{2}}{a^{6} x}+\frac {4 B b}{a^{5} x}+\frac {2 A b}{a^{5} x^{2}}-\frac {B}{2 a^{4} x^{2}}-\frac {A}{3 a^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 193, normalized size = 1.16 \begin {gather*} -\frac {2 \, A a^{5} - 60 \, {\left (B a b^{4} - 2 \, A b^{5}\right )} x^{5} - 150 \, {\left (B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{4} - 110 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{3} - 15 \, {\left (B a^{4} b - 2 \, A a^{3} b^{2}\right )} x^{2} + 3 \, {\left (B a^{5} - 2 \, A a^{4} b\right )} x}{6 \, {\left (a^{6} b^{3} x^{6} + 3 \, a^{7} b^{2} x^{5} + 3 \, a^{8} b x^{4} + a^{9} x^{3}\right )}} - \frac {10 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} \log \left (b x + a\right )}{a^{7}} + \frac {10 \, {\left (B a b^{2} - 2 \, A b^{3}\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 = 1.19, size = 195, normalized size = 1.17 \begin {gather*} \frac {20\,b^2\,\mathrm {atanh}\left (\frac {10\,b^2\,\left (2\,A\,b-B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (20\,A\,b^3-10\,B\,a\,b^2\right )}\right )\,\left (2\,A\,b-B\,a\right )}{a^7}-\frac {\frac {A}{3\,a}-\frac {x\,\left (2\,A\,b-B\,a\right )}{2\,a^2}+\frac {55\,b^2\,x^3\,\left (2\,A\,b-B\,a\right )}{3\,a^4}+\frac {25\,b^3\,x^4\,\left (2\,A\,b-B\,a\right )}{a^5}+\frac {10\,b^4\,x^5\,\left (2\,A\,b-B\,a\right )}{a^6}+\frac {5\,b\,x^2\,\left (2\,A\,b-B\,a\right )}{2\,a^3}}{a^3\,x^3+3\,a^2\,b\,x^4+3\,a\,b^2\,x^5+b^3\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.94, size = 291, normalized size = 1.75 \begin {gather*} \frac {- 2 A a^{5} + x^{5} \left (- 120 A b^{5} + 60 B a b^{4}\right ) + x^{4} \left (- 300 A a b^{4} + 150 B a^{2} b^{3}\right ) + x^{3} \left (- 220 A a^{2} b^{3} + 110 B a^{3} b^{2}\right ) + x^{2} \left (- 30 A a^{3} b^{2} + 15 B a^{4} b\right ) + x \left (6 A a^{4} b - 3 B a^{5}\right )}{6 a^{9} x^{3} + 18 a^{8} b x^{4} + 18 a^{7} b^{2} x^{5} + 6 a^{6} b^{3} x^{6}} + \frac {10 b^{2} \left (- 2 A b + B a\right ) \log {\left (x + \frac {- 20 A a b^{3} + 10 B a^{2} b^{2} - 10 a b^{2} \left (- 2 A b + B a\right )}{- 40 A b^{4} + 20 B a b^{3}} \right )}}{a^{7}} - \frac {10 b^{2} \left (- 2 A b + B a\right ) \log {\left (x + \frac {- 20 A a b^{3} + 10 B a^{2} b^{2} + 10 a b^{2} \left (- 2 A b + B a\right )}{- 40 A b^{4} + 20 B a b^{3}} \right )}}{a^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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