Optimal. Leaf size=71 \[ -\frac {a^4 B}{4 x^4}-\frac {4 a^3 b B}{3 x^3}-\frac {3 a^2 b^2 B}{x^2}-\frac {A (a+b x)^5}{5 a x^5}-\frac {4 a b^3 B}{x}+b^4 B \log (x) \]
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Rubi [A] time = 0.03, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 78, 43} \[ -\frac {3 a^2 b^2 B}{x^2}-\frac {4 a^3 b B}{3 x^3}-\frac {a^4 B}{4 x^4}-\frac {A (a+b x)^5}{5 a x^5}-\frac {4 a b^3 B}{x}+b^4 B \log (x) \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rule 78
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^6} \, dx &=\int \frac {(a+b x)^4 (A+B x)}{x^6} \, dx\\ &=-\frac {A (a+b x)^5}{5 a x^5}+B \int \frac {(a+b x)^4}{x^5} \, dx\\ &=-\frac {A (a+b x)^5}{5 a x^5}+B \int \left (\frac {a^4}{x^5}+\frac {4 a^3 b}{x^4}+\frac {6 a^2 b^2}{x^3}+\frac {4 a b^3}{x^2}+\frac {b^4}{x}\right ) \, dx\\ &=-\frac {a^4 B}{4 x^4}-\frac {4 a^3 b B}{3 x^3}-\frac {3 a^2 b^2 B}{x^2}-\frac {4 a b^3 B}{x}-\frac {A (a+b x)^5}{5 a x^5}+b^4 B \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 87, normalized size = 1.23 \[ b^4 B \log (x)-\frac {3 a^4 (4 A+5 B x)+20 a^3 b x (3 A+4 B x)+60 a^2 b^2 x^2 (2 A+3 B x)+120 a b^3 x^3 (A+2 B x)+60 A b^4 x^4}{60 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 101, normalized size = 1.42 \[ \frac {60 \, B b^{4} x^{5} \log \relax (x) - 12 \, A a^{4} - 60 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} - 60 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} - 40 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} - 15 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{60 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 99, normalized size = 1.39 \[ B b^{4} \log \left ({\left | x \right |}\right ) - \frac {12 \, A a^{4} + 60 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 60 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 40 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 15 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{60 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 100, normalized size = 1.41 \[ B \,b^{4} \ln \relax (x )-\frac {A \,b^{4}}{x}-\frac {4 B a \,b^{3}}{x}-\frac {2 A a \,b^{3}}{x^{2}}-\frac {3 B \,a^{2} b^{2}}{x^{2}}-\frac {2 A \,a^{2} b^{2}}{x^{3}}-\frac {4 B \,a^{3} b}{3 x^{3}}-\frac {A \,a^{3} b}{x^{4}}-\frac {B \,a^{4}}{4 x^{4}}-\frac {A \,a^{4}}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 98, normalized size = 1.38 \[ B b^{4} \log \relax (x) - \frac {12 \, A a^{4} + 60 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 60 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 40 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 15 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{60 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 94, normalized size = 1.32 \[ B\,b^4\,\ln \relax (x)-\frac {x\,\left (\frac {B\,a^4}{4}+A\,b\,a^3\right )+\frac {A\,a^4}{5}+x^3\,\left (3\,B\,a^2\,b^2+2\,A\,a\,b^3\right )+x^2\,\left (\frac {4\,B\,a^3\,b}{3}+2\,A\,a^2\,b^2\right )+x^4\,\left (A\,b^4+4\,B\,a\,b^3\right )}{x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.75, size = 105, normalized size = 1.48 \[ B b^{4} \log {\relax (x )} + \frac {- 12 A a^{4} + x^{4} \left (- 60 A b^{4} - 240 B a b^{3}\right ) + x^{3} \left (- 120 A a b^{3} - 180 B a^{2} b^{2}\right ) + x^{2} \left (- 120 A a^{2} b^{2} - 80 B a^{3} b\right ) + x \left (- 60 A a^{3} b - 15 B a^{4}\right )}{60 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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