Optimal. Leaf size=68 \[ \frac {a^4 \log (x)}{b^5}-\frac {a^4 \log (a x+b)}{b^5}+\frac {a^3}{b^4 x}-\frac {a^2}{2 b^3 x^2}+\frac {a}{3 b^2 x^3}-\frac {1}{4 b x^4} \]
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Rubi [A] time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {263, 44} \[ -\frac {a^2}{2 b^3 x^2}+\frac {a^3}{b^4 x}+\frac {a^4 \log (x)}{b^5}-\frac {a^4 \log (a x+b)}{b^5}+\frac {a}{3 b^2 x^3}-\frac {1}{4 b x^4} \]
Antiderivative was successfully verified.
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Rule 44
Rule 263
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right ) x^6} \, dx &=\int \frac {1}{x^5 (b+a x)} \, dx\\ &=\int \left (\frac {1}{b x^5}-\frac {a}{b^2 x^4}+\frac {a^2}{b^3 x^3}-\frac {a^3}{b^4 x^2}+\frac {a^4}{b^5 x}-\frac {a^5}{b^5 (b+a x)}\right ) \, dx\\ &=-\frac {1}{4 b x^4}+\frac {a}{3 b^2 x^3}-\frac {a^2}{2 b^3 x^2}+\frac {a^3}{b^4 x}+\frac {a^4 \log (x)}{b^5}-\frac {a^4 \log (b+a x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 68, normalized size = 1.00 \[ \frac {a^4 \log (x)}{b^5}-\frac {a^4 \log (a x+b)}{b^5}+\frac {a^3}{b^4 x}-\frac {a^2}{2 b^3 x^2}+\frac {a}{3 b^2 x^3}-\frac {1}{4 b x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 65, normalized size = 0.96 \[ -\frac {12 \, a^{4} x^{4} \log \left (a x + b\right ) - 12 \, a^{4} x^{4} \log \relax (x) - 12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 4 \, a b^{3} x + 3 \, b^{4}}{12 \, b^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 67, normalized size = 0.99 \[ -\frac {a^{4} \log \left ({\left | a x + b \right |}\right )}{b^{5}} + \frac {a^{4} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {12 \, a^{3} b x^{3} - 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x - 3 \, b^{4}}{12 \, b^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.93 \[ \frac {a^{4} \ln \relax (x )}{b^{5}}-\frac {a^{4} \ln \left (a x +b \right )}{b^{5}}+\frac {a^{3}}{b^{4} x}-\frac {a^{2}}{2 b^{3} x^{2}}+\frac {a}{3 b^{2} x^{3}}-\frac {1}{4 b \,x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 62, normalized size = 0.91 \[ -\frac {a^{4} \log \left (a x + b\right )}{b^{5}} + \frac {a^{4} \log \relax (x)}{b^{5}} + \frac {12 \, a^{3} x^{3} - 6 \, a^{2} b x^{2} + 4 \, a b^{2} x - 3 \, b^{3}}{12 \, b^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 60, normalized size = 0.88 \[ -\frac {-a^3\,b\,x^3+\frac {a^2\,b^2\,x^2}{2}-\frac {a\,b^3\,x}{3}+\frac {b^4}{4}}{b^5\,x^4}-\frac {2\,a^4\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 56, normalized size = 0.82 \[ \frac {a^{4} \left (\log {\relax (x )} - \log {\left (x + \frac {b}{a} \right )}\right )}{b^{5}} + \frac {12 a^{3} x^{3} - 6 a^{2} b x^{2} + 4 a b^{2} x - 3 b^{3}}{12 b^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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