Optimal. Leaf size=84 \[ \frac {5 a^4 \log (x)}{b^6}-\frac {5 a^4 \log (a x+b)}{b^6}+\frac {a^4}{b^5 (a x+b)}+\frac {4 a^3}{b^5 x}-\frac {3 a^2}{2 b^4 x^2}+\frac {2 a}{3 b^3 x^3}-\frac {1}{4 b^2 x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 84, 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} \[ -\frac {3 a^2}{2 b^4 x^2}+\frac {a^4}{b^5 (a x+b)}+\frac {4 a^3}{b^5 x}+\frac {5 a^4 \log (x)}{b^6}-\frac {5 a^4 \log (a x+b)}{b^6}+\frac {2 a}{3 b^3 x^3}-\frac {1}{4 b^2 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 263
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^7} \, dx &=\int \frac {1}{x^5 (b+a x)^2} \, dx\\ &=\int \left (\frac {1}{b^2 x^5}-\frac {2 a}{b^3 x^4}+\frac {3 a^2}{b^4 x^3}-\frac {4 a^3}{b^5 x^2}+\frac {5 a^4}{b^6 x}-\frac {a^5}{b^5 (b+a x)^2}-\frac {5 a^5}{b^6 (b+a x)}\right ) \, dx\\ &=-\frac {1}{4 b^2 x^4}+\frac {2 a}{3 b^3 x^3}-\frac {3 a^2}{2 b^4 x^2}+\frac {4 a^3}{b^5 x}+\frac {a^4}{b^5 (b+a x)}+\frac {5 a^4 \log (x)}{b^6}-\frac {5 a^4 \log (b+a x)}{b^6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 79, normalized size = 0.94 \[ \frac {-60 a^4 \log (a x+b)+60 a^4 \log (x)+\frac {b \left (60 a^4 x^4+30 a^3 b x^3-10 a^2 b^2 x^2+5 a b^3 x-3 b^4\right )}{x^4 (a x+b)}}{12 b^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.01, size = 108, normalized size = 1.29 \[ \frac {60 \, a^{4} b x^{4} + 30 \, a^{3} b^{2} x^{3} - 10 \, a^{2} b^{3} x^{2} + 5 \, a b^{4} x - 3 \, b^{5} - 60 \, {\left (a^{5} x^{5} + a^{4} b x^{4}\right )} \log \left (a x + b\right ) + 60 \, {\left (a^{5} x^{5} + a^{4} b x^{4}\right )} \log \relax (x)}{12 \, {\left (a b^{6} x^{5} + b^{7} x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 86, normalized size = 1.02 \[ -\frac {5 \, a^{4} \log \left ({\left | a x + b \right |}\right )}{b^{6}} + \frac {5 \, a^{4} \log \left ({\left | x \right |}\right )}{b^{6}} + \frac {60 \, a^{4} b x^{4} + 30 \, a^{3} b^{2} x^{3} - 10 \, a^{2} b^{3} x^{2} + 5 \, a b^{4} x - 3 \, b^{5}}{12 \, {\left (a x + b\right )} b^{6} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 79, normalized size = 0.94 \[ \frac {a^{4}}{\left (a x +b \right ) b^{5}}+\frac {5 a^{4} \ln \relax (x )}{b^{6}}-\frac {5 a^{4} \ln \left (a x +b \right )}{b^{6}}+\frac {4 a^{3}}{b^{5} x}-\frac {3 a^{2}}{2 b^{4} x^{2}}+\frac {2 a}{3 b^{3} x^{3}}-\frac {1}{4 b^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.01, size = 86, normalized size = 1.02 \[ \frac {60 \, a^{4} x^{4} + 30 \, a^{3} b x^{3} - 10 \, a^{2} b^{2} x^{2} + 5 \, a b^{3} x - 3 \, b^{4}}{12 \, {\left (a b^{5} x^{5} + b^{6} x^{4}\right )}} - \frac {5 \, a^{4} \log \left (a x + b\right )}{b^{6}} + \frac {5 \, a^{4} \log \relax (x)}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.10, size = 79, normalized size = 0.94 \[ \frac {\frac {5\,a^3\,x^3}{2\,b^4}-\frac {5\,a^2\,x^2}{6\,b^3}-\frac {1}{4\,b}+\frac {5\,a^4\,x^4}{b^5}+\frac {5\,a\,x}{12\,b^2}}{a\,x^5+b\,x^4}-\frac {10\,a^4\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.37, size = 80, normalized size = 0.95 \[ \frac {5 a^{4} \left (\log {\relax (x )} - \log {\left (x + \frac {b}{a} \right )}\right )}{b^{6}} + \frac {60 a^{4} x^{4} + 30 a^{3} b x^{3} - 10 a^{2} b^{2} x^{2} + 5 a b^{3} x - 3 b^{4}}{12 a b^{5} x^{5} + 12 b^{6} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________