Optimal. Leaf size=57 \[ -\frac {3 b \log (x)}{a^4}+\frac {3 b \log (a+b x)}{a^4}-\frac {2 b}{a^3 (a+b x)}-\frac {1}{a^3 x}-\frac {b}{2 a^2 (a+b x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {44} \[ -\frac {2 b}{a^3 (a+b x)}-\frac {b}{2 a^2 (a+b x)^2}-\frac {3 b \log (x)}{a^4}+\frac {3 b \log (a+b x)}{a^4}-\frac {1}{a^3 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rubi steps
\begin {align*} \int \frac {1}{x^2 (a+b x)^3} \, dx &=\int \left (\frac {1}{a^3 x^2}-\frac {3 b}{a^4 x}+\frac {b^2}{a^2 (a+b x)^3}+\frac {2 b^2}{a^3 (a+b x)^2}+\frac {3 b^2}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac {1}{a^3 x}-\frac {b}{2 a^2 (a+b x)^2}-\frac {2 b}{a^3 (a+b x)}-\frac {3 b \log (x)}{a^4}+\frac {3 b \log (a+b x)}{a^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 53, normalized size = 0.93 \[ -\frac {\frac {a \left (2 a^2+9 a b x+6 b^2 x^2\right )}{x (a+b x)^2}-6 b \log (a+b x)+6 b \log (x)}{2 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 109, normalized size = 1.91 \[ -\frac {6 \, a b^{2} x^{2} + 9 \, a^{2} b x + 2 \, a^{3} - 6 \, {\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \log \left (b x + a\right ) + 6 \, {\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \log \relax (x)}{2 \, {\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.07, size = 60, normalized size = 1.05 \[ \frac {3 \, b \log \left ({\left | b x + a \right |}\right )}{a^{4}} - \frac {3 \, b \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {6 \, a b^{2} x^{2} + 9 \, a^{2} b x + 2 \, a^{3}}{2 \, {\left (b x + a\right )}^{2} a^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 56, normalized size = 0.98 \[ -\frac {b}{2 \left (b x +a \right )^{2} a^{2}}-\frac {2 b}{\left (b x +a \right ) a^{3}}-\frac {3 b \ln \relax (x )}{a^{4}}+\frac {3 b \ln \left (b x +a \right )}{a^{4}}-\frac {1}{a^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.32, size = 69, normalized size = 1.21 \[ -\frac {6 \, b^{2} x^{2} + 9 \, a b x + 2 \, a^{2}}{2 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} + \frac {3 \, b \log \left (b x + a\right )}{a^{4}} - \frac {3 \, b \log \relax (x)}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.11, size = 63, normalized size = 1.11 \[ \frac {6\,b\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^4}-\frac {\frac {1}{a}+\frac {3\,b^2\,x^2}{a^3}+\frac {9\,b\,x}{2\,a^2}}{a^2\,x+2\,a\,b\,x^2+b^2\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.40, size = 66, normalized size = 1.16 \[ \frac {- 2 a^{2} - 9 a b x - 6 b^{2} x^{2}}{2 a^{5} x + 4 a^{4} b x^{2} + 2 a^{3} b^{2} x^{3}} + \frac {3 b \left (- \log {\relax (x )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________