Optimal. Leaf size=72 \[ \frac{6 c^2 \log (x)}{b^5}-\frac{6 c^2 \log (b+c x)}{b^5}+\frac{3 c (b+2 c x)}{b^4 \left (b x+c x^2\right )}-\frac{b+2 c x}{2 b^2 \left (b x+c x^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0192287, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {614, 615} \[ \frac{6 c^2 \log (x)}{b^5}-\frac{6 c^2 \log (b+c x)}{b^5}+\frac{3 c (b+2 c x)}{b^4 \left (b x+c x^2\right )}-\frac{b+2 c x}{2 b^2 \left (b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 614
Rule 615
Rubi steps
\begin{align*} \int \frac{1}{\left (b x+c x^2\right )^3} \, dx &=-\frac{b+2 c x}{2 b^2 \left (b x+c x^2\right )^2}-\frac{(3 c) \int \frac{1}{\left (b x+c x^2\right )^2} \, dx}{b^2}\\ &=-\frac{b+2 c x}{2 b^2 \left (b x+c x^2\right )^2}+\frac{3 c (b+2 c x)}{b^4 \left (b x+c x^2\right )}+\frac{\left (6 c^2\right ) \int \frac{1}{b x+c x^2} \, dx}{b^4}\\ &=-\frac{b+2 c x}{2 b^2 \left (b x+c x^2\right )^2}+\frac{3 c (b+2 c x)}{b^4 \left (b x+c x^2\right )}+\frac{6 c^2 \log (x)}{b^5}-\frac{6 c^2 \log (b+c x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0459729, size = 68, normalized size = 0.94 \[ \frac{\frac{b \left (4 b^2 c x-b^3+18 b c^2 x^2+12 c^3 x^3\right )}{x^2 (b+c x)^2}-12 c^2 \log (b+c x)+12 c^2 \log (x)}{2 b^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.051, size = 73, normalized size = 1. \begin{align*} -{\frac{1}{2\,{b}^{3}{x}^{2}}}+6\,{\frac{{c}^{2}\ln \left ( x \right ) }{{b}^{5}}}+3\,{\frac{c}{{b}^{4}x}}-6\,{\frac{{c}^{2}\ln \left ( cx+b \right ) }{{b}^{5}}}+3\,{\frac{{c}^{2}}{{b}^{4} \left ( cx+b \right ) }}+{\frac{{c}^{2}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.12503, size = 116, normalized size = 1.61 \begin{align*} \frac{12 \, c^{3} x^{3} + 18 \, b c^{2} x^{2} + 4 \, b^{2} c x - b^{3}}{2 \,{\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} - \frac{6 \, c^{2} \log \left (c x + b\right )}{b^{5}} + \frac{6 \, c^{2} \log \left (x\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.69016, size = 269, normalized size = 3.74 \begin{align*} \frac{12 \, b c^{3} x^{3} + 18 \, b^{2} c^{2} x^{2} + 4 \, b^{3} c x - b^{4} - 12 \,{\left (c^{4} x^{4} + 2 \, b c^{3} x^{3} + b^{2} c^{2} x^{2}\right )} \log \left (c x + b\right ) + 12 \,{\left (c^{4} x^{4} + 2 \, b c^{3} x^{3} + b^{2} c^{2} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.50587, size = 78, normalized size = 1.08 \begin{align*} \frac{- b^{3} + 4 b^{2} c x + 18 b c^{2} x^{2} + 12 c^{3} x^{3}}{2 b^{6} x^{2} + 4 b^{5} c x^{3} + 2 b^{4} c^{2} x^{4}} + \frac{6 c^{2} \left (\log{\left (x \right )} - \log{\left (\frac{b}{c} + x \right )}\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.29595, size = 99, normalized size = 1.38 \begin{align*} -\frac{6 \, c^{2} \log \left ({\left | c x + b \right |}\right )}{b^{5}} + \frac{6 \, c^{2} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac{12 \, c^{3} x^{3} + 18 \, b c^{2} x^{2} + 4 \, b^{2} c x - b^{3}}{2 \,{\left (c x^{2} + b x\right )}^{2} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]