Optimal. Leaf size=85 \[ \frac{2 A b-a B}{a^3 x}+\frac{b (A b-a B)}{a^3 (a+b x)}+\frac{b \log (x) (3 A b-2 a B)}{a^4}-\frac{b (3 A b-2 a B) \log (a+b x)}{a^4}-\frac{A}{2 a^2 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0651152, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {77} \[ \frac{2 A b-a B}{a^3 x}+\frac{b (A b-a B)}{a^3 (a+b x)}+\frac{b \log (x) (3 A b-2 a B)}{a^4}-\frac{b (3 A b-2 a B) \log (a+b x)}{a^4}-\frac{A}{2 a^2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{x^3 (a+b x)^2} \, dx &=\int \left (\frac{A}{a^2 x^3}+\frac{-2 A b+a B}{a^3 x^2}-\frac{b (-3 A b+2 a B)}{a^4 x}+\frac{b^2 (-A b+a B)}{a^3 (a+b x)^2}+\frac{b^2 (-3 A b+2 a B)}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac{A}{2 a^2 x^2}+\frac{2 A b-a B}{a^3 x}+\frac{b (A b-a B)}{a^3 (a+b x)}+\frac{b (3 A b-2 a B) \log (x)}{a^4}-\frac{b (3 A b-2 a B) \log (a+b x)}{a^4}\\ \end{align*}
Mathematica [A] time = 0.0682777, size = 85, normalized size = 1. \[ \frac{-\frac{a \left (a^2 (A+2 B x)+a b x (4 B x-3 A)-6 A b^2 x^2\right )}{x^2 (a+b x)}+2 b \log (x) (3 A b-2 a B)+2 b (2 a B-3 A b) \log (a+b x)}{2 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 107, normalized size = 1.3 \begin{align*} -{\frac{A}{2\,{a}^{2}{x}^{2}}}+2\,{\frac{Ab}{{a}^{3}x}}-{\frac{B}{{a}^{2}x}}+3\,{\frac{A\ln \left ( x \right ){b}^{2}}{{a}^{4}}}-2\,{\frac{bB\ln \left ( x \right ) }{{a}^{3}}}-3\,{\frac{{b}^{2}\ln \left ( bx+a \right ) A}{{a}^{4}}}+2\,{\frac{b\ln \left ( bx+a \right ) B}{{a}^{3}}}+{\frac{A{b}^{2}}{{a}^{3} \left ( bx+a \right ) }}-{\frac{Bb}{{a}^{2} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.58002, size = 134, normalized size = 1.58 \begin{align*} -\frac{A a^{2} + 2 \,{\left (2 \, B a b - 3 \, A b^{2}\right )} x^{2} +{\left (2 \, B a^{2} - 3 \, A a b\right )} x}{2 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} + \frac{{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (b x + a\right )}{a^{4}} - \frac{{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (x\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.00095, size = 321, normalized size = 3.78 \begin{align*} -\frac{A a^{3} + 2 \,{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2} +{\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} x - 2 \,{\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{3} +{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 2 \,{\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{3} +{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.947209, size = 184, normalized size = 2.16 \begin{align*} - \frac{A a^{2} + x^{2} \left (- 6 A b^{2} + 4 B a b\right ) + x \left (- 3 A a b + 2 B a^{2}\right )}{2 a^{4} x^{2} + 2 a^{3} b x^{3}} - \frac{b \left (- 3 A b + 2 B a\right ) \log{\left (x + \frac{- 3 A a b^{2} + 2 B a^{2} b - a b \left (- 3 A b + 2 B a\right )}{- 6 A b^{3} + 4 B a b^{2}} \right )}}{a^{4}} + \frac{b \left (- 3 A b + 2 B a\right ) \log{\left (x + \frac{- 3 A a b^{2} + 2 B a^{2} b + a b \left (- 3 A b + 2 B a\right )}{- 6 A b^{3} + 4 B a b^{2}} \right )}}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1921, size = 176, normalized size = 2.07 \begin{align*} -\frac{{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{4} b} - \frac{\frac{B a b^{4}}{b x + a} - \frac{A b^{5}}{b x + a}}{a^{3} b^{3}} - \frac{2 \, B a b - 5 \, A b^{2} - \frac{2 \,{\left (B a^{2} b^{2} - 3 \, A a b^{3}\right )}}{{\left (b x + a\right )} b}}{2 \, a^{4}{\left (\frac{a}{b x + a} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]