Optimal. Leaf size=140 \[ -\frac{b^2 (4 A b-3 a B)}{a^5 (a+b x)}-\frac{b^2 (A b-a B)}{2 a^4 (a+b x)^2}-\frac{2 b^2 \log (x) (5 A b-3 a B)}{a^6}+\frac{2 b^2 (5 A b-3 a B) \log (a+b x)}{a^6}+\frac{3 A b-a B}{2 a^4 x^2}-\frac{3 b (2 A b-a B)}{a^5 x}-\frac{A}{3 a^3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.122405, antiderivative size = 140, 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} \[ -\frac{b^2 (4 A b-3 a B)}{a^5 (a+b x)}-\frac{b^2 (A b-a B)}{2 a^4 (a+b x)^2}-\frac{2 b^2 \log (x) (5 A b-3 a B)}{a^6}+\frac{2 b^2 (5 A b-3 a B) \log (a+b x)}{a^6}+\frac{3 A b-a B}{2 a^4 x^2}-\frac{3 b (2 A b-a B)}{a^5 x}-\frac{A}{3 a^3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{x^4 (a+b x)^3} \, dx &=\int \left (\frac{A}{a^3 x^4}+\frac{-3 A b+a B}{a^4 x^3}-\frac{3 b (-2 A b+a B)}{a^5 x^2}+\frac{2 b^2 (-5 A b+3 a B)}{a^6 x}-\frac{b^3 (-A b+a B)}{a^4 (a+b x)^3}-\frac{b^3 (-4 A b+3 a B)}{a^5 (a+b x)^2}-\frac{2 b^3 (-5 A b+3 a B)}{a^6 (a+b x)}\right ) \, dx\\ &=-\frac{A}{3 a^3 x^3}+\frac{3 A b-a B}{2 a^4 x^2}-\frac{3 b (2 A b-a B)}{a^5 x}-\frac{b^2 (A b-a B)}{2 a^4 (a+b x)^2}-\frac{b^2 (4 A b-3 a B)}{a^5 (a+b x)}-\frac{2 b^2 (5 A b-3 a B) \log (x)}{a^6}+\frac{2 b^2 (5 A b-3 a B) \log (a+b x)}{a^6}\\ \end{align*}
Mathematica [A] time = 0.126271, size = 129, normalized size = 0.92 \[ \frac{\frac{a \left (2 a^2 b^2 x^2 (27 B x-10 A)+a^3 b x (5 A+12 B x)+a^4 (-(2 A+3 B x))+18 a b^3 x^3 (2 B x-5 A)-60 A b^4 x^4\right )}{x^3 (a+b x)^2}+12 b^2 \log (x) (3 a B-5 A b)+12 b^2 (5 A b-3 a B) \log (a+b x)}{6 a^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.011, size = 168, normalized size = 1.2 \begin{align*} -{\frac{A}{3\,{a}^{3}{x}^{3}}}+{\frac{3\,Ab}{2\,{a}^{4}{x}^{2}}}-{\frac{B}{2\,{a}^{3}{x}^{2}}}-6\,{\frac{A{b}^{2}}{{a}^{5}x}}+3\,{\frac{Bb}{{a}^{4}x}}-10\,{\frac{A\ln \left ( x \right ){b}^{3}}{{a}^{6}}}+6\,{\frac{{b}^{2}B\ln \left ( x \right ) }{{a}^{5}}}-4\,{\frac{A{b}^{3}}{{a}^{5} \left ( bx+a \right ) }}+3\,{\frac{B{b}^{2}}{{a}^{4} \left ( bx+a \right ) }}-{\frac{A{b}^{3}}{2\,{a}^{4} \left ( bx+a \right ) ^{2}}}+{\frac{B{b}^{2}}{2\,{a}^{3} \left ( bx+a \right ) ^{2}}}+10\,{\frac{{b}^{3}\ln \left ( bx+a \right ) A}{{a}^{6}}}-6\,{\frac{{b}^{2}\ln \left ( bx+a \right ) B}{{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.09656, size = 221, normalized size = 1.58 \begin{align*} -\frac{2 \, A a^{4} - 12 \,{\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} x^{4} - 18 \,{\left (3 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{3} - 4 \,{\left (3 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{2} +{\left (3 \, B a^{4} - 5 \, A a^{3} b\right )} x}{6 \,{\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}} - \frac{2 \,{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (b x + a\right )}{a^{6}} + \frac{2 \,{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (x\right )}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.01509, size = 551, normalized size = 3.94 \begin{align*} -\frac{2 \, A a^{5} - 12 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} - 18 \,{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 4 \,{\left (3 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} +{\left (3 \, B a^{5} - 5 \, A a^{4} b\right )} x + 12 \,{\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 2 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} +{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) - 12 \,{\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 2 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} +{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{6} b^{2} x^{5} + 2 \, a^{7} b x^{4} + a^{8} x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.2909, size = 262, normalized size = 1.87 \begin{align*} \frac{- 2 A a^{4} + x^{4} \left (- 60 A b^{4} + 36 B a b^{3}\right ) + x^{3} \left (- 90 A a b^{3} + 54 B a^{2} b^{2}\right ) + x^{2} \left (- 20 A a^{2} b^{2} + 12 B a^{3} b\right ) + x \left (5 A a^{3} b - 3 B a^{4}\right )}{6 a^{7} x^{3} + 12 a^{6} b x^{4} + 6 a^{5} b^{2} x^{5}} + \frac{2 b^{2} \left (- 5 A b + 3 B a\right ) \log{\left (x + \frac{- 10 A a b^{3} + 6 B a^{2} b^{2} - 2 a b^{2} \left (- 5 A b + 3 B a\right )}{- 20 A b^{4} + 12 B a b^{3}} \right )}}{a^{6}} - \frac{2 b^{2} \left (- 5 A b + 3 B a\right ) \log{\left (x + \frac{- 10 A a b^{3} + 6 B a^{2} b^{2} + 2 a b^{2} \left (- 5 A b + 3 B a\right )}{- 20 A b^{4} + 12 B a b^{3}} \right )}}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19276, size = 213, normalized size = 1.52 \begin{align*} \frac{2 \,{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{6}} - \frac{2 \,{\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{6} b} - \frac{2 \, A a^{5} - 12 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} - 18 \,{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 4 \,{\left (3 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} +{\left (3 \, B a^{5} - 5 \, A a^{4} b\right )} x}{6 \,{\left (b x + a\right )}^{2} a^{6} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]