Optimal. Leaf size=30 \[ \frac{a}{3 b^2 (a+b x)^3}-\frac{1}{2 b^2 (a+b x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0128605, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ \frac{a}{3 b^2 (a+b x)^3}-\frac{1}{2 b^2 (a+b x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin{align*} \int \frac{x}{(a+b x)^4} \, dx &=\int \left (-\frac{a}{b (a+b x)^4}+\frac{1}{b (a+b x)^3}\right ) \, dx\\ &=\frac{a}{3 b^2 (a+b x)^3}-\frac{1}{2 b^2 (a+b x)^2}\\ \end{align*}
Mathematica [A] time = 0.0065877, size = 20, normalized size = 0.67 \[ -\frac{a+3 b x}{6 b^2 (a+b x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 27, normalized size = 0.9 \begin{align*}{\frac{a}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{1}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03351, size = 58, normalized size = 1.93 \begin{align*} -\frac{3 \, b x + a}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.51679, size = 89, normalized size = 2.97 \begin{align*} -\frac{3 \, b x + a}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.539855, size = 44, normalized size = 1.47 \begin{align*} - \frac{a + 3 b x}{6 a^{3} b^{2} + 18 a^{2} b^{3} x + 18 a b^{4} x^{2} + 6 b^{5} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20428, size = 24, normalized size = 0.8 \begin{align*} -\frac{3 \, b x + a}{6 \,{\left (b x + a\right )}^{3} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]