Optimal. Leaf size=24 \[ -\frac{(-a-b x)^{-n} (a+b x)^n}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0030462, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {23, 30} \[ -\frac{(-a-b x)^{-n} (a+b x)^n}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 23
Rule 30
Rubi steps
\begin{align*} \int \frac{(-a-b x)^{-n} (a+b x)^n}{x^2} \, dx &=\left ((-a-b x)^{-n} (a+b x)^n\right ) \int \frac{1}{x^2} \, dx\\ &=-\frac{(-a-b x)^{-n} (a+b x)^n}{x}\\ \end{align*}
Mathematica [A] time = 0.0027004, size = 24, normalized size = 1. \[ -\frac{(-a-b x)^{-n} (a+b x)^n}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 25, normalized size = 1. \begin{align*} -{\frac{ \left ( bx+a \right ) ^{n}}{x \left ( -bx-a \right ) ^{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.50331, size = 11, normalized size = 0.46 \begin{align*} -\frac{\left (-1\right )^{n}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.83193, size = 19, normalized size = 0.79 \begin{align*} -\frac{\cos \left (\pi n\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 18.5831, size = 44, normalized size = 1.83 \begin{align*} \begin{cases} - \frac{\left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{x} + \frac{b \left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{a} & \text{for}\: a \neq 0 \\- \frac{\left (-1\right )^{- n}}{x} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n}}{{\left (-b x - a\right )}^{n} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]