Optimal. Leaf size=22 \[ \log (x) (-a-b x)^{-n} (a+b x)^n \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0042418, antiderivative size = 22, 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, 29} \[ \log (x) (-a-b x)^{-n} (a+b x)^n \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 23
Rule 29
Rubi steps
\begin{align*} \int \frac{(-a-b x)^{-n} (a+b x)^n}{x} \, dx &=\left ((-a-b x)^{-n} (a+b x)^n\right ) \int \frac{1}{x} \, dx\\ &=(-a-b x)^{-n} (a+b x)^n \log (x)\\ \end{align*}
Mathematica [A] time = 0.0024547, size = 22, normalized size = 1. \[ \log (x) (-a-b x)^{-n} (a+b x)^n \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.026, size = 56, normalized size = 2.6 \begin{align*} \ln \left ( x \right ) \left ( bx+a \right ) ^{n}{{\rm e}^{-n \left ( i\pi \, \left ({\it csgn} \left ( i \left ( bx+a \right ) \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ( i \left ( bx+a \right ) \right ) \right ) ^{2}+i\pi +\ln \left ( bx+a \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.22875, size = 8, normalized size = 0.36 \begin{align*} \left (-1\right )^{n} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.86752, size = 24, normalized size = 1.09 \begin{align*} \cos \left (\pi n\right ) \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 28.6126, size = 46, normalized size = 2.09 \begin{align*} \begin{cases} e^{- i \pi n} \log{\left (-1 + \frac{b \left (\frac{a}{b} + x\right )}{a} \right )} & \text{for}\: \frac{\left |{b \left (\frac{a}{b} + x\right )}\right |}{\left |{a}\right |} > 1 \\e^{- i \pi n} \log{\left (1 - \frac{b \left (\frac{a}{b} + x\right )}{a} \right )} & \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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]