Optimal. Leaf size=26 \[ -\frac{(-a-b x)^{-n} (a+b x)^n}{2 x^2} \]
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Rubi [A] time = 0.0030823, antiderivative size = 26, 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} \[ -\frac{(-a-b x)^{-n} (a+b x)^n}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 23
Rule 30
Rubi steps
\begin{align*} \int \frac{(-a-b x)^{-n} (a+b x)^n}{x^3} \, dx &=\left ((-a-b x)^{-n} (a+b x)^n\right ) \int \frac{1}{x^3} \, dx\\ &=-\frac{(-a-b x)^{-n} (a+b x)^n}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0025799, size = 26, normalized size = 1. \[ -\frac{(-a-b x)^{-n} (a+b x)^n}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 25, normalized size = 1. \begin{align*} -{\frac{ \left ( bx+a \right ) ^{n}}{2\,{x}^{2} \left ( -bx-a \right ) ^{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06961, size = 11, normalized size = 0.42 \begin{align*} -\frac{\left (-1\right )^{n}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84959, size = 27, normalized size = 1.04 \begin{align*} -\frac{\cos \left (\pi n\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 22.4169, size = 20, normalized size = 0.77 \begin{align*} - \frac{\left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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