Optimal. Leaf size=24 \[ -\frac {(-a-b x)^{-n} (a+b x)^n}{x} \]
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Rubi [A] time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, 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.
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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*}
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Mathematica [A] time = 0.00, size = 24, normalized size = 1.00 \[ -\frac {(-a-b x)^{-n} (a+b x)^n}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.16, size = 9, normalized size = 0.38 \[ -\frac {\cos \left (\pi n\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n}}{{\left (-b x - a\right )}^{n} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 1.04 \[ -\frac {\left (-b x -a \right )^{-n} \left (b x +a \right )^{n}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 8, normalized size = 0.33 \[ -\frac {\left (-1\right )^{n}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 24, normalized size = 1.00 \[ -\frac {{\left (a+b\,x\right )}^n}{x\,{\left (-a-b\,x\right )}^n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.70, size = 17, normalized size = 0.71 \[ - \frac {\left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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