Optimal. Leaf size=28 \[ -\frac {x^{-1+n} (a+b x)^{1-n}}{a (1-n)} \]
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Rubi [A]
time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37}
\begin {gather*} -\frac {x^{n-1} (a+b x)^{1-n}}{a (1-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int x^{-2+n} (a+b x)^{-n} \, dx &=-\frac {x^{-1+n} (a+b x)^{1-n}}{a (1-n)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 25, normalized size = 0.89 \begin {gather*} \frac {x^{-1+n} (a+b x)^{1-n}}{a (-1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 29, normalized size = 1.04
method | result | size |
gosper | \(\frac {x^{-1+n} \left (b x +a \right ) \left (b x +a \right )^{-n}}{a \left (-1+n \right )}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.12, size = 33, normalized size = 1.18 \begin {gather*} \frac {{\left (b x^{2} + a x\right )} x^{n - 2}}{{\left (a n - a\right )} {\left (b x + a\right )}^{n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs.
\(2 (19) = 38\).
time = 201.00, size = 85, normalized size = 3.04 \begin {gather*} \begin {cases} - \frac {1}{b x} & \text {for}\: a = 0 \wedge n = 1 \\- \frac {x^{n} \left (b x\right )^{- n}}{x} & \text {for}\: a = 0 \\\frac {\log {\left (x \right )}}{a} - \frac {\log {\left (\frac {a}{b} + x \right )}}{a} & \text {for}\: n = 1 \\\frac {a x^{n}}{a n x \left (a + b x\right )^{n} - a x \left (a + b x\right )^{n}} + \frac {b x x^{n}}{a n x \left (a + b x\right )^{n} - a x \left (a + b x\right )^{n}} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.35, size = 29, normalized size = 1.04 \begin {gather*} \frac {x^n\,\left (a+b\,x\right )}{a\,x\,\left (n-1\right )\,{\left (a+b\,x\right )}^n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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