Optimal. Leaf size=64 \[ -\frac {x^{-2+n} (a+b x)^{1-n}}{a (2-n)}+\frac {b x^{-1+n} (a+b x)^{1-n}}{a^2 (1-n) (2-n)} \]
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Rubi [A]
time = 0.01, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37}
\begin {gather*} \frac {b x^{n-1} (a+b x)^{1-n}}{a^2 (1-n) (2-n)}-\frac {x^{n-2} (a+b x)^{1-n}}{a (2-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int x^{-3+n} (a+b x)^{-n} \, dx &=-\frac {x^{-2+n} (a+b x)^{1-n}}{a (2-n)}-\frac {b \int x^{-2+n} (a+b x)^{-n} \, dx}{a (2-n)}\\ &=-\frac {x^{-2+n} (a+b x)^{1-n}}{a (2-n)}+\frac {b x^{-1+n} (a+b x)^{1-n}}{a^2 (1-n) (2-n)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 39, normalized size = 0.61 \begin {gather*} \frac {x^{-2+n} (a+b x)^{1-n} (a (-1+n)+b x)}{a^2 (-2+n) (-1+n)} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 190.94, size = 344, normalized size = 5.38 \begin {gather*} \text {Piecewise}\left [\left \{\left \{-\frac {x^{-2+n} \left (b x\right )^{-n}}{2},a\text {==}0\right \},\left \{\frac {-a+b x \left (\text {Log}\left [\frac {a+b x}{b}\right ]-\text {Log}\left [x\right ]\right )}{a^2 x},n\text {==}1\right \},\left \{\frac {a \left (1+\text {Log}\left [x\right ]-\text {Log}\left [\frac {a+b x}{b}\right ]\right )+b x \left (\text {Log}\left [x\right ]-\text {Log}\left [\frac {a+b x}{b}\right ]\right )}{a^2 \left (a+b x\right )},n\text {==}2\right \}\right \},-\frac {a^2 x^n}{2 a^2 x^2 \left (a+b x\right )^n-3 a^2 n x^2 \left (a+b x\right )^n+a^2 n^2 x^2 \left (a+b x\right )^n}+\frac {a^2 n x^n}{2 a^2 x^2 \left (a+b x\right )^n-3 a^2 n x^2 \left (a+b x\right )^n+a^2 n^2 x^2 \left (a+b x\right )^n}+\frac {a b n x x^n}{2 a^2 x^2 \left (a+b x\right )^n-3 a^2 n x^2 \left (a+b x\right )^n+a^2 n^2 x^2 \left (a+b x\right )^n}+\frac {b^2 x^2 x^n}{2 a^2 x^2 \left (a+b x\right )^n-3 a^2 n x^2 \left (a+b x\right )^n+a^2 n^2 x^2 \left (a+b x\right )^n}\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 44, normalized size = 0.69
method | result | size |
gosper | \(\frac {x^{-2+n} \left (a n +b x -a \right ) \left (b x +a \right ) \left (b x +a \right )^{-n}}{\left (-2+n \right ) \left (-1+n \right ) a^{2}}\) | \(44\) |
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 = 0.32, size = 64, normalized size = 1.00 \begin {gather*} \frac {{\left (a b n x^{2} + b^{2} x^{3} + {\left (a^{2} n - a^{2}\right )} x\right )} x^{n - 3}}{{\left (a^{2} n^{2} - 3 \, a^{2} n + 2 \, a^{2}\right )} {\left (b x + a\right )}^{n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
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 = 0.45, size = 80, normalized size = 1.25 \begin {gather*} \frac {\frac {x\,x^{n-3}\,\left (n-1\right )}{n^2-3\,n+2}+\frac {b^2\,x^{n-3}\,x^3}{a^2\,\left (n^2-3\,n+2\right )}+\frac {b\,n\,x^{n-3}\,x^2}{a\,\left (n^2-3\,n+2\right )}}{{\left (a+b\,x\right )}^n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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