Optimal. Leaf size=39 \[ \frac {(a+b x)^{n+2}}{b^2 (n+2)}-\frac {a (a+b x)^{n+1}}{b^2 (n+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {43} \begin {gather*} \frac {(a+b x)^{n+2}}{b^2 (n+2)}-\frac {a (a+b x)^{n+1}}{b^2 (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int x (a+b x)^n \, dx &=\int \left (-\frac {a (a+b x)^n}{b}+\frac {(a+b x)^{1+n}}{b}\right ) \, dx\\ &=-\frac {a (a+b x)^{1+n}}{b^2 (1+n)}+\frac {(a+b x)^{2+n}}{b^2 (2+n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.85 \begin {gather*} \frac {(a+b x)^{n+1} (b (n+1) x-a)}{b^2 (n+1) (n+2)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int x (a+b x)^n \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.14, size = 53, normalized size = 1.36 \begin {gather*} \frac {{\left (a b n x + {\left (b^{2} n + b^{2}\right )} x^{2} - a^{2}\right )} {\left (b x + a\right )}^{n}}{b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 76, normalized size = 1.95 \begin {gather*} \frac {{\left (b x + a\right )}^{n} b^{2} n x^{2} + {\left (b x + a\right )}^{n} a b n x + {\left (b x + a\right )}^{n} b^{2} x^{2} - {\left (b x + a\right )}^{n} a^{2}}{b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 0.92 \begin {gather*} -\frac {\left (-x n b -b x +a \right ) \left (b x +a \right )^{n +1}}{\left (n^{2}+3 n +2\right ) b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 42, normalized size = 1.08 \begin {gather*} \frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 94, normalized size = 2.41 \begin {gather*} \left \{\begin {array}{cl} -\frac {a\,\ln \left (a+b\,x\right )-b\,x}{b^2} & \text {\ if\ \ }n=-1\\ \frac {\ln \left (a+b\,x\right )+\frac {a}{a+b\,x}}{b^2} & \text {\ if\ \ }n=-2\\ \frac {2\,\left (\frac {{\left (a+b\,x\right )}^{n+2}}{2\,n+4}-\frac {a\,{\left (a+b\,x\right )}^{n+1}}{2\,n+2}\right )}{b^2} & \text {\ if\ \ }n\neq -1\wedge n\neq -2 \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.70, size = 201, normalized size = 5.15 \begin {gather*} \begin {cases} \frac {a^{n} x^{2}}{2} & \text {for}\: b = 0 \\\frac {a \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac {a}{a b^{2} + b^{3} x} + \frac {b x \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text {for}\: n = -2 \\- \frac {a \log {\left (\frac {a}{b} + x \right )}}{b^{2}} + \frac {x}{b} & \text {for}\: n = -1 \\- \frac {a^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {a b n x \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {b^{2} n x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {b^{2} x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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