Optimal. Leaf size=44 \[ \frac {a^2 x^2}{2}+\frac {b^2 x^{2 (1+n)}}{2 (1+n)}+\frac {2 a b x^{2+n}}{2+n} \]
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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {276}
\begin {gather*} \frac {a^2 x^2}{2}+\frac {2 a b x^{n+2}}{n+2}+\frac {b^2 x^{2 (n+1)}}{2 (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rubi steps
\begin {align*} \int x \left (a+b x^n\right )^2 \, dx &=\int \left (a^2 x+2 a b x^{1+n}+b^2 x^{1+2 n}\right ) \, dx\\ &=\frac {a^2 x^2}{2}+\frac {b^2 x^{2 (1+n)}}{2 (1+n)}+\frac {2 a b x^{2+n}}{2+n}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 37, normalized size = 0.84 \begin {gather*} \frac {1}{2} x^2 \left (a^2+\frac {4 a b x^n}{2+n}+\frac {b^2 x^{2 n}}{1+n}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 43, normalized size = 0.98
method | result | size |
risch | \(\frac {a^{2} x^{2}}{2}+\frac {b^{2} x^{2} x^{2 n}}{2+2 n}+\frac {2 a b \,x^{2} x^{n}}{2+n}\) | \(43\) |
norman | \(\frac {a^{2} x^{2}}{2}+\frac {b^{2} x^{2} {\mathrm e}^{2 n \ln \left (x \right )}}{2+2 n}+\frac {2 a b \,x^{2} {\mathrm e}^{n \ln \left (x \right )}}{2+n}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 40, normalized size = 0.91 \begin {gather*} \frac {1}{2} \, a^{2} x^{2} + \frac {b^{2} x^{2 \, n + 2}}{2 \, {\left (n + 1\right )}} + \frac {2 \, a b x^{n + 2}}{n + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 72, normalized size = 1.64 \begin {gather*} \frac {{\left (b^{2} n + 2 \, b^{2}\right )} x^{2} x^{2 \, n} + 4 \, {\left (a b n + a b\right )} x^{2} x^{n} + {\left (a^{2} n^{2} + 3 \, a^{2} n + 2 \, a^{2}\right )} x^{2}}{2 \, {\left (n^{2} + 3 \, n + 2\right )}} \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. 201 vs.
\(2 (36) = 72\).
time = 0.16, size = 201, normalized size = 4.57 \begin {gather*} \begin {cases} \frac {a^{2} x^{2}}{2} + 2 a b \log {\left (x \right )} - \frac {b^{2}}{2 x^{2}} & \text {for}\: n = -2 \\\frac {a^{2} x^{2}}{2} + 2 a b x + b^{2} \log {\left (x \right )} & \text {for}\: n = -1 \\\frac {a^{2} n^{2} x^{2}}{2 n^{2} + 6 n + 4} + \frac {3 a^{2} n x^{2}}{2 n^{2} + 6 n + 4} + \frac {2 a^{2} x^{2}}{2 n^{2} + 6 n + 4} + \frac {4 a b n x^{2} x^{n}}{2 n^{2} + 6 n + 4} + \frac {4 a b x^{2} x^{n}}{2 n^{2} + 6 n + 4} + \frac {b^{2} n x^{2} x^{2 n}}{2 n^{2} + 6 n + 4} + \frac {2 b^{2} x^{2} x^{2 n}}{2 n^{2} + 6 n + 4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs.
\(2 (40) = 80\).
time = 1.20, size = 87, normalized size = 1.98 \begin {gather*} \frac {b^{2} n x^{2} x^{2 \, n} + 4 \, a b n x^{2} x^{n} + a^{2} n^{2} x^{2} + 2 \, b^{2} x^{2} x^{2 \, n} + 4 \, a b x^{2} x^{n} + 3 \, a^{2} n x^{2} + 2 \, a^{2} x^{2}}{2 \, {\left (n^{2} + 3 \, n + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.24, size = 43, normalized size = 0.98 \begin {gather*} \frac {a^2\,x^2}{2}+\frac {b^2\,x^{2\,n}\,x^2}{2\,n+2}+\frac {2\,a\,b\,x^n\,x^2}{n+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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