Optimal. Leaf size=45 \[ \frac {a^2 x^{2 n}}{2 n}+\frac {2 a b x^{3 n}}{3 n}+\frac {b^2 x^{4 n}}{4 n} \]
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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {272, 45}
\begin {gather*} \frac {a^2 x^{2 n}}{2 n}+\frac {2 a b x^{3 n}}{3 n}+\frac {b^2 x^{4 n}}{4 n} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^{-1+2 n} \left (a+b x^n\right )^2 \, dx &=\frac {\text {Subst}\left (\int x (a+b x)^2 \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {a^2 x^{2 n}}{2 n}+\frac {2 a b x^{3 n}}{3 n}+\frac {b^2 x^{4 n}}{4 n}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 35, normalized size = 0.78 \begin {gather*} \frac {x^{2 n} \left (6 a^2+8 a b x^n+3 b^2 x^{2 n}\right )}{12 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 40, normalized size = 0.89
method | result | size |
risch | \(\frac {a^{2} x^{2 n}}{2 n}+\frac {2 a b \,x^{3 n}}{3 n}+\frac {b^{2} x^{4 n}}{4 n}\) | \(40\) |
norman | \(\frac {a^{2} {\mathrm e}^{2 n \ln \left (x \right )}}{2 n}+\frac {b^{2} {\mathrm e}^{4 n \ln \left (x \right )}}{4 n}+\frac {2 a b \,{\mathrm e}^{3 n \ln \left (x \right )}}{3 n}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 39, normalized size = 0.87 \begin {gather*} \frac {b^{2} x^{4 \, n}}{4 \, n} + \frac {2 \, a b x^{3 \, n}}{3 \, n} + \frac {a^{2} x^{2 \, n}}{2 \, n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 35, normalized size = 0.78 \begin {gather*} \frac {3 \, b^{2} x^{4 \, n} + 8 \, a b x^{3 \, n} + 6 \, a^{2} x^{2 \, n}}{12 \, n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.12, size = 44, normalized size = 0.98 \begin {gather*} \begin {cases} \frac {a^{2} x^{2 n}}{2 n} + \frac {2 a b x^{3 n}}{3 n} + \frac {b^{2} x^{4 n}}{4 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{2} \log {\left (x \right )} & \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 = 1.29, size = 33, normalized size = 0.73 \begin {gather*} \frac {x^{2\,n}\,\left (6\,a^2+3\,b^2\,x^{2\,n}+8\,a\,b\,x^n\right )}{12\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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