Optimal. Leaf size=32 \[ \frac {2 a b x^n}{n}+\frac {b^2 x^{2 n}}{2 n}+a^2 \log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45}
\begin {gather*} a^2 \log (x)+\frac {2 a b x^n}{n}+\frac {b^2 x^{2 n}}{2 n} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b x^n\right )^2}{x} \, dx &=\frac {\text {Subst}\left (\int \frac {(a+b x)^2}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {2 a b x^n}{n}+\frac {b^2 x^{2 n}}{2 n}+a^2 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 32, normalized size = 1.00 \begin {gather*} \frac {b x^n \left (4 a+b x^n\right )}{2 n}+\frac {a^2 \log \left (x^n\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 31, normalized size = 0.97
method | result | size |
derivativedivides | \(\frac {\frac {b^{2} x^{2 n}}{2}+2 a b \,x^{n}+a^{2} \ln \left (x^{n}\right )}{n}\) | \(31\) |
default | \(\frac {\frac {b^{2} x^{2 n}}{2}+2 a b \,x^{n}+a^{2} \ln \left (x^{n}\right )}{n}\) | \(31\) |
risch | \(\frac {2 a b \,x^{n}}{n}+\frac {b^{2} x^{2 n}}{2 n}+a^{2} \ln \left (x \right )\) | \(31\) |
norman | \(a^{2} \ln \left (x \right )+\frac {b^{2} {\mathrm e}^{2 n \ln \left (x \right )}}{2 n}+\frac {2 a b \,{\mathrm e}^{n \ln \left (x \right )}}{n}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 34, normalized size = 1.06 \begin {gather*} \frac {a^{2} \log \left (x^{n}\right )}{n} + \frac {b^{2} x^{2 \, n} + 4 \, a b x^{n}}{2 \, n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.55, size = 30, normalized size = 0.94 \begin {gather*} \frac {2 \, a^{2} n \log \left (x\right ) + b^{2} x^{2 \, n} + 4 \, a b x^{n}}{2 \, n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 36, normalized size = 1.12 \begin {gather*} \begin {cases} a^{2} \log {\left (x \right )} + \frac {2 a b x^{n}}{n} + \frac {b^{2} x^{2 n}}{2 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 = 1.18, size = 30, normalized size = 0.94 \begin {gather*} \frac {b^2\,x^{2\,n}+4\,a\,b\,x^n+2\,a^2\,n\,\ln \left (x\right )}{2\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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