Optimal. Leaf size=50 \[ \frac {3 a^2 b x^n}{n}+\frac {3 a b^2 x^{2 n}}{2 n}+\frac {b^3 x^{3 n}}{3 n}+a^3 \log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 50, 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^3 \log (x)+\frac {3 a^2 b x^n}{n}+\frac {3 a b^2 x^{2 n}}{2 n}+\frac {b^3 x^{3 n}}{3 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 )^3}{x} \, dx &=\frac {\text {Subst}\left (\int \frac {(a+b x)^3}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (3 a^2 b+\frac {a^3}{x}+3 a b^2 x+b^3 x^2\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {3 a^2 b x^n}{n}+\frac {3 a b^2 x^{2 n}}{2 n}+\frac {b^3 x^{3 n}}{3 n}+a^3 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 46, normalized size = 0.92 \begin {gather*} \frac {b x^n \left (18 a^2+9 a b x^n+2 b^2 x^{2 n}\right )}{6 n}+\frac {a^3 \log \left (x^n\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 44, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {\frac {b^{3} x^{3 n}}{3}+\frac {3 a \,b^{2} x^{2 n}}{2}+3 a^{2} b \,x^{n}+a^{3} \ln \left (x^{n}\right )}{n}\) | \(44\) |
default | \(\frac {\frac {b^{3} x^{3 n}}{3}+\frac {3 a \,b^{2} x^{2 n}}{2}+3 a^{2} b \,x^{n}+a^{3} \ln \left (x^{n}\right )}{n}\) | \(44\) |
risch | \(\frac {3 a^{2} b \,x^{n}}{n}+\frac {3 a \,b^{2} x^{2 n}}{2 n}+\frac {b^{3} x^{3 n}}{3 n}+a^{3} \ln \left (x \right )\) | \(47\) |
norman | \(a^{3} \ln \left (x \right )+\frac {b^{3} {\mathrm e}^{3 n \ln \left (x \right )}}{3 n}+\frac {3 a \,b^{2} {\mathrm e}^{2 n \ln \left (x \right )}}{2 n}+\frac {3 a^{2} b \,{\mathrm e}^{n \ln \left (x \right )}}{n}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 48, normalized size = 0.96 \begin {gather*} \frac {a^{3} \log \left (x^{n}\right )}{n} + \frac {2 \, b^{3} x^{3 \, n} + 9 \, a b^{2} x^{2 \, n} + 18 \, a^{2} b x^{n}}{6 \, n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 44, normalized size = 0.88 \begin {gather*} \frac {6 \, a^{3} n \log \left (x\right ) + 2 \, b^{3} x^{3 \, n} + 9 \, a b^{2} x^{2 \, n} + 18 \, a^{2} b x^{n}}{6 \, n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 53, normalized size = 1.06 \begin {gather*} \begin {cases} a^{3} \log {\left (x \right )} + \frac {3 a^{2} b x^{n}}{n} + \frac {3 a b^{2} x^{2 n}}{2 n} + \frac {b^{3} x^{3 n}}{3 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{3} \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 = 0.00, size = 46, normalized size = 0.92 \begin {gather*} a^3\,\ln \left (x\right )+\frac {b^3\,x^{3\,n}}{3\,n}+\frac {3\,a^2\,b\,x^n}{n}+\frac {3\,a\,b^2\,x^{2\,n}}{2\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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