Optimal. Leaf size=48 \[ -\frac {a^3 x^{-2 n}}{2 n}-\frac {3 a^2 b x^{-n}}{n}+3 a b^2 \log (x)+\frac {b^3 x^n}{n} \]
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Rubi [A] time = 0.02, antiderivative size = 48, 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 = {266, 43} \[ -\frac {3 a^2 b x^{-n}}{n}-\frac {a^3 x^{-2 n}}{2 n}+3 a b^2 \log (x)+\frac {b^3 x^n}{n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1-2 n} \left (a+b x^n\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^3}{x^3} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (b^3+\frac {a^3}{x^3}+\frac {3 a^2 b}{x^2}+\frac {3 a b^2}{x}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^3 x^{-2 n}}{2 n}-\frac {3 a^2 b x^{-n}}{n}+\frac {b^3 x^n}{n}+3 a b^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 44, normalized size = 0.92 \[ \frac {-\frac {1}{2} a^3 x^{-2 n}-3 a^2 b x^{-n}+3 a b^2 n \log (x)+b^3 x^n}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 51, normalized size = 1.06 \[ \frac {6 \, a b^{2} n x^{2 \, n} \log \relax (x) + 2 \, b^{3} x^{3 \, n} - 6 \, a^{2} b x^{n} - a^{3}}{2 \, n x^{2 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 51, normalized size = 1.06 \[ \frac {6 \, a b^{2} n x^{2 \, n} \log \relax (x) + 2 \, b^{3} x^{3 \, n} - 6 \, a^{2} b x^{n} - a^{3}}{2 \, n x^{2 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 61, normalized size = 1.27 \[ \left (3 a \,b^{2} {\mathrm e}^{2 n \ln \relax (x )} \ln \relax (x )-\frac {3 a^{2} b \,{\mathrm e}^{n \ln \relax (x )}}{n}+\frac {b^{3} {\mathrm e}^{3 n \ln \relax (x )}}{n}-\frac {a^{3}}{2 n}\right ) {\mathrm e}^{-2 n \ln \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 48, normalized size = 1.00 \[ 3 \, a b^{2} \log \relax (x) + \frac {b^{3} x^{n}}{n} - \frac {a^{3}}{2 \, n x^{2 \, n}} - \frac {3 \, a^{2} b}{n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 48, normalized size = 1.00 \[ \frac {b^3\,x^n}{n}+3\,a\,b^2\,\ln \relax (x)-\frac {a^3}{2\,n\,x^{2\,n}}-\frac {3\,a^2\,b}{n\,x^n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 117.45, size = 318, normalized size = 6.62 \[ \begin {cases} a^{3} x + 6 a^{2} b \sqrt {x} + 3 a b^{2} \log {\relax (x )} - \frac {2 b^{3}}{\sqrt {x}} & \text {for}\: n = - \frac {1}{2} \\\left (a + b\right )^{3} \log {\relax (x )} & \text {for}\: n = 0 \\- \frac {2 a^{3} n}{4 n^{2} x^{2 n} + 2 n x^{2 n}} - \frac {a^{3}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} - \frac {12 a^{2} b n x^{n}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} - \frac {6 a^{2} b x^{n}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} + \frac {12 a b^{2} n^{2} x^{2 n} \log {\relax (x )}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} + \frac {6 a b^{2} n x^{2 n} \log {\relax (x )}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} + \frac {6 a b^{2} n x^{2 n}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} + \frac {4 b^{3} n x^{3 n}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} + \frac {2 b^{3} x^{3 n}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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