Optimal. Leaf size=63 \[ \frac {a^3 x^{3 n}}{3 n}+\frac {3 a^2 b x^{4 n}}{4 n}+\frac {3 a b^2 x^{5 n}}{5 n}+\frac {b^3 x^{6 n}}{6 n} \]
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Rubi [A] time = 0.03, antiderivative size = 63, 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} \[ \frac {3 a^2 b x^{4 n}}{4 n}+\frac {a^3 x^{3 n}}{3 n}+\frac {3 a b^2 x^{5 n}}{5 n}+\frac {b^3 x^{6 n}}{6 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1+3 n} \left (a+b x^n\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int x^2 (a+b x)^3 \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^3 x^2+3 a^2 b x^3+3 a b^2 x^4+b^3 x^5\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {a^3 x^{3 n}}{3 n}+\frac {3 a^2 b x^{4 n}}{4 n}+\frac {3 a b^2 x^{5 n}}{5 n}+\frac {b^3 x^{6 n}}{6 n}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 48, normalized size = 0.76 \[ \frac {x^{3 n} \left (20 a^3+45 a^2 b x^n+36 a b^2 x^{2 n}+10 b^3 x^{3 n}\right )}{60 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 48, normalized size = 0.76 \[ \frac {10 \, b^{3} x^{6 \, n} + 36 \, a b^{2} x^{5 \, n} + 45 \, a^{2} b x^{4 \, n} + 20 \, a^{3} x^{3 \, n}}{60 \, n} \]
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 \[ \int {\left (b x^{n} + a\right )}^{3} x^{3 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 56, normalized size = 0.89 \[ \frac {a^{3} x^{3 n}}{3 n}+\frac {3 a^{2} b \,x^{4 n}}{4 n}+\frac {3 a \,b^{2} x^{5 n}}{5 n}+\frac {b^{3} x^{6 n}}{6 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 55, normalized size = 0.87 \[ \frac {b^{3} x^{6 \, n}}{6 \, n} + \frac {3 \, a b^{2} x^{5 \, n}}{5 \, n} + \frac {3 \, a^{2} b x^{4 \, n}}{4 \, n} + \frac {a^{3} x^{3 \, n}}{3 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 55, normalized size = 0.87 \[ \frac {a^3\,x^{3\,n}}{3\,n}+\frac {b^3\,x^{6\,n}}{6\,n}+\frac {3\,a^2\,b\,x^{4\,n}}{4\,n}+\frac {3\,a\,b^2\,x^{5\,n}}{5\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 26.44, size = 61, normalized size = 0.97 \[ \begin {cases} \frac {a^{3} x^{3 n}}{3 n} + \frac {3 a^{2} b x^{4 n}}{4 n} + \frac {3 a b^{2} x^{5 n}}{5 n} + \frac {b^{3} x^{6 n}}{6 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{3} \log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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