Optimal. Leaf size=63 \[ \frac {a^3 x^{4 n}}{4 n}+\frac {3 a^2 b x^{5 n}}{5 n}+\frac {a b^2 x^{6 n}}{2 n}+\frac {b^3 x^{7 n}}{7 n} \]
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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} \[ \frac {3 a^2 b x^{5 n}}{5 n}+\frac {a^3 x^{4 n}}{4 n}+\frac {a b^2 x^{6 n}}{2 n}+\frac {b^3 x^{7 n}}{7 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1+4 n} \left (a+b x^n\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int x^3 (a+b x)^3 \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^3 x^3+3 a^2 b x^4+3 a b^2 x^5+b^3 x^6\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {a^3 x^{4 n}}{4 n}+\frac {3 a^2 b x^{5 n}}{5 n}+\frac {a b^2 x^{6 n}}{2 n}+\frac {b^3 x^{7 n}}{7 n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 0.76 \[ \frac {x^{4 n} \left (35 a^3+84 a^2 b x^n+70 a b^2 x^{2 n}+20 b^3 x^{3 n}\right )}{140 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 48, normalized size = 0.76 \[ \frac {20 \, b^{3} x^{7 \, n} + 70 \, a b^{2} x^{6 \, n} + 84 \, a^{2} b x^{5 \, n} + 35 \, a^{3} x^{4 \, n}}{140 \, n} \]
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 \[ \int {\left (b x^{n} + a\right )}^{3} x^{4 \, n - 1}\,{d x} \]
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 \[ \frac {a^{3} x^{4 n}}{4 n}+\frac {3 a^{2} b \,x^{5 n}}{5 n}+\frac {a \,b^{2} x^{6 n}}{2 n}+\frac {b^{3} x^{7 n}}{7 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 55, normalized size = 0.87 \[ \frac {b^{3} x^{7 \, n}}{7 \, n} + \frac {a b^{2} x^{6 \, n}}{2 \, n} + \frac {3 \, a^{2} b x^{5 \, n}}{5 \, n} + \frac {a^{3} x^{4 \, n}}{4 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 55, normalized size = 0.87 \[ \frac {a^3\,x^{4\,n}}{4\,n}+\frac {b^3\,x^{7\,n}}{7\,n}+\frac {3\,a^2\,b\,x^{5\,n}}{5\,n}+\frac {a\,b^2\,x^{6\,n}}{2\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 26.85, size = 60, normalized size = 0.95 \[ \begin {cases} \frac {a^{3} x^{4 n}}{4 n} + \frac {3 a^{2} b x^{5 n}}{5 n} + \frac {a b^{2} x^{6 n}}{2 n} + \frac {b^{3} x^{7 n}}{7 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{3} \log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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