Optimal. Leaf size=75 \[ \frac {a^3 x^{m+1}}{m+1}+\frac {3 a^2 b x^{m+n+1}}{m+n+1}+\frac {3 a b^2 x^{m+2 n+1}}{m+2 n+1}+\frac {b^3 x^{m+3 n+1}}{m+3 n+1} \]
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Rubi [A] time = 0.03, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \frac {3 a^2 b x^{m+n+1}}{m+n+1}+\frac {a^3 x^{m+1}}{m+1}+\frac {3 a b^2 x^{m+2 n+1}}{m+2 n+1}+\frac {b^3 x^{m+3 n+1}}{m+3 n+1} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int x^m \left (a+b x^n\right )^3 \, dx &=\int \left (a^3 x^m+3 a^2 b x^{m+n}+3 a b^2 x^{m+2 n}+b^3 x^{m+3 n}\right ) \, dx\\ &=\frac {a^3 x^{1+m}}{1+m}+\frac {3 a^2 b x^{1+m+n}}{1+m+n}+\frac {3 a b^2 x^{1+m+2 n}}{1+m+2 n}+\frac {b^3 x^{1+m+3 n}}{1+m+3 n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 67, normalized size = 0.89 \[ x^{m+1} \left (\frac {a^3}{m+1}+\frac {3 a^2 b x^n}{m+n+1}+\frac {3 a b^2 x^{2 n}}{m+2 n+1}+\frac {b^3 x^{3 n}}{m+3 n+1}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 362, normalized size = 4.83 \[ \frac {{\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3} + 2 \, {\left (b^{3} m + b^{3}\right )} n^{2} + 3 \, {\left (b^{3} m^{2} + 2 \, b^{3} m + b^{3}\right )} n\right )} x x^{m} x^{3 \, n} + 3 \, {\left (a b^{2} m^{3} + 3 \, a b^{2} m^{2} + 3 \, a b^{2} m + a b^{2} + 3 \, {\left (a b^{2} m + a b^{2}\right )} n^{2} + 4 \, {\left (a b^{2} m^{2} + 2 \, a b^{2} m + a b^{2}\right )} n\right )} x x^{m} x^{2 \, n} + 3 \, {\left (a^{2} b m^{3} + 3 \, a^{2} b m^{2} + 3 \, a^{2} b m + a^{2} b + 6 \, {\left (a^{2} b m + a^{2} b\right )} n^{2} + 5 \, {\left (a^{2} b m^{2} + 2 \, a^{2} b m + a^{2} b\right )} n\right )} x x^{m} x^{n} + {\left (a^{3} m^{3} + 6 \, a^{3} n^{3} + 3 \, a^{3} m^{2} + 3 \, a^{3} m + a^{3} + 11 \, {\left (a^{3} m + a^{3}\right )} n^{2} + 6 \, {\left (a^{3} m^{2} + 2 \, a^{3} m + a^{3}\right )} n\right )} x x^{m}}{m^{4} + 6 \, {\left (m + 1\right )} n^{3} + 4 \, m^{3} + 11 \, {\left (m^{2} + 2 \, m + 1\right )} n^{2} + 6 \, m^{2} + 6 \, {\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} n + 4 \, m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 622, normalized size = 8.29 \[ \frac {b^{3} m^{3} x x^{m} x^{3 \, n} + 3 \, b^{3} m^{2} n x x^{m} x^{3 \, n} + 2 \, b^{3} m n^{2} x x^{m} x^{3 \, n} + 3 \, a b^{2} m^{3} x x^{m} x^{2 \, n} + 12 \, a b^{2} m^{2} n x x^{m} x^{2 \, n} + 9 \, a b^{2} m n^{2} x x^{m} x^{2 \, n} + 3 \, a^{2} b m^{3} x x^{m} x^{n} + 15 \, a^{2} b m^{2} n x x^{m} x^{n} + 18 \, a^{2} b m n^{2} x x^{m} x^{n} + a^{3} m^{3} x x^{m} + 6 \, a^{3} m^{2} n x x^{m} + 11 \, a^{3} m n^{2} x x^{m} + 6 \, a^{3} n^{3} x x^{m} + 3 \, b^{3} m^{2} x x^{m} x^{3 \, n} + 6 \, b^{3} m n x x^{m} x^{3 \, n} + 2 \, b^{3} n^{2} x x^{m} x^{3 \, n} + 9 \, a b^{2} m^{2} x x^{m} x^{2 \, n} + 24 \, a b^{2} m n x x^{m} x^{2 \, n} + 9 \, a b^{2} n^{2} x x^{m} x^{2 \, n} + 9 \, a^{2} b m^{2} x x^{m} x^{n} + 30 \, a^{2} b m n x x^{m} x^{n} + 18 \, a^{2} b n^{2} x x^{m} x^{n} + 3 \, a^{3} m^{2} x x^{m} + 12 \, a^{3} m n x x^{m} + 11 \, a^{3} n^{2} x x^{m} + 3 \, b^{3} m x x^{m} x^{3 \, n} + 3 \, b^{3} n x x^{m} x^{3 \, n} + 9 \, a b^{2} m x x^{m} x^{2 \, n} + 12 \, a b^{2} n x x^{m} x^{2 \, n} + 9 \, a^{2} b m x x^{m} x^{n} + 15 \, a^{2} b n x x^{m} x^{n} + 3 \, a^{3} m x x^{m} + 6 \, a^{3} n x x^{m} + b^{3} x x^{m} x^{3 \, n} + 3 \, a b^{2} x x^{m} x^{2 \, n} + 3 \, a^{2} b x x^{m} x^{n} + a^{3} x x^{m}}{m^{4} + 6 \, m^{3} n + 11 \, m^{2} n^{2} + 6 \, m n^{3} + 4 \, m^{3} + 18 \, m^{2} n + 22 \, m n^{2} + 6 \, n^{3} + 6 \, m^{2} + 18 \, m n + 11 \, n^{2} + 4 \, m + 6 \, n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 92, normalized size = 1.23 \[ \frac {3 a^{2} b x \,{\mathrm e}^{m \ln \relax (x )} {\mathrm e}^{n \ln \relax (x )}}{m +n +1}+\frac {3 a \,b^{2} x \,{\mathrm e}^{m \ln \relax (x )} {\mathrm e}^{2 n \ln \relax (x )}}{m +2 n +1}+\frac {b^{3} x \,{\mathrm e}^{m \ln \relax (x )} {\mathrm e}^{3 n \ln \relax (x )}}{m +3 n +1}+\frac {a^{3} x \,{\mathrm e}^{m \ln \relax (x )}}{m +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 75, normalized size = 1.00 \[ \frac {b^{3} x^{m + 3 \, n + 1}}{m + 3 \, n + 1} + \frac {3 \, a b^{2} x^{m + 2 \, n + 1}}{m + 2 \, n + 1} + \frac {3 \, a^{2} b x^{m + n + 1}}{m + n + 1} + \frac {a^{3} x^{m + 1}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.53, size = 77, normalized size = 1.03 \[ \frac {a^3\,x\,x^m}{m+1}+\frac {b^3\,x\,x^m\,x^{3\,n}}{m+3\,n+1}+\frac {3\,a^2\,b\,x\,x^m\,x^n}{m+n+1}+\frac {3\,a\,b^2\,x\,x^m\,x^{2\,n}}{m+2\,n+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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