Optimal. Leaf size=70 \[ a^2 c x+\frac {a x^{n+1} (a d+2 b c)}{n+1}+\frac {b x^{2 n+1} (2 a d+b c)}{2 n+1}+\frac {b^2 d x^{3 n+1}}{3 n+1} \]
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Rubi [A] time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {373} \[ a^2 c x+\frac {a x^{n+1} (a d+2 b c)}{n+1}+\frac {b x^{2 n+1} (2 a d+b c)}{2 n+1}+\frac {b^2 d x^{3 n+1}}{3 n+1} \]
Antiderivative was successfully verified.
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Rule 373
Rubi steps
\begin {align*} \int \left (a+b x^n\right )^2 \left (c+d x^n\right ) \, dx &=\int \left (a^2 c+a (2 b c+a d) x^n+b (b c+2 a d) x^{2 n}+b^2 d x^{3 n}\right ) \, dx\\ &=a^2 c x+\frac {a (2 b c+a d) x^{1+n}}{1+n}+\frac {b (b c+2 a d) x^{1+2 n}}{1+2 n}+\frac {b^2 d x^{1+3 n}}{1+3 n}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 70, normalized size = 1.00 \[ \frac {d x \left (a+b x^n\right )^3-x \left (a^2+\frac {2 a b x^n}{n+1}+\frac {b^2 x^{2 n}}{2 n+1}\right ) (a d-b (3 c n+c))}{3 b n+b} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.08, size = 175, normalized size = 2.50 \[ \frac {{\left (2 \, b^{2} d n^{2} + 3 \, b^{2} d n + b^{2} d\right )} x x^{3 \, n} + {\left (b^{2} c + 2 \, a b d + 3 \, {\left (b^{2} c + 2 \, a b d\right )} n^{2} + 4 \, {\left (b^{2} c + 2 \, a b d\right )} n\right )} x x^{2 \, n} + {\left (2 \, a b c + a^{2} d + 6 \, {\left (2 \, a b c + a^{2} d\right )} n^{2} + 5 \, {\left (2 \, a b c + a^{2} d\right )} n\right )} x x^{n} + {\left (6 \, a^{2} c n^{3} + 11 \, a^{2} c n^{2} + 6 \, a^{2} c n + a^{2} c\right )} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 232, normalized size = 3.31 \[ \frac {6 \, a^{2} c n^{3} x + 2 \, b^{2} d n^{2} x x^{3 \, n} + 3 \, b^{2} c n^{2} x x^{2 \, n} + 6 \, a b d n^{2} x x^{2 \, n} + 12 \, a b c n^{2} x x^{n} + 6 \, a^{2} d n^{2} x x^{n} + 11 \, a^{2} c n^{2} x + 3 \, b^{2} d n x x^{3 \, n} + 4 \, b^{2} c n x x^{2 \, n} + 8 \, a b d n x x^{2 \, n} + 10 \, a b c n x x^{n} + 5 \, a^{2} d n x x^{n} + 6 \, a^{2} c n x + b^{2} d x x^{3 \, n} + b^{2} c x x^{2 \, n} + 2 \, a b d x x^{2 \, n} + 2 \, a b c x x^{n} + a^{2} d x x^{n} + a^{2} c x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 74, normalized size = 1.06 \[ \frac {b^{2} d x \,{\mathrm e}^{3 n \ln \relax (x )}}{3 n +1}+a^{2} c x +\frac {\left (a d +2 b c \right ) a x \,{\mathrm e}^{n \ln \relax (x )}}{n +1}+\frac {\left (2 a d +b c \right ) b x \,{\mathrm e}^{2 n \ln \relax (x )}}{2 n +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 94, normalized size = 1.34 \[ a^{2} c x + \frac {b^{2} d x^{3 \, n + 1}}{3 \, n + 1} + \frac {b^{2} c x^{2 \, n + 1}}{2 \, n + 1} + \frac {2 \, a b d x^{2 \, n + 1}}{2 \, n + 1} + \frac {2 \, a b c x^{n + 1}}{n + 1} + \frac {a^{2} d x^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.53, size = 71, normalized size = 1.01 \[ a^2\,c\,x+\frac {x\,x^{2\,n}\,\left (c\,b^2+2\,a\,d\,b\right )}{2\,n+1}+\frac {x\,x^n\,\left (d\,a^2+2\,b\,c\,a\right )}{n+1}+\frac {b^2\,d\,x\,x^{3\,n}}{3\,n+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.74, size = 726, normalized size = 10.37 \[ \begin {cases} a^{2} c x + a^{2} d \log {\relax (x )} + 2 a b c \log {\relax (x )} - \frac {2 a b d}{x} - \frac {b^{2} c}{x} - \frac {b^{2} d}{2 x^{2}} & \text {for}\: n = -1 \\a^{2} c x + 2 a^{2} d \sqrt {x} + 4 a b c \sqrt {x} + 2 a b d \log {\relax (x )} + b^{2} c \log {\relax (x )} - \frac {2 b^{2} d}{\sqrt {x}} & \text {for}\: n = - \frac {1}{2} \\a^{2} c x + \frac {3 a^{2} d x^{\frac {2}{3}}}{2} + 3 a b c x^{\frac {2}{3}} + 6 a b d \sqrt [3]{x} + 3 b^{2} c \sqrt [3]{x} + b^{2} d \log {\relax (x )} & \text {for}\: n = - \frac {1}{3} \\\frac {6 a^{2} c n^{3} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {11 a^{2} c n^{2} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {6 a^{2} c n x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {a^{2} c x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {6 a^{2} d n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {5 a^{2} d n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {a^{2} d x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {12 a b c n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {10 a b c n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {2 a b c x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {6 a b d n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {8 a b d n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {2 a b d x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {3 b^{2} c n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {4 b^{2} c n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {b^{2} c x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {2 b^{2} d n^{2} x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {3 b^{2} d n x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {b^{2} d x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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