Optimal. Leaf size=40 \[ \frac {x^{n+1} (a d+b c)}{n+1}+a c x+\frac {b d x^{2 n+1}}{2 n+1} \]
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Rubi [A] time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {373} \[ \frac {x^{n+1} (a d+b c)}{n+1}+a c x+\frac {b d x^{2 n+1}}{2 n+1} \]
Antiderivative was successfully verified.
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Rule 373
Rubi steps
\begin {align*} \int \left (a+b x^n\right ) \left (c+d x^n\right ) \, dx &=\int \left (a c+(b c+a d) x^n+b d x^{2 n}\right ) \, dx\\ &=a c x+\frac {(b c+a d) x^{1+n}}{1+n}+\frac {b d x^{1+2 n}}{1+2 n}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 37, normalized size = 0.92 \[ x \left (\frac {x^n (a d+b c)}{n+1}+a c+\frac {b d x^{2 n}}{2 n+1}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 69, normalized size = 1.72 \[ \frac {{\left (b d n + b d\right )} x x^{2 \, n} + {\left (b c + a d + 2 \, {\left (b c + a d\right )} n\right )} x x^{n} + {\left (2 \, a c n^{2} + 3 \, a c n + a c\right )} x}{2 \, n^{2} + 3 \, n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 83, normalized size = 2.08 \[ \frac {2 \, a c n^{2} x + b d n x x^{2 \, n} + 2 \, b c n x x^{n} + 2 \, a d n x x^{n} + 3 \, a c n x + b d x x^{2 \, n} + b c x x^{n} + a d x x^{n} + a c x}{2 \, n^{2} + 3 \, n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 43, normalized size = 1.08 \[ \frac {b d x \,{\mathrm e}^{2 n \ln \relax (x )}}{2 n +1}+a c x +\frac {\left (a d +b c \right ) x \,{\mathrm e}^{n \ln \relax (x )}}{n +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 48, normalized size = 1.20 \[ a c x + \frac {b d x^{2 \, n + 1}}{2 \, n + 1} + \frac {b c x^{n + 1}}{n + 1} + \frac {a d x^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.48, size = 38, normalized size = 0.95 \[ a\,c\,x+\frac {x\,x^n\,\left (a\,d+b\,c\right )}{n+1}+\frac {b\,d\,x\,x^{2\,n}}{2\,n+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 236, normalized size = 5.90 \[ \begin {cases} a c x + a d \log {\relax (x )} + b c \log {\relax (x )} - \frac {b d}{x} & \text {for}\: n = -1 \\a c x + 2 a d \sqrt {x} + 2 b c \sqrt {x} + b d \log {\relax (x )} & \text {for}\: n = - \frac {1}{2} \\\frac {2 a c n^{2} x}{2 n^{2} + 3 n + 1} + \frac {3 a c n x}{2 n^{2} + 3 n + 1} + \frac {a c x}{2 n^{2} + 3 n + 1} + \frac {2 a d n x x^{n}}{2 n^{2} + 3 n + 1} + \frac {a d x x^{n}}{2 n^{2} + 3 n + 1} + \frac {2 b c n x x^{n}}{2 n^{2} + 3 n + 1} + \frac {b c x x^{n}}{2 n^{2} + 3 n + 1} + \frac {b d n x x^{2 n}}{2 n^{2} + 3 n + 1} + \frac {b d x x^{2 n}}{2 n^{2} + 3 n + 1} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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