Optimal. Leaf size=41 \[ a c x+\frac {a d x^n}{n}+\frac {b c x^{n+1}}{n+1}+\frac {b d x^{2 n}}{2 n} \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1891, 14} \[ a c x+\frac {a d x^n}{n}+\frac {b c x^{n+1}}{n+1}+\frac {b d x^{2 n}}{2 n} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1891
Rubi steps
\begin {align*} \int \left (c+d x^{-1+n}\right ) \left (a+b x^n\right ) \, dx &=c \int \left (a+b x^n\right ) \, dx+d \int x^{-1+n} \left (a+b x^n\right ) \, dx\\ &=a c x+\frac {b c x^{1+n}}{1+n}+d \int \left (a x^{-1+n}+b x^{-1+2 n}\right ) \, dx\\ &=a c x+\frac {a d x^n}{n}+\frac {b d x^{2 n}}{2 n}+\frac {b c x^{1+n}}{1+n}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 42, normalized size = 1.02 \[ \frac {2 a \left (c n x+d x^n\right )+b x^n \left (\frac {2 c n x}{n+1}+d x^n\right )}{2 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 56, normalized size = 1.37 \[ \frac {2 \, {\left (a c n^{2} + a c n\right )} x + {\left (b d n + b d\right )} x^{2 \, n} + 2 \, {\left (b c n x + a d n + a d\right )} x^{n}}{2 \, {\left (n^{2} + n\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 65, normalized size = 1.59 \[ \frac {2 \, a c n^{2} x + 2 \, b c n x x^{n} + 2 \, a c n x + b d n x^{2 \, n} + 2 \, a d n x^{n} + b d x^{2 \, n} + 2 \, a d x^{n}}{2 \, {\left (n^{2} + n\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 45, normalized size = 1.10 \[ \frac {b c x \,{\mathrm e}^{n \ln \relax (x )}}{n +1}+a c x +\frac {a d \,{\mathrm e}^{n \ln \relax (x )}}{n}+\frac {b d \,{\mathrm e}^{2 n \ln \relax (x )}}{2 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 39, normalized size = 0.95 \[ a c x + \frac {b d x^{2 \, n}}{2 \, n} + \frac {b c x^{n + 1}}{n + 1} + \frac {a d x^{n}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.06, size = 38, normalized size = 0.93 \[ a\,c\,x+\frac {a\,d\,x^n}{n}+\frac {b\,d\,x^{2\,n}}{2\,n}+\frac {b\,c\,x\,x^n}{n+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.03, size = 163, normalized size = 3.98 \[ \begin {cases} a c x - \frac {a d}{x} + b c \log {\relax (x )} - \frac {b d}{2 x^{2}} & \text {for}\: n = -1 \\\left (a + b\right ) \left (c x + d \log {\relax (x )}\right ) & \text {for}\: n = 0 \\\frac {2 a c n^{2} x}{2 n^{2} + 2 n} + \frac {2 a c n x}{2 n^{2} + 2 n} + \frac {2 a d n x^{n}}{2 n^{2} + 2 n} + \frac {2 a d x^{n}}{2 n^{2} + 2 n} + \frac {2 b c n x x^{n}}{2 n^{2} + 2 n} + \frac {b d n x^{2 n}}{2 n^{2} + 2 n} + \frac {b d x^{2 n}}{2 n^{2} + 2 n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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