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 = 0.0227337, antiderivative size = 41, normalized size of antiderivative = 1., 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 1891
Rule 14
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*}
Mathematica [A] time = 0.0962162, 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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Maple [A] time = 0.011, size = 45, normalized size = 1.1 \begin{align*} acx+{\frac{ad{{\rm e}^{n\ln \left ( x \right ) }}}{n}}+{\frac{bcx{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}}+{\frac{bd \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48533, size = 128, normalized size = 3.12 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.50908, size = 163, normalized size = 3.98 \begin{align*} \begin{cases} a c x - \frac{a d}{x} + b c \log{\left (x \right )} - \frac{b d}{2 x^{2}} & \text{for}\: n = -1 \\\left (a + b\right ) \left (c x + d \log{\left (x \right )}\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05632, size = 88, normalized size = 2.15 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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