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 = 0.0460881, 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 \[ a c x+\frac{a d x^n}{n}+\frac{b c x^{n+1}}{n+1}+\frac{b d x^{2 n}}{2 n} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^(-1 + n))*(a + b*x^n),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a d x^{n}}{n} + \frac{b c x^{n + 1}}{n + 1} + \frac{b d x^{2 n}}{2 n} + c \int a\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c+d*x**(-1+n))*(a+b*x**n),x)
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Mathematica [A] time = 0.092453, 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.
[In] Integrate[(c + d*x^(-1 + n))*(a + b*x^n),x]
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Maple [A] time = 0.021, size = 45, normalized size = 1.1 \[ acx+{\frac{ad{{\rm e}^{n\ln \left ( x \right ) }}}{n}}+{\frac{bcx{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}}+{\frac{bd \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c+d*x^(-1+n))*(a+b*x^n),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^(n - 1) + c)*(b*x^n + a),x, algorithm="maxima")
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Fricas [A] time = 0.224454, size = 76, normalized size = 1.85 \[ \frac{2 \,{\left (a c n^{2} + a c n\right )} x +{\left (b d n + b d\right )} x^{2 \, n} + 2 \,{\left (b c n x + a d n + a d\right )} x^{n}}{2 \,{\left (n^{2} + n\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^(n - 1) + c)*(b*x^n + a),x, algorithm="fricas")
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Sympy [A] time = 2.41195, size = 163, normalized size = 3.98 \[ \begin{cases} a c x - \frac{a d}{x} + b c \log{\left (x \right )} - \frac{b d}{2 x^{2}} & \text{for}\: n = -1 \\\left (a + b\right ) \left (c x + d \log{\left (x \right )}\right ) & \text{for}\: n = 0 \\\frac{2 a c n^{2} x}{2 n^{2} + 2 n} + \frac{2 a c n x}{2 n^{2} + 2 n} + \frac{2 a d n x^{n}}{2 n^{2} + 2 n} + \frac{2 a d x^{n}}{2 n^{2} + 2 n} + \frac{2 b c n x x^{n}}{2 n^{2} + 2 n} + \frac{b d n x^{2 n}}{2 n^{2} + 2 n} + \frac{b d x^{2 n}}{2 n^{2} + 2 n} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c+d*x**(-1+n))*(a+b*x**n),x)
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GIAC/XCAS [A] time = 0.219808, size = 99, normalized size = 2.41 \[ \frac{2 \, a c n^{2} x + 2 \, b c n x e^{\left (n{\rm ln}\left (x\right )\right )} + 2 \, a c n x + b d n e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 2 \, a d n e^{\left (n{\rm ln}\left (x\right )\right )} + b d e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 2 \, a d e^{\left (n{\rm ln}\left (x\right )\right )}}{2 \,{\left (n^{2} + n\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^(n - 1) + c)*(b*x^n + a),x, algorithm="giac")
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