Optimal. Leaf size=35 \[ \frac{\left (a x+\frac{b x^2}{2}+c\right )^{n+1}}{n+1}+a x+\frac{b x^2}{2} \]
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Rubi [A] time = 0.0200527, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{\left (a x+\frac{b x^2}{2}+c\right )^{n+1}}{n+1}+a x+\frac{b x^2}{2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*(1 + (c + a*x + (b*x^2)/2)^n),x]
[Out]
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Rubi in Sympy [A] time = 2.38454, size = 29, normalized size = 0.83 \[ a x + \frac{b x^{2}}{2} + c + \frac{\left (a x + \frac{b x^{2}}{2} + c\right )^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(1+(c+a*x+1/2*b*x**2)**n),x)
[Out]
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Mathematica [B] time = 0.066281, size = 73, normalized size = 2.09 \[ \frac{2 c \left (a x+\frac{b x^2}{2}+c\right )^n+b x^2 \left (\left (a x+\frac{b x^2}{2}+c\right )^n+n+1\right )+2 a x \left (\left (a x+\frac{b x^2}{2}+c\right )^n+n+1\right )}{2 (n+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*(1 + (c + a*x + (b*x^2)/2)^n),x]
[Out]
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Maple [A] time = 0.003, size = 33, normalized size = 0.9 \[ c+ax+{\frac{b{x}^{2}}{2}}+{\frac{1}{1+n} \left ( c+ax+{\frac{b{x}^{2}}{2}} \right ) ^{1+n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(1+(c+a*x+1/2*b*x^2)^n),x)
[Out]
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Maxima [A] time = 0.96775, size = 73, normalized size = 2.09 \[ \frac{1}{2} \, b x^{2} + a x + \frac{{\left (b x^{2} + 2 \, a x + 2 \, c\right )}{\left (b x^{2} + 2 \, a x + 2 \, c\right )}^{n}}{2^{n + 1} n + 2^{n + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*((1/2*b*x^2 + a*x + c)^n + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.272704, size = 70, normalized size = 2. \[ \frac{{\left (b n + b\right )} x^{2} +{\left (b x^{2} + 2 \, a x + 2 \, c\right )}{\left (\frac{1}{2} \, b x^{2} + a x + c\right )}^{n} + 2 \,{\left (a n + a\right )} x}{2 \,{\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*((1/2*b*x^2 + a*x + c)^n + 1),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(1+(c+a*x+1/2*b*x**2)**n),x)
[Out]
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GIAC/XCAS [A] time = 0.26976, size = 115, normalized size = 3.29 \[ \frac{b n x^{2} + b x^{2} e^{\left (n{\rm ln}\left (\frac{1}{2} \, b x^{2} + a x + c\right )\right )} + 2 \, a n x + b x^{2} + 2 \, a x e^{\left (n{\rm ln}\left (\frac{1}{2} \, b x^{2} + a x + c\right )\right )} + 2 \, a x + 2 \, c e^{\left (n{\rm ln}\left (\frac{1}{2} \, b x^{2} + a x + c\right )\right )}}{2 \,{\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*((1/2*b*x^2 + a*x + c)^n + 1),x, algorithm="giac")
[Out]