3.209 \(\int (a+b x) \left (1+\left (c+a x+\frac{b x^2}{2}\right )^n\right ) \, dx\)

Optimal. Leaf size=35 \[ \frac{\left (a x+\frac{b x^2}{2}+c\right )^{n+1}}{n+1}+a x+\frac{b x^2}{2} \]

[Out]

a*x + (b*x^2)/2 + (c + a*x + (b*x^2)/2)^(1 + n)/(1 + n)

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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]

a*x + (b*x^2)/2 + (c + a*x + (b*x^2)/2)^(1 + n)/(1 + n)

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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]

a*x + b*x**2/2 + c + (a*x + b*x**2/2 + c)**(n + 1)/(n + 1)

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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]

(2*c*(c + a*x + (b*x^2)/2)^n + 2*a*x*(1 + n + (c + a*x + (b*x^2)/2)^n) + b*x^2*(
1 + n + (c + a*x + (b*x^2)/2)^n))/(2*(1 + n))

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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]

c+a*x+1/2*b*x^2+(c+a*x+1/2*b*x^2)^(1+n)/(1+n)

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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]

1/2*b*x^2 + a*x + (b*x^2 + 2*a*x + 2*c)*(b*x^2 + 2*a*x + 2*c)^n/(2^(n + 1)*n + 2
^(n + 1))

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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]

1/2*((b*n + b)*x^2 + (b*x^2 + 2*a*x + 2*c)*(1/2*b*x^2 + a*x + c)^n + 2*(a*n + a)
*x)/(n + 1)

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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]

Timed 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]

1/2*(b*n*x^2 + b*x^2*e^(n*ln(1/2*b*x^2 + a*x + c)) + 2*a*n*x + b*x^2 + 2*a*x*e^(
n*ln(1/2*b*x^2 + a*x + c)) + 2*a*x + 2*c*e^(n*ln(1/2*b*x^2 + a*x + c)))/(n + 1)