3.2295 \(\int (1+x)^3 (a+b x) (1-x+x^2)^3 \, dx\)

Optimal. Leaf size=60 \[ \frac{a x^{10}}{10}+\frac{3 a x^7}{7}+\frac{3 a x^4}{4}+a x+\frac{b x^{11}}{11}+\frac{3 b x^8}{8}+\frac{3 b x^5}{5}+\frac{b x^2}{2} \]

[Out]

a*x + (b*x^2)/2 + (3*a*x^4)/4 + (3*b*x^5)/5 + (3*a*x^7)/7 + (3*b*x^8)/8 + (a*x^10)/10 + (b*x^11)/11

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Rubi [A]  time = 0.037747, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {771} \[ \frac{a x^{10}}{10}+\frac{3 a x^7}{7}+\frac{3 a x^4}{4}+a x+\frac{b x^{11}}{11}+\frac{3 b x^8}{8}+\frac{3 b x^5}{5}+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)^3*(a + b*x)*(1 - x + x^2)^3,x]

[Out]

a*x + (b*x^2)/2 + (3*a*x^4)/4 + (3*b*x^5)/5 + (3*a*x^7)/7 + (3*b*x^8)/8 + (a*x^10)/10 + (b*x^11)/11

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (1+x)^3 (a+b x) \left (1-x+x^2\right )^3 \, dx &=\int \left (a+b x+3 a x^3+3 b x^4+3 a x^6+3 b x^7+a x^9+b x^{10}\right ) \, dx\\ &=a x+\frac{b x^2}{2}+\frac{3 a x^4}{4}+\frac{3 b x^5}{5}+\frac{3 a x^7}{7}+\frac{3 b x^8}{8}+\frac{a x^{10}}{10}+\frac{b x^{11}}{11}\\ \end{align*}

Mathematica [A]  time = 0.00217, size = 60, normalized size = 1. \[ \frac{a x^{10}}{10}+\frac{3 a x^7}{7}+\frac{3 a x^4}{4}+a x+\frac{b x^{11}}{11}+\frac{3 b x^8}{8}+\frac{3 b x^5}{5}+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^3*(a + b*x)*(1 - x + x^2)^3,x]

[Out]

a*x + (b*x^2)/2 + (3*a*x^4)/4 + (3*b*x^5)/5 + (3*a*x^7)/7 + (3*b*x^8)/8 + (a*x^10)/10 + (b*x^11)/11

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Maple [A]  time = 0.002, size = 47, normalized size = 0.8 \begin{align*} ax+{\frac{b{x}^{2}}{2}}+{\frac{3\,a{x}^{4}}{4}}+{\frac{3\,b{x}^{5}}{5}}+{\frac{3\,a{x}^{7}}{7}}+{\frac{3\,b{x}^{8}}{8}}+{\frac{a{x}^{10}}{10}}+{\frac{b{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^3*(b*x+a)*(x^2-x+1)^3,x)

[Out]

a*x+1/2*b*x^2+3/4*a*x^4+3/5*b*x^5+3/7*a*x^7+3/8*b*x^8+1/10*a*x^10+1/11*b*x^11

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Maxima [A]  time = 1.04165, size = 62, normalized size = 1.03 \begin{align*} \frac{1}{11} \, b x^{11} + \frac{1}{10} \, a x^{10} + \frac{3}{8} \, b x^{8} + \frac{3}{7} \, a x^{7} + \frac{3}{5} \, b x^{5} + \frac{3}{4} \, a x^{4} + \frac{1}{2} \, b x^{2} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^3*(b*x+a)*(x^2-x+1)^3,x, algorithm="maxima")

[Out]

1/11*b*x^11 + 1/10*a*x^10 + 3/8*b*x^8 + 3/7*a*x^7 + 3/5*b*x^5 + 3/4*a*x^4 + 1/2*b*x^2 + a*x

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Fricas [A]  time = 1.11977, size = 126, normalized size = 2.1 \begin{align*} \frac{1}{11} x^{11} b + \frac{1}{10} x^{10} a + \frac{3}{8} x^{8} b + \frac{3}{7} x^{7} a + \frac{3}{5} x^{5} b + \frac{3}{4} x^{4} a + \frac{1}{2} x^{2} b + x a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^3*(b*x+a)*(x^2-x+1)^3,x, algorithm="fricas")

[Out]

1/11*x^11*b + 1/10*x^10*a + 3/8*x^8*b + 3/7*x^7*a + 3/5*x^5*b + 3/4*x^4*a + 1/2*x^2*b + x*a

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Sympy [A]  time = 0.069017, size = 56, normalized size = 0.93 \begin{align*} \frac{a x^{10}}{10} + \frac{3 a x^{7}}{7} + \frac{3 a x^{4}}{4} + a x + \frac{b x^{11}}{11} + \frac{3 b x^{8}}{8} + \frac{3 b x^{5}}{5} + \frac{b x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**3*(b*x+a)*(x**2-x+1)**3,x)

[Out]

a*x**10/10 + 3*a*x**7/7 + 3*a*x**4/4 + a*x + b*x**11/11 + 3*b*x**8/8 + 3*b*x**5/5 + b*x**2/2

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Giac [A]  time = 1.13949, size = 62, normalized size = 1.03 \begin{align*} \frac{1}{11} \, b x^{11} + \frac{1}{10} \, a x^{10} + \frac{3}{8} \, b x^{8} + \frac{3}{7} \, a x^{7} + \frac{3}{5} \, b x^{5} + \frac{3}{4} \, a x^{4} + \frac{1}{2} \, b x^{2} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^3*(b*x+a)*(x^2-x+1)^3,x, algorithm="giac")

[Out]

1/11*b*x^11 + 1/10*a*x^10 + 3/8*b*x^8 + 3/7*a*x^7 + 3/5*b*x^5 + 3/4*a*x^4 + 1/2*b*x^2 + a*x