3.2294 \(\int (1+x)^4 (a+b x) (1-x+x^2)^4 \, dx\)

Optimal. Leaf size=73 \[ \frac{a x^{13}}{13}+\frac{2 a x^{10}}{5}+\frac{6 a x^7}{7}+a x^4+a x+\frac{b x^{14}}{14}+\frac{4 b x^{11}}{11}+\frac{3 b x^8}{4}+\frac{4 b x^5}{5}+\frac{b x^2}{2} \]

[Out]

a*x + (b*x^2)/2 + a*x^4 + (4*b*x^5)/5 + (6*a*x^7)/7 + (3*b*x^8)/4 + (2*a*x^10)/5 + (4*b*x^11)/11 + (a*x^13)/13
+ (b*x^14)/14

________________________________________________________________________________________

Rubi [A]  time = 0.0529385, antiderivative size = 73, 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} \[ \frac{a x^{13}}{13}+\frac{2 a x^{10}}{5}+\frac{6 a x^7}{7}+a x^4+a x+\frac{b x^{14}}{14}+\frac{4 b x^{11}}{11}+\frac{3 b x^8}{4}+\frac{4 b x^5}{5}+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)^4*(a + b*x)*(1 - x + x^2)^4,x]

[Out]

a*x + (b*x^2)/2 + a*x^4 + (4*b*x^5)/5 + (6*a*x^7)/7 + (3*b*x^8)/4 + (2*a*x^10)/5 + (4*b*x^11)/11 + (a*x^13)/13
+ (b*x^14)/14

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)^4 (a+b x) \left (1-x+x^2\right )^4 \, dx &=\int \left (a+b x+4 a x^3+4 b x^4+6 a x^6+6 b x^7+4 a x^9+4 b x^{10}+a x^{12}+b x^{13}\right ) \, dx\\ &=a x+\frac{b x^2}{2}+a x^4+\frac{4 b x^5}{5}+\frac{6 a x^7}{7}+\frac{3 b x^8}{4}+\frac{2 a x^{10}}{5}+\frac{4 b x^{11}}{11}+\frac{a x^{13}}{13}+\frac{b x^{14}}{14}\\ \end{align*}

Mathematica [A]  time = 0.0025908, size = 73, normalized size = 1. \[ \frac{a x^{13}}{13}+\frac{2 a x^{10}}{5}+\frac{6 a x^7}{7}+a x^4+a x+\frac{b x^{14}}{14}+\frac{4 b x^{11}}{11}+\frac{3 b x^8}{4}+\frac{4 b x^5}{5}+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^4*(a + b*x)*(1 - x + x^2)^4,x]

[Out]

a*x + (b*x^2)/2 + a*x^4 + (4*b*x^5)/5 + (6*a*x^7)/7 + (3*b*x^8)/4 + (2*a*x^10)/5 + (4*b*x^11)/11 + (a*x^13)/13
+ (b*x^14)/14

________________________________________________________________________________________

Maple [A]  time = 0., size = 58, normalized size = 0.8 \begin{align*} ax+{\frac{b{x}^{2}}{2}}+a{x}^{4}+{\frac{4\,b{x}^{5}}{5}}+{\frac{6\,a{x}^{7}}{7}}+{\frac{3\,b{x}^{8}}{4}}+{\frac{2\,a{x}^{10}}{5}}+{\frac{4\,b{x}^{11}}{11}}+{\frac{a{x}^{13}}{13}}+{\frac{b{x}^{14}}{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^4*(b*x+a)*(x^2-x+1)^4,x)

[Out]

a*x+1/2*b*x^2+a*x^4+4/5*b*x^5+6/7*a*x^7+3/4*b*x^8+2/5*a*x^10+4/11*b*x^11+1/13*a*x^13+1/14*b*x^14

________________________________________________________________________________________

Maxima [A]  time = 0.973341, size = 77, normalized size = 1.05 \begin{align*} \frac{1}{14} \, b x^{14} + \frac{1}{13} \, a x^{13} + \frac{4}{11} \, b x^{11} + \frac{2}{5} \, a x^{10} + \frac{3}{4} \, b x^{8} + \frac{6}{7} \, a x^{7} + \frac{4}{5} \, b x^{5} + 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)^4*(b*x+a)*(x^2-x+1)^4,x, algorithm="maxima")

[Out]

1/14*b*x^14 + 1/13*a*x^13 + 4/11*b*x^11 + 2/5*a*x^10 + 3/4*b*x^8 + 6/7*a*x^7 + 4/5*b*x^5 + a*x^4 + 1/2*b*x^2 +
a*x

________________________________________________________________________________________

Fricas [A]  time = 1.13461, size = 157, normalized size = 2.15 \begin{align*} \frac{1}{14} x^{14} b + \frac{1}{13} x^{13} a + \frac{4}{11} x^{11} b + \frac{2}{5} x^{10} a + \frac{3}{4} x^{8} b + \frac{6}{7} x^{7} a + \frac{4}{5} x^{5} b + 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)^4*(b*x+a)*(x^2-x+1)^4,x, algorithm="fricas")

[Out]

1/14*x^14*b + 1/13*x^13*a + 4/11*x^11*b + 2/5*x^10*a + 3/4*x^8*b + 6/7*x^7*a + 4/5*x^5*b + x^4*a + 1/2*x^2*b +
x*a

________________________________________________________________________________________

Sympy [A]  time = 0.074587, size = 70, normalized size = 0.96 \begin{align*} \frac{a x^{13}}{13} + \frac{2 a x^{10}}{5} + \frac{6 a x^{7}}{7} + a x^{4} + a x + \frac{b x^{14}}{14} + \frac{4 b x^{11}}{11} + \frac{3 b x^{8}}{4} + \frac{4 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)**4*(b*x+a)*(x**2-x+1)**4,x)

[Out]

a*x**13/13 + 2*a*x**10/5 + 6*a*x**7/7 + a*x**4 + a*x + b*x**14/14 + 4*b*x**11/11 + 3*b*x**8/4 + 4*b*x**5/5 + b
*x**2/2

________________________________________________________________________________________

Giac [A]  time = 1.13004, size = 77, normalized size = 1.05 \begin{align*} \frac{1}{14} \, b x^{14} + \frac{1}{13} \, a x^{13} + \frac{4}{11} \, b x^{11} + \frac{2}{5} \, a x^{10} + \frac{3}{4} \, b x^{8} + \frac{6}{7} \, a x^{7} + \frac{4}{5} \, b x^{5} + 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)^4*(b*x+a)*(x^2-x+1)^4,x, algorithm="giac")

[Out]

1/14*b*x^14 + 1/13*a*x^13 + 4/11*b*x^11 + 2/5*a*x^10 + 3/4*b*x^8 + 6/7*a*x^7 + 4/5*b*x^5 + a*x^4 + 1/2*b*x^2 +
a*x