3.23.95 \(\int (1+x)^3 (a+b x) (1-x+x^2)^3 \, dx\) [2295]

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

[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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Rubi [A]
time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, 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 = {785} \begin {gather*} \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 {gather*}

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 785

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*}

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Mathematica [A]
time = 0.00, size = 60, normalized size = 1.00 \begin {gather*} 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 {gather*}

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 = 0.04, size = 47, normalized size = 0.78

method	result	size
gosper	\(a x +\frac {1}{2} b \,x^{2}+\frac {3}{4} a \,x^{4}+\frac {3}{5} b \,x^{5}+\frac {3}{7} a \,x^{7}+\frac {3}{8} b \,x^{8}+\frac {1}{10} a \,x^{10}+\frac {1}{11} b \,x^{11}\)	\(47\)
default	\(a x +\frac {1}{2} b \,x^{2}+\frac {3}{4} a \,x^{4}+\frac {3}{5} b \,x^{5}+\frac {3}{7} a \,x^{7}+\frac {3}{8} b \,x^{8}+\frac {1}{10} a \,x^{10}+\frac {1}{11} b \,x^{11}\)	\(47\)
norman	\(a x +\frac {1}{2} b \,x^{2}+\frac {3}{4} a \,x^{4}+\frac {3}{5} b \,x^{5}+\frac {3}{7} a \,x^{7}+\frac {3}{8} b \,x^{8}+\frac {1}{10} a \,x^{10}+\frac {1}{11} b \,x^{11}\)	\(47\)
risch	\(a x +\frac {1}{2} b \,x^{2}+\frac {3}{4} a \,x^{4}+\frac {3}{5} b \,x^{5}+\frac {3}{7} a \,x^{7}+\frac {3}{8} b \,x^{8}+\frac {1}{10} a \,x^{10}+\frac {1}{11} b \,x^{11}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.28, size = 46, normalized size = 0.77 \begin {gather*} \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 {gather*}

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 = 3.32, size = 46, normalized size = 0.77 \begin {gather*} \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 {gather*}

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*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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Sympy [A]
time = 0.01, size = 56, normalized size = 0.93 \begin {gather*} \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 {gather*}

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 = 1.62, size = 46, normalized size = 0.77 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.03, size = 46, normalized size = 0.77 \begin {gather*} \frac {b\,x^{11}}{11}+\frac {a\,x^{10}}{10}+\frac {3\,b\,x^8}{8}+\frac {3\,a\,x^7}{7}+\frac {3\,b\,x^5}{5}+\frac {3\,a\,x^4}{4}+\frac {b\,x^2}{2}+a\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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