3.3 \(\int (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3)^3 \, dx\)

Optimal. Leaf size=14 \[ \frac {(a+b x)^{10}}{10 b} \]

[Out]

1/10*(b*x+a)^10/b

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2059, 32} \[ \frac {(a+b x)^{10}}{10 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x)^10/(10*b)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2059

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[u^p, x] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x]
&& IntegerQ[p]

Rubi steps

\begin {align*} \int \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^3 \, dx &=\int (a+b x)^9 \, dx\\ &=\frac {(a+b x)^{10}}{10 b}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \[ \frac {(a+b x)^{10}}{10 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x)^10/(10*b)

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fricas [B]  time = 0.46, size = 97, normalized size = 6.93 \[ \frac {1}{10} x^{10} b^{9} + x^{9} b^{8} a + \frac {9}{2} x^{8} b^{7} a^{2} + 12 x^{7} b^{6} a^{3} + 21 x^{6} b^{5} a^{4} + \frac {126}{5} x^{5} b^{4} a^{5} + 21 x^{4} b^{3} a^{6} + 12 x^{3} b^{2} a^{7} + \frac {9}{2} x^{2} b a^{8} + x a^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*b^9 + x^9*b^8*a + 9/2*x^8*b^7*a^2 + 12*x^7*b^6*a^3 + 21*x^6*b^5*a^4 + 126/5*x^5*b^4*a^5 + 21*x^4*b^3
*a^6 + 12*x^3*b^2*a^7 + 9/2*x^2*b*a^8 + x*a^9

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giac [B]  time = 0.25, size = 97, normalized size = 6.93 \[ \frac {1}{10} \, b^{9} x^{10} + a b^{8} x^{9} + \frac {9}{2} \, a^{2} b^{7} x^{8} + 12 \, a^{3} b^{6} x^{7} + 21 \, a^{4} b^{5} x^{6} + \frac {126}{5} \, a^{5} b^{4} x^{5} + 21 \, a^{6} b^{3} x^{4} + 12 \, a^{7} b^{2} x^{3} + \frac {9}{2} \, a^{8} b x^{2} + a^{9} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^9*x^10 + a*b^8*x^9 + 9/2*a^2*b^7*x^8 + 12*a^3*b^6*x^7 + 21*a^4*b^5*x^6 + 126/5*a^5*b^4*x^5 + 21*a^6*b^3
*x^4 + 12*a^7*b^2*x^3 + 9/2*a^8*b*x^2 + a^9*x

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maple [B]  time = 0.00, size = 98, normalized size = 7.00 \[ \frac {1}{10} b^{9} x^{10}+a \,b^{8} x^{9}+\frac {9}{2} a^{2} b^{7} x^{8}+12 a^{3} b^{6} x^{7}+21 a^{4} b^{5} x^{6}+\frac {126}{5} a^{5} b^{4} x^{5}+21 a^{6} b^{3} x^{4}+12 a^{7} b^{2} x^{3}+\frac {9}{2} a^{8} b \,x^{2}+a^{9} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^9*x^10+a*b^8*x^9+9/2*a^2*b^7*x^8+12*a^3*b^6*x^7+21*a^4*b^5*x^6+126/5*a^5*b^4*x^5+21*a^6*b^3*x^4+12*a^7*
b^2*x^3+9/2*a^8*b*x^2+a^9*x

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maxima [B]  time = 0.69, size = 216, normalized size = 15.43 \[ \frac {1}{10} \, b^{9} x^{10} + a b^{8} x^{9} + \frac {27}{8} \, a^{2} b^{7} x^{8} + \frac {27}{7} \, a^{3} b^{6} x^{7} + \frac {27}{4} \, a^{6} b^{3} x^{4} + a^{9} x + \frac {3}{4} \, {\left (b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2}\right )} a^{6} + \frac {9}{10} \, {\left (5 \, b^{3} x^{6} + 18 \, a b^{2} x^{5}\right )} a^{4} b^{2} + \frac {3}{70} \, {\left (10 \, b^{6} x^{7} + 70 \, a b^{5} x^{6} + 126 \, a^{2} b^{4} x^{5} + 210 \, a^{4} b^{2} x^{3} + 21 \, {\left (4 \, b^{3} x^{5} + 15 \, a b^{2} x^{4}\right )} a^{2} b\right )} a^{3} + \frac {9}{56} \, {\left (7 \, b^{6} x^{8} + 48 \, a b^{5} x^{7} + 84 \, a^{2} b^{4} x^{6}\right )} a^{2} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^9*x^10 + a*b^8*x^9 + 27/8*a^2*b^7*x^8 + 27/7*a^3*b^6*x^7 + 27/4*a^6*b^3*x^4 + a^9*x + 3/4*(b^3*x^4 + 4*
a*b^2*x^3 + 6*a^2*b*x^2)*a^6 + 9/10*(5*b^3*x^6 + 18*a*b^2*x^5)*a^4*b^2 + 3/70*(10*b^6*x^7 + 70*a*b^5*x^6 + 126
*a^2*b^4*x^5 + 210*a^4*b^2*x^3 + 21*(4*b^3*x^5 + 15*a*b^2*x^4)*a^2*b)*a^3 + 9/56*(7*b^6*x^8 + 48*a*b^5*x^7 + 8
4*a^2*b^4*x^6)*a^2*b

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mupad [B]  time = 2.07, size = 97, normalized size = 6.93 \[ a^9\,x+\frac {9\,a^8\,b\,x^2}{2}+12\,a^7\,b^2\,x^3+21\,a^6\,b^3\,x^4+\frac {126\,a^5\,b^4\,x^5}{5}+21\,a^4\,b^5\,x^6+12\,a^3\,b^6\,x^7+\frac {9\,a^2\,b^7\,x^8}{2}+a\,b^8\,x^9+\frac {b^9\,x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^9*x + (b^9*x^10)/10 + (9*a^8*b*x^2)/2 + a*b^8*x^9 + 12*a^7*b^2*x^3 + 21*a^6*b^3*x^4 + (126*a^5*b^4*x^5)/5 +
21*a^4*b^5*x^6 + 12*a^3*b^6*x^7 + (9*a^2*b^7*x^8)/2

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sympy [B]  time = 0.09, size = 107, normalized size = 7.64 \[ a^{9} x + \frac {9 a^{8} b x^{2}}{2} + 12 a^{7} b^{2} x^{3} + 21 a^{6} b^{3} x^{4} + \frac {126 a^{5} b^{4} x^{5}}{5} + 21 a^{4} b^{5} x^{6} + 12 a^{3} b^{6} x^{7} + \frac {9 a^{2} b^{7} x^{8}}{2} + a b^{8} x^{9} + \frac {b^{9} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**9*x + 9*a**8*b*x**2/2 + 12*a**7*b**2*x**3 + 21*a**6*b**3*x**4 + 126*a**5*b**4*x**5/5 + 21*a**4*b**5*x**6 +
12*a**3*b**6*x**7 + 9*a**2*b**7*x**8/2 + a*b**8*x**9 + b**9*x**10/10

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