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]

(a + b*x)^10/(10*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0074009, antiderivative size = 14, normalized size of antiderivative = 1., 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 2059

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

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]

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*}

Mathematica [A]  time = 0.0009253, size = 14, normalized size = 1. \[ \frac{(a+b x)^{10}}{10 b} \]

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)

________________________________________________________________________________________

Maple [B]  time = 0.002, size = 98, normalized size = 7. \begin{align*}{\frac{{b}^{9}{x}^{10}}{10}}+a{b}^{8}{x}^{9}+{\frac{9\,{a}^{2}{b}^{7}{x}^{8}}{2}}+12\,{a}^{3}{b}^{6}{x}^{7}+21\,{a}^{4}{b}^{5}{x}^{6}+{\frac{126\,{a}^{5}{b}^{4}{x}^{5}}{5}}+21\,{a}^{6}{b}^{3}{x}^{4}+12\,{a}^{7}{b}^{2}{x}^{3}+{\frac{9\,{a}^{8}b{x}^{2}}{2}}+{a}^{9}x \end{align*}

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

________________________________________________________________________________________

Maxima [B]  time = 1.16856, size = 292, normalized size = 20.86 \begin{align*} \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 \end{align*}

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

________________________________________________________________________________________

Fricas [B]  time = 1.11428, size = 212, normalized size = 15.14 \begin{align*} \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} \end{align*}

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

________________________________________________________________________________________

Sympy [B]  time = 0.083674, size = 107, normalized size = 7.64 \begin{align*} 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} \end{align*}

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

________________________________________________________________________________________

Giac [B]  time = 1.06108, size = 131, normalized size = 9.36 \begin{align*} \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 \end{align*}

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