### 3.1489 $$\int (a^2+2 a b x+b^2 x^2)^3 \, dx$$

Optimal. Leaf size=14 $\frac{(a+b x)^7}{7 b}$

[Out]

(a + b*x)^7/(7*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0021366, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.111, Rules used = {27, 32} $\frac{(a+b x)^7}{7 b}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(a + b*x)^7/(7*b)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && 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^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 \, dx\\ &=\frac{(a+b x)^7}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.0010708, size = 14, normalized size = 1. $\frac{(a+b x)^7}{7 b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a + b*x)^7/(7*b)

________________________________________________________________________________________

Maple [B]  time = 0.04, size = 65, normalized size = 4.6 \begin{align*}{\frac{{b}^{6}{x}^{7}}{7}}+a{b}^{5}{x}^{6}+3\,{a}^{2}{b}^{4}{x}^{5}+5\,{a}^{3}{b}^{3}{x}^{4}+5\,{a}^{4}{b}^{2}{x}^{3}+3\,{a}^{5}b{x}^{2}+{a}^{6}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^6*x^7+a*b^5*x^6+3*a^2*b^4*x^5+5*a^3*b^3*x^4+5*a^4*b^2*x^3+3*a^5*b*x^2+a^6*x

________________________________________________________________________________________

Maxima [B]  time = 1.15505, size = 131, normalized size = 9.36 \begin{align*} \frac{1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + \frac{12}{5} \, a^{2} b^{4} x^{5} + 2 \, a^{3} b^{3} x^{4} + a^{6} x +{\left (b^{2} x^{3} + 3 \, a b x^{2}\right )} a^{4} + \frac{1}{5} \,{\left (3 \, b^{4} x^{5} + 15 \, a b^{3} x^{4} + 20 \, a^{2} b^{2} x^{3}\right )} a^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^6*x^7 + a*b^5*x^6 + 12/5*a^2*b^4*x^5 + 2*a^3*b^3*x^4 + a^6*x + (b^2*x^3 + 3*a*b*x^2)*a^4 + 1/5*(3*b^4*x^
5 + 15*a*b^3*x^4 + 20*a^2*b^2*x^3)*a^2

________________________________________________________________________________________

Fricas [B]  time = 1.57417, size = 128, normalized size = 9.14 \begin{align*} \frac{1}{7} x^{7} b^{6} + x^{6} b^{5} a + 3 x^{5} b^{4} a^{2} + 5 x^{4} b^{3} a^{3} + 5 x^{3} b^{2} a^{4} + 3 x^{2} b a^{5} + x a^{6} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*b^6 + x^6*b^5*a + 3*x^5*b^4*a^2 + 5*x^4*b^3*a^3 + 5*x^3*b^2*a^4 + 3*x^2*b*a^5 + x*a^6

________________________________________________________________________________________

Sympy [B]  time = 0.085929, size = 66, normalized size = 4.71 \begin{align*} a^{6} x + 3 a^{5} b x^{2} + 5 a^{4} b^{2} x^{3} + 5 a^{3} b^{3} x^{4} + 3 a^{2} b^{4} x^{5} + a b^{5} x^{6} + \frac{b^{6} x^{7}}{7} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**6*x + 3*a**5*b*x**2 + 5*a**4*b**2*x**3 + 5*a**3*b**3*x**4 + 3*a**2*b**4*x**5 + a*b**5*x**6 + b**6*x**7/7

________________________________________________________________________________________

Giac [B]  time = 1.14211, size = 86, normalized size = 6.14 \begin{align*} \frac{1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + 3 \, a^{2} b^{4} x^{5} + 5 \, a^{3} b^{3} x^{4} + 5 \, a^{4} b^{2} x^{3} + 3 \, a^{5} b x^{2} + a^{6} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^6*x^7 + a*b^5*x^6 + 3*a^2*b^4*x^5 + 5*a^3*b^3*x^4 + 5*a^4*b^2*x^3 + 3*a^5*b*x^2 + a^6*x