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

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

[Out]

(a + b*x)^7/(7*b)

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Rubi [A]  time = 0.007188, 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} \[ \frac{(a+b x)^7}{7 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x)^7/(7*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 )^2 \, dx &=\int (a+b x)^6 \, dx\\ &=\frac{(a+b x)^7}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.0010834, size = 14, normalized size = 1. \[ \frac{(a+b x)^7}{7 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x)^7/(7*b)

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Maple [B]  time = 0., 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^3*x^3+3*a*b^2*x^2+3*a^2*b*x+a^3)^2,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

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Maxima [B]  time = 1.16221, size = 134, normalized size = 9.57 \begin{align*} \frac{1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + \frac{9}{5} \, a^{2} b^{4} x^{5} + 3 \, a^{4} b^{2} x^{3} + a^{6} x + \frac{1}{2} \,{\left (b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2}\right )} a^{3} + \frac{3}{10} \,{\left (4 \, b^{3} x^{5} + 15 \, a b^{2} x^{4}\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)^2,x, algorithm="maxima")

[Out]

1/7*b^6*x^7 + a*b^5*x^6 + 9/5*a^2*b^4*x^5 + 3*a^4*b^2*x^3 + a^6*x + 1/2*(b^3*x^4 + 4*a*b^2*x^3 + 6*a^2*b*x^2)*
a^3 + 3/10*(4*b^3*x^5 + 15*a*b^2*x^4)*a^2*b

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Fricas [B]  time = 1.08018, 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^3*x^3+3*a*b^2*x^2+3*a^2*b*x+a^3)^2,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

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Sympy [B]  time = 0.073416, 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**3*x**3+3*a*b**2*x**2+3*a**2*b*x+a**3)**2,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

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Giac [B]  time = 1.06185, 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^3*x^3+3*a*b^2*x^2+3*a^2*b*x+a^3)^2,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