3.1900 \(\int (a+b x) (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=14 \[ \frac{(a+b x)^4}{4 b} \]

[Out]

(a + b*x)^4/(4*b)

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Rubi [A]  time = 0.0023068, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {27, 32} \[ \frac{(a+b x)^4}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x)^4/(4*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 (a+b x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^3 \, dx\\ &=\frac{(a+b x)^4}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.0011029, size = 14, normalized size = 1. \[ \frac{(a+b x)^4}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x)^4/(4*b)

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Maple [B]  time = 0., size = 32, normalized size = 2.3 \begin{align*}{\frac{{b}^{3}{x}^{4}}{4}}+{b}^{2}a{x}^{3}+{\frac{3\,{a}^{2}b{x}^{2}}{2}}+{a}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^3*x^4+b^2*a*x^3+3/2*a^2*b*x^2+a^3*x

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Maxima [A]  time = 0.952269, size = 31, normalized size = 2.21 \begin{align*} \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{2}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b^2*x^2 + 2*a*b*x + a^2)^2/b

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Fricas [B]  time = 1.34458, size = 66, normalized size = 4.71 \begin{align*} \frac{1}{4} x^{4} b^{3} + x^{3} b^{2} a + \frac{3}{2} x^{2} b a^{2} + x a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*b^3 + x^3*b^2*a + 3/2*x^2*b*a^2 + x*a^3

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Sympy [B]  time = 0.060346, size = 32, normalized size = 2.29 \begin{align*} a^{3} x + \frac{3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac{b^{3} x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4

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Giac [B]  time = 1.136, size = 42, normalized size = 3. \begin{align*} \frac{1}{4} \, b^{3} x^{4} + a b^{2} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^3*x^4 + a*b^2*x^3 + 3/2*a^2*b*x^2 + a^3*x