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

Optimal. Leaf size=14 \[ \frac{(a+b x)^3}{3 b} \]

[Out]

(a + b*x)^3/(3*b)

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Rubi [A]  time = 0.0014801, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {32} \[ \frac{(a+b x)^3}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2,x]

[Out]

(a + b*x)^3/(3*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]

Rubi steps

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

Mathematica [A]  time = 0.0012379, size = 14, normalized size = 1. \[ \frac{(a+b x)^3}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2,x]

[Out]

(a + b*x)^3/(3*b)

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Maple [A]  time = 0., size = 13, normalized size = 0.9 \begin{align*}{\frac{ \left ( bx+a \right ) ^{3}}{3\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b*x+a)^3/b

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Maxima [A]  time = 1.03112, size = 27, normalized size = 1.93 \begin{align*} \frac{1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2*x^3 + a*b*x^2 + a^2*x

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Fricas [A]  time = 1.32787, size = 42, normalized size = 3. \begin{align*} \frac{1}{3} x^{3} b^{2} + x^{2} b a + x a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*b^2 + x^2*b*a + x*a^2

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Sympy [B]  time = 0.067018, size = 19, normalized size = 1.36 \begin{align*} a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x + a*b*x**2 + b**2*x**3/3

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Giac [A]  time = 1.21265, size = 16, normalized size = 1.14 \begin{align*} \frac{{\left (b x + a\right )}^{3}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b*x + a)^3/b