3.1250 \(\int (c+d x)^2 \, dx\)

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

[Out]

(c + d*x)^3/(3*d)

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Rubi [A]  time = 0.0015332, 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} \[ \frac{(c+d x)^3}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2,x]

[Out]

(c + d*x)^3/(3*d)

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 (c+d x)^2 \, dx &=\frac{(c+d x)^3}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0013158, size = 14, normalized size = 1. \[ \frac{(c+d x)^3}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2,x]

[Out]

(c + d*x)^3/(3*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2,x)

[Out]

1/3*(d*x+c)^3/d

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Maxima [A]  time = 0.950799, size = 27, normalized size = 1.93 \begin{align*} \frac{1}{3} \, d^{2} x^{3} + c d x^{2} + c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2,x, algorithm="maxima")

[Out]

1/3*d^2*x^3 + c*d*x^2 + c^2*x

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Fricas [A]  time = 1.76396, size = 42, normalized size = 3. \begin{align*} \frac{1}{3} x^{3} d^{2} + x^{2} d c + x c^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2,x, algorithm="fricas")

[Out]

1/3*x^3*d^2 + x^2*d*c + x*c^2

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Sympy [B]  time = 0.056934, size = 19, normalized size = 1.36 \begin{align*} c^{2} x + c d x^{2} + \frac{d^{2} x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2,x)

[Out]

c**2*x + c*d*x**2 + d**2*x**3/3

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Giac [A]  time = 1.07417, size = 16, normalized size = 1.14 \begin{align*} \frac{{\left (d x + c\right )}^{3}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2,x, algorithm="giac")

[Out]

1/3*(d*x + c)^3/d