3.1263 \(\int (c+d x)^3 \, dx\)

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

[Out]

(c + d*x)^4/(4*d)

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Rubi [A]  time = 0.0015485, 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} \[ \frac{(c+d x)^4}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3,x]

[Out]

(c + d*x)^4/(4*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)^3 \, dx &=\frac{(c+d x)^4}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.0014879, size = 14, normalized size = 1. \[ \frac{(c+d x)^4}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3,x]

[Out]

(c + d*x)^4/(4*d)

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Maple [A]  time = 0., size = 13, normalized size = 0.9 \begin{align*}{\frac{ \left ( dx+c \right ) ^{4}}{4\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3,x)

[Out]

1/4*(d*x+c)^4/d

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Maxima [B]  time = 0.966924, size = 42, normalized size = 3. \begin{align*} \frac{1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac{3}{2} \, c^{2} d x^{2} + c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3,x, algorithm="maxima")

[Out]

1/4*d^3*x^4 + c*d^2*x^3 + 3/2*c^2*d*x^2 + c^3*x

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Fricas [B]  time = 1.76399, size = 66, normalized size = 4.71 \begin{align*} \frac{1}{4} x^{4} d^{3} + x^{3} d^{2} c + \frac{3}{2} x^{2} d c^{2} + x c^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3,x, algorithm="fricas")

[Out]

1/4*x^4*d^3 + x^3*d^2*c + 3/2*x^2*d*c^2 + x*c^3

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Sympy [B]  time = 0.059863, size = 32, normalized size = 2.29 \begin{align*} c^{3} x + \frac{3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac{d^{3} x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3,x)

[Out]

c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4

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Giac [A]  time = 1.05997, size = 16, normalized size = 1.14 \begin{align*} \frac{{\left (d x + c\right )}^{4}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3,x, algorithm="giac")

[Out]

1/4*(d*x + c)^4/d