\(\int (c+d x)^2 \, dx\) [1250]

Optimal result

Integrand size = 7, antiderivative size = 14 \[ \int (c+d x)^2 \, dx=\frac {(c+d x)^3}{3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \[ \int (c+d x)^2 \, dx=\frac {(c+d x)^3}{3 d} \]

[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*} \text {integral}& = \frac {(c+d x)^3}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int (c+d x)^2 \, dx=\frac {(c+d x)^3}{3 d} \]

[In]

Integrate[(c + d*x)^2,x]

[Out]

(c + d*x)^3/(3*d)

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {\left (d x +c \right )^{3}}{3 d}\)	\(13\)
gosper	\(\frac {1}{3} d^{2} x^{3}+c d \,x^{2}+c^{2} x\)	\(21\)
norman	\(\frac {1}{3} d^{2} x^{3}+c d \,x^{2}+c^{2} x\)	\(21\)
parallelrisch	\(\frac {1}{3} d^{2} x^{3}+c d \,x^{2}+c^{2} x\)	\(21\)
risch	\(\frac {d^{2} x^{3}}{3}+c d \,x^{2}+c^{2} x +\frac {c^{3}}{3 d}\)	\(29\)

[In]

int((d*x+c)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int (c+d x)^2 \, dx=\frac {1}{3} \, d^{2} x^{3} + c d x^{2} + c^{2} x \]

[In]

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

[Out]

1/3*d^2*x^3 + c*d*x^2 + c^2*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (8) = 16\).

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int (c+d x)^2 \, dx=c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3} \]

[In]

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

[Out]

c**2*x + c*d*x**2 + d**2*x**3/3

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int (c+d x)^2 \, dx=\frac {1}{3} \, d^{2} x^{3} + c d x^{2} + c^{2} x \]

[In]

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

[Out]

1/3*d^2*x^3 + c*d*x^2 + c^2*x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int (c+d x)^2 \, dx=\frac {{\left (d x + c\right )}^{3}}{3 \, d} \]

[In]

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

[Out]

1/3*(d*x + c)^3/d

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int (c+d x)^2 \, dx=c^2\,x+c\,d\,x^2+\frac {d^2\,x^3}{3} \]

[In]

int((c + d*x)^2,x)

[Out]

c^2*x + (d^2*x^3)/3 + c*d*x^2