3.13.63 \(\int (c+d x)^3 \, dx\) [1263]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {(c+d x)^4}{4 d} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {(c+d x)^4}{4 d} \end {gather*}

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 = 0.12, size = 13, normalized size = 0.93

method	result	size
default	\(\frac {\left (d x +c \right )^{4}}{4 d}\)	\(13\)
gosper	\(\frac {1}{4} d^{3} x^{4}+c \,d^{2} x^{3}+\frac {3}{2} c^{2} d \,x^{2}+c^{3} x\)	\(32\)
norman	\(\frac {1}{4} d^{3} x^{4}+c \,d^{2} x^{3}+\frac {3}{2} c^{2} d \,x^{2}+c^{3} x\)	\(32\)
risch	\(\frac {d^{3} x^{4}}{4}+c \,d^{2} x^{3}+\frac {3 c^{2} d \,x^{2}}{2}+c^{3} x +\frac {c^{4}}{4 d}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).
time = 0.28, size = 31, normalized size = 2.21 \begin {gather*} \frac {1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac {3}{2} \, c^{2} d x^{2} + c^{3} x \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).
time = 0.85, size = 31, normalized size = 2.21 \begin {gather*} \frac {1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac {3}{2} \, c^{2} d x^{2} + c^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (8) = 16\).
time = 0.01, size = 32, normalized size = 2.29 \begin {gather*} c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4} \end {gather*}

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 = 0.83, size = 12, normalized size = 0.86 \begin {gather*} \frac {{\left (d x + c\right )}^{4}}{4 \, d} \end {gather*}

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

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Mupad [B]
time = 0.04, size = 31, normalized size = 2.21 \begin {gather*} c^3\,x+\frac {3\,c^2\,d\,x^2}{2}+c\,d^2\,x^3+\frac {d^3\,x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3,x)

[Out]

c^3*x + (d^3*x^4)/4 + (3*c^2*d*x^2)/2 + c*d^2*x^3

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