\(\int (3 a b+3 b^2 x+3 b c x^2+c^2 x^3) \, dx\) [11]

Optimal result

Integrand size = 25, antiderivative size = 32 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right ) \, dx=3 a b x+\frac {3 b^2 x^2}{2}+b c x^3+\frac {c^2 x^4}{4} \]

[Out]

3*a*b*x+3/2*b^2*x^2+b*c*x^3+1/4*c^2*x^4

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right ) \, dx=3 a b x+\frac {3 b^2 x^2}{2}+b c x^3+\frac {c^2 x^4}{4} \]

[In]

Int[3*a*b + 3*b^2*x + 3*b*c*x^2 + c^2*x^3,x]

[Out]

3*a*b*x + (3*b^2*x^2)/2 + b*c*x^3 + (c^2*x^4)/4

Rubi steps \begin{align*} \text {integral}& = 3 a b x+\frac {3 b^2 x^2}{2}+b c x^3+\frac {c^2 x^4}{4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right ) \, dx=3 a b x+\frac {3 b^2 x^2}{2}+b c x^3+\frac {c^2 x^4}{4} \]

[In]

Integrate[3*a*b + 3*b^2*x + 3*b*c*x^2 + c^2*x^3,x]

[Out]

3*a*b*x + (3*b^2*x^2)/2 + b*c*x^3 + (c^2*x^4)/4

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91

method	result	size
gosper	\(3 a b x +\frac {3}{2} b^{2} x^{2}+b c \,x^{3}+\frac {1}{4} c^{2} x^{4}\)	\(29\)
default	\(3 a b x +\frac {3}{2} b^{2} x^{2}+b c \,x^{3}+\frac {1}{4} c^{2} x^{4}\)	\(29\)
norman	\(3 a b x +\frac {3}{2} b^{2} x^{2}+b c \,x^{3}+\frac {1}{4} c^{2} x^{4}\)	\(29\)
risch	\(3 a b x +\frac {3}{2} b^{2} x^{2}+b c \,x^{3}+\frac {1}{4} c^{2} x^{4}\)	\(29\)
parallelrisch	\(3 a b x +\frac {3}{2} b^{2} x^{2}+b c \,x^{3}+\frac {1}{4} c^{2} x^{4}\)	\(29\)
parts	\(3 a b x +\frac {3}{2} b^{2} x^{2}+b c \,x^{3}+\frac {1}{4} c^{2} x^{4}\)	\(29\)

[In]

int(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b,x,method=_RETURNVERBOSE)

[Out]

3*a*b*x+3/2*b^2*x^2+b*c*x^3+1/4*c^2*x^4

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right ) \, dx=\frac {1}{4} \, c^{2} x^{4} + b c x^{3} + \frac {3}{2} \, b^{2} x^{2} + 3 \, a b x \]

[In]

integrate(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b,x, algorithm="fricas")

[Out]

1/4*c^2*x^4 + b*c*x^3 + 3/2*b^2*x^2 + 3*a*b*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right ) \, dx=3 a b x + \frac {3 b^{2} x^{2}}{2} + b c x^{3} + \frac {c^{2} x^{4}}{4} \]

[In]

integrate(c**2*x**3+3*b*c*x**2+3*b**2*x+3*a*b,x)

[Out]

3*a*b*x + 3*b**2*x**2/2 + b*c*x**3 + c**2*x**4/4

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right ) \, dx=\frac {1}{4} \, c^{2} x^{4} + b c x^{3} + \frac {3}{2} \, b^{2} x^{2} + 3 \, a b x \]

[In]

integrate(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b,x, algorithm="maxima")

[Out]

1/4*c^2*x^4 + b*c*x^3 + 3/2*b^2*x^2 + 3*a*b*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right ) \, dx=\frac {1}{4} \, c^{2} x^{4} + b c x^{3} + \frac {3}{2} \, b^{2} x^{2} + 3 \, a b x \]

[In]

integrate(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b,x, algorithm="giac")

[Out]

1/4*c^2*x^4 + b*c*x^3 + 3/2*b^2*x^2 + 3*a*b*x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right ) \, dx=\frac {3\,b^2\,x^2}{2}+b\,c\,x^3+3\,a\,b\,x+\frac {c^2\,x^4}{4} \]

[In]

int(3*a*b + 3*b^2*x + c^2*x^3 + 3*b*c*x^2,x)

[Out]

(3*b^2*x^2)/2 + (c^2*x^4)/4 + 3*a*b*x + b*c*x^3