\(\int (a+b x+c x^2+d x^3) \, dx\) [1891]

Optimal result

Integrand size = 15, antiderivative size = 28 \[ \int \left (a+b x+c x^2+d x^3\right ) \, dx=a x+\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {d x^4}{4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 28, 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 = {} \[ \int \left (a+b x+c x^2+d x^3\right ) \, dx=a x+\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {d x^4}{4} \]

[In]

Int[a + b*x + c*x^2 + d*x^3,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \left (a+b x+c x^2+d x^3\right ) \, dx=a x+\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {d x^4}{4} \]

[In]

Integrate[a + b*x + c*x^2 + d*x^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82

method	result	size
gosper	\(a x +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}+\frac {1}{4} d \,x^{4}\)	\(23\)
default	\(a x +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}+\frac {1}{4} d \,x^{4}\)	\(23\)
norman	\(a x +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}+\frac {1}{4} d \,x^{4}\)	\(23\)
risch	\(a x +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}+\frac {1}{4} d \,x^{4}\)	\(23\)
parallelrisch	\(a x +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}+\frac {1}{4} d \,x^{4}\)	\(23\)
parts	\(a x +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}+\frac {1}{4} d \,x^{4}\)	\(23\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \left (a+b x+c x^2+d x^3\right ) \, dx=\frac {1}{4} \, d x^{4} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \left (a+b x+c x^2+d x^3\right ) \, dx=a x + \frac {b x^{2}}{2} + \frac {c x^{3}}{3} + \frac {d x^{4}}{4} \]

[In]

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

[Out]

a*x + b*x**2/2 + c*x**3/3 + d*x**4/4

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \left (a+b x+c x^2+d x^3\right ) \, dx=\frac {1}{4} \, d x^{4} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \left (a+b x+c x^2+d x^3\right ) \, dx=\frac {1}{4} \, d x^{4} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \left (a+b x+c x^2+d x^3\right ) \, dx=\frac {d\,x^4}{4}+\frac {c\,x^3}{3}+\frac {b\,x^2}{2}+a\,x \]

[In]

int(a + b*x + c*x^2 + d*x^3,x)

[Out]

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