\(\int (a x^2+b x^3+c x^4) \, dx\) [3]

Optimal result

Integrand size = 16, antiderivative size = 25 \[ \int \left (a x^2+b x^3+c x^4\right ) \, dx=\frac {a x^3}{3}+\frac {b x^4}{4}+\frac {c x^5}{5} \]

[Out]

1/3*a*x^3+1/4*b*x^4+1/5*c*x^5

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, 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 = {} \[ \int \left (a x^2+b x^3+c x^4\right ) \, dx=\frac {a x^3}{3}+\frac {b x^4}{4}+\frac {c x^5}{5} \]

[In]

Int[a*x^2 + b*x^3 + c*x^4,x]

[Out]

(a*x^3)/3 + (b*x^4)/4 + (c*x^5)/5

Rubi steps \begin{align*} \text {integral}& = \frac {a x^3}{3}+\frac {b x^4}{4}+\frac {c x^5}{5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \left (a x^2+b x^3+c x^4\right ) \, dx=\frac {a x^3}{3}+\frac {b x^4}{4}+\frac {c x^5}{5} \]

[In]

Integrate[a*x^2 + b*x^3 + c*x^4,x]

[Out]

(a*x^3)/3 + (b*x^4)/4 + (c*x^5)/5

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80

method	result	size
gosper	\(\frac {x^{3} \left (12 c \,x^{2}+15 b x +20 a \right )}{60}\)	\(20\)
default	\(\frac {1}{3} a \,x^{3}+\frac {1}{4} b \,x^{4}+\frac {1}{5} c \,x^{5}\)	\(20\)
norman	\(\frac {1}{3} a \,x^{3}+\frac {1}{4} b \,x^{4}+\frac {1}{5} c \,x^{5}\)	\(20\)
risch	\(\frac {1}{3} a \,x^{3}+\frac {1}{4} b \,x^{4}+\frac {1}{5} c \,x^{5}\)	\(20\)
parallelrisch	\(\frac {1}{3} a \,x^{3}+\frac {1}{4} b \,x^{4}+\frac {1}{5} c \,x^{5}\)	\(20\)
parts	\(\frac {1}{3} a \,x^{3}+\frac {1}{4} b \,x^{4}+\frac {1}{5} c \,x^{5}\)	\(20\)

[In]

int(c*x^4+b*x^3+a*x^2,x,method=_RETURNVERBOSE)

[Out]

1/60*x^3*(12*c*x^2+15*b*x+20*a)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \left (a x^2+b x^3+c x^4\right ) \, dx=\frac {1}{5} \, c x^{5} + \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \]

[In]

integrate(c*x^4+b*x^3+a*x^2,x, algorithm="fricas")

[Out]

1/5*c*x^5 + 1/4*b*x^4 + 1/3*a*x^3

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \left (a x^2+b x^3+c x^4\right ) \, dx=\frac {a x^{3}}{3} + \frac {b x^{4}}{4} + \frac {c x^{5}}{5} \]

[In]

integrate(c*x**4+b*x**3+a*x**2,x)

[Out]

a*x**3/3 + b*x**4/4 + c*x**5/5

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \left (a x^2+b x^3+c x^4\right ) \, dx=\frac {1}{5} \, c x^{5} + \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \]

[In]

integrate(c*x^4+b*x^3+a*x^2,x, algorithm="maxima")

[Out]

1/5*c*x^5 + 1/4*b*x^4 + 1/3*a*x^3

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \left (a x^2+b x^3+c x^4\right ) \, dx=\frac {1}{5} \, c x^{5} + \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \]

[In]

integrate(c*x^4+b*x^3+a*x^2,x, algorithm="giac")

[Out]

1/5*c*x^5 + 1/4*b*x^4 + 1/3*a*x^3

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \left (a x^2+b x^3+c x^4\right ) \, dx=\frac {x^3\,\left (12\,c\,x^2+15\,b\,x+20\,a\right )}{60} \]

[In]

int(a*x^2 + b*x^3 + c*x^4,x)

[Out]

(x^3*(20*a + 15*b*x + 12*c*x^2))/60