\(\int x (a+b x+c x^2) \, dx\) [2356]

Optimal result

Integrand size = 12, antiderivative size = 25 \[ \int x \left (a+b x+c x^2\right ) \, dx=\frac {a x^2}{2}+\frac {b x^3}{3}+\frac {c x^4}{4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {14} \[ \int x \left (a+b x+c x^2\right ) \, dx=\frac {a x^2}{2}+\frac {b x^3}{3}+\frac {c x^4}{4} \]

[In]

Int[x*(a + b*x + c*x^2),x]

[Out]

(a*x^2)/2 + (b*x^3)/3 + (c*x^4)/4

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (a x+b x^2+c x^3\right ) \, dx \\ & = \frac {a x^2}{2}+\frac {b x^3}{3}+\frac {c x^4}{4} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[x*(a + b*x + c*x^2),x]

[Out]

(a*x^2)/2 + (b*x^3)/3 + (c*x^4)/4

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80

method	result	size
gosper	\(\frac {1}{2} a \,x^{2}+\frac {1}{3} b \,x^{3}+\frac {1}{4} c \,x^{4}\)	\(20\)
default	\(\frac {1}{2} a \,x^{2}+\frac {1}{3} b \,x^{3}+\frac {1}{4} c \,x^{4}\)	\(20\)
norman	\(\frac {1}{2} a \,x^{2}+\frac {1}{3} b \,x^{3}+\frac {1}{4} c \,x^{4}\)	\(20\)
risch	\(\frac {1}{2} a \,x^{2}+\frac {1}{3} b \,x^{3}+\frac {1}{4} c \,x^{4}\)	\(20\)
parallelrisch	\(\frac {1}{2} a \,x^{2}+\frac {1}{3} b \,x^{3}+\frac {1}{4} c \,x^{4}\)	\(20\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x \left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, c x^{4} + \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} \]

[In]

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

[Out]

1/4*c*x^4 + 1/3*b*x^3 + 1/2*a*x^2

Sympy [A] (verification not implemented)

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

[In]

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

[Out]

a*x**2/2 + b*x**3/3 + c*x**4/4

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x \left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, c x^{4} + \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} \]

[In]

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

[Out]

1/4*c*x^4 + 1/3*b*x^3 + 1/2*a*x^2

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x \left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, c x^{4} + \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} \]

[In]

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

[Out]

1/4*c*x^4 + 1/3*b*x^3 + 1/2*a*x^2

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x \left (a+b x+c x^2\right ) \, dx=\frac {x^2\,\left (3\,c\,x^2+4\,b\,x+6\,a\right )}{12} \]

[In]

int(x*(a + b*x + c*x^2),x)

[Out]

(x^2*(6*a + 4*b*x + 3*c*x^2))/12