\(\int x (b x^2+c x^4) \, dx\) [135]

Optimal result

Integrand size = 13, antiderivative size = 17 \[ \int x \left (b x^2+c x^4\right ) \, dx=\frac {b x^4}{4}+\frac {c x^6}{6} \]

[Out]

1/4*b*x^4+1/6*c*x^6

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14} \[ \int x \left (b x^2+c x^4\right ) \, dx=\frac {b x^4}{4}+\frac {c x^6}{6} \]

[In]

Int[x*(b*x^2 + c*x^4),x]

[Out]

(b*x^4)/4 + (c*x^6)/6

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 (b x^3+c x^5\right ) \, dx \\ & = \frac {b x^4}{4}+\frac {c x^6}{6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int x \left (b x^2+c x^4\right ) \, dx=\frac {b x^4}{4}+\frac {c x^6}{6} \]

[In]

Integrate[x*(b*x^2 + c*x^4),x]

[Out]

(b*x^4)/4 + (c*x^6)/6

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
default	\(\frac {1}{4} b \,x^{4}+\frac {1}{6} c \,x^{6}\)	\(14\)
norman	\(\frac {1}{4} b \,x^{4}+\frac {1}{6} c \,x^{6}\)	\(14\)
risch	\(\frac {1}{4} b \,x^{4}+\frac {1}{6} c \,x^{6}\)	\(14\)
parallelrisch	\(\frac {1}{4} b \,x^{4}+\frac {1}{6} c \,x^{6}\)	\(14\)
gosper	\(\frac {x^{4} \left (2 c \,x^{2}+3 b \right )}{12}\)	\(16\)

[In]

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

[Out]

1/4*b*x^4+1/6*c*x^6

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int x \left (b x^2+c x^4\right ) \, dx=\frac {1}{6} \, c x^{6} + \frac {1}{4} \, b x^{4} \]

[In]

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

[Out]

1/6*c*x^6 + 1/4*b*x^4

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int x \left (b x^2+c x^4\right ) \, dx=\frac {b x^{4}}{4} + \frac {c x^{6}}{6} \]

[In]

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

[Out]

b*x**4/4 + c*x**6/6

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int x \left (b x^2+c x^4\right ) \, dx=\frac {1}{6} \, c x^{6} + \frac {1}{4} \, b x^{4} \]

[In]

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

[Out]

1/6*c*x^6 + 1/4*b*x^4

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int x \left (b x^2+c x^4\right ) \, dx=\frac {1}{6} \, c x^{6} + \frac {1}{4} \, b x^{4} \]

[In]

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

[Out]

1/6*c*x^6 + 1/4*b*x^4

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int x \left (b x^2+c x^4\right ) \, dx=\frac {c\,x^6}{6}+\frac {b\,x^4}{4} \]

[In]

int(x*(b*x^2 + c*x^4),x)

[Out]

(b*x^4)/4 + (c*x^6)/6