\(\int x^2 (a+b x^2+c x^4) \, dx\) [814]

Optimal result

Integrand size = 16, antiderivative size = 25 \[ \int x^2 \left (a+b x^2+c x^4\right ) \, dx=\frac {a x^3}{3}+\frac {b x^5}{5}+\frac {c x^7}{7} \]

[Out]

1/3*a*x^3+1/5*b*x^5+1/7*c*x^7

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14} \[ \int x^2 \left (a+b x^2+c x^4\right ) \, dx=\frac {a x^3}{3}+\frac {b x^5}{5}+\frac {c x^7}{7} \]

[In]

Int[x^2*(a + b*x^2 + c*x^4),x]

[Out]

(a*x^3)/3 + (b*x^5)/5 + (c*x^7)/7

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^2+b x^4+c x^6\right ) \, dx \\ & = \frac {a x^3}{3}+\frac {b x^5}{5}+\frac {c x^7}{7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b x^2+c x^4\right ) \, dx=\frac {a x^3}{3}+\frac {b x^5}{5}+\frac {c x^7}{7} \]

[In]

Integrate[x^2*(a + b*x^2 + c*x^4),x]

[Out]

(a*x^3)/3 + (b*x^5)/5 + (c*x^7)/7

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80

method	result	size
gosper	\(\frac {1}{3} a \,x^{3}+\frac {1}{5} b \,x^{5}+\frac {1}{7} c \,x^{7}\)	\(20\)
default	\(\frac {1}{3} a \,x^{3}+\frac {1}{5} b \,x^{5}+\frac {1}{7} c \,x^{7}\)	\(20\)
norman	\(\frac {1}{3} a \,x^{3}+\frac {1}{5} b \,x^{5}+\frac {1}{7} c \,x^{7}\)	\(20\)
risch	\(\frac {1}{3} a \,x^{3}+\frac {1}{5} b \,x^{5}+\frac {1}{7} c \,x^{7}\)	\(20\)
parallelrisch	\(\frac {1}{3} a \,x^{3}+\frac {1}{5} b \,x^{5}+\frac {1}{7} c \,x^{7}\)	\(20\)

[In]

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

[Out]

1/3*a*x^3+1/5*b*x^5+1/7*c*x^7

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x^2 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{7} \, c x^{7} + \frac {1}{5} \, b x^{5} + \frac {1}{3} \, a x^{3} \]

[In]

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

[Out]

1/7*c*x^7 + 1/5*b*x^5 + 1/3*a*x^3

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x^2 \left (a+b x^2+c x^4\right ) \, dx=\frac {a x^{3}}{3} + \frac {b x^{5}}{5} + \frac {c x^{7}}{7} \]

[In]

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

[Out]

a*x**3/3 + b*x**5/5 + c*x**7/7

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x^2 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{7} \, c x^{7} + \frac {1}{5} \, b x^{5} + \frac {1}{3} \, a x^{3} \]

[In]

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

[Out]

1/7*c*x^7 + 1/5*b*x^5 + 1/3*a*x^3

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x^2 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{7} \, c x^{7} + \frac {1}{5} \, b x^{5} + \frac {1}{3} \, a x^{3} \]

[In]

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

[Out]

1/7*c*x^7 + 1/5*b*x^5 + 1/3*a*x^3

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x^2 \left (a+b x^2+c x^4\right ) \, dx=\frac {c\,x^7}{7}+\frac {b\,x^5}{5}+\frac {a\,x^3}{3} \]

[In]

int(x^2*(a + b*x^2 + c*x^4),x)

[Out]

(a*x^3)/3 + (b*x^5)/5 + (c*x^7)/7