\(\int (a+b x^2+c x^4) \, dx\) [816]

Optimal result

Integrand size = 12, antiderivative size = 20 \[ \int \left (a+b x^2+c x^4\right ) \, dx=a x+\frac {b x^3}{3}+\frac {c x^5}{5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, 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 = {} \[ \int \left (a+b x^2+c x^4\right ) \, dx=a x+\frac {b x^3}{3}+\frac {c x^5}{5} \]

[In]

Int[a + b*x^2 + c*x^4,x]

[Out]

a*x + (b*x^3)/3 + (c*x^5)/5

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2+c x^4\right ) \, dx=a x+\frac {b x^3}{3}+\frac {c x^5}{5} \]

[In]

Integrate[a + b*x^2 + c*x^4,x]

[Out]

a*x + (b*x^3)/3 + (c*x^5)/5

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85

method	result	size
gosper	\(a x +\frac {1}{3} b \,x^{3}+\frac {1}{5} c \,x^{5}\)	\(17\)
default	\(a x +\frac {1}{3} b \,x^{3}+\frac {1}{5} c \,x^{5}\)	\(17\)
norman	\(a x +\frac {1}{3} b \,x^{3}+\frac {1}{5} c \,x^{5}\)	\(17\)
risch	\(a x +\frac {1}{3} b \,x^{3}+\frac {1}{5} c \,x^{5}\)	\(17\)
parallelrisch	\(a x +\frac {1}{3} b \,x^{3}+\frac {1}{5} c \,x^{5}\)	\(17\)
parts	\(a x +\frac {1}{3} b \,x^{3}+\frac {1}{5} c \,x^{5}\)	\(17\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{5} \, c x^{5} + \frac {1}{3} \, b x^{3} + a x \]

[In]

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

[Out]

1/5*c*x^5 + 1/3*b*x^3 + a*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \left (a+b x^2+c x^4\right ) \, dx=a x + \frac {b x^{3}}{3} + \frac {c x^{5}}{5} \]

[In]

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

[Out]

a*x + b*x**3/3 + c*x**5/5

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{5} \, c x^{5} + \frac {1}{3} \, b x^{3} + a x \]

[In]

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

[Out]

1/5*c*x^5 + 1/3*b*x^3 + a*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{5} \, c x^{5} + \frac {1}{3} \, b x^{3} + a x \]

[In]

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

[Out]

1/5*c*x^5 + 1/3*b*x^3 + a*x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \left (a+b x^2+c x^4\right ) \, dx=\frac {c\,x^5}{5}+\frac {b\,x^3}{3}+a\,x \]

[In]

int(a + b*x^2 + c*x^4,x)

[Out]

a*x + (b*x^3)/3 + (c*x^5)/5