\(\int (a x+b x^3+c x^5) \, dx\) [68]

Optimal result

Integrand size = 14, antiderivative size = 25 \[ \int \left (a x+b x^3+c x^5\right ) \, dx=\frac {a x^2}{2}+\frac {b x^4}{4}+\frac {c x^6}{6} \]

[Out]

1/2*a*x^2+1/4*b*x^4+1/6*c*x^6

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 = {} \[ \int \left (a x+b x^3+c x^5\right ) \, dx=\frac {a x^2}{2}+\frac {b x^4}{4}+\frac {c x^6}{6} \]

[In]

Int[a*x + b*x^3 + c*x^5,x]

[Out]

(a*x^2)/2 + (b*x^4)/4 + (c*x^6)/6

Rubi steps \begin{align*} \text {integral}& = \frac {a x^2}{2}+\frac {b x^4}{4}+\frac {c x^6}{6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \left (a x+b x^3+c x^5\right ) \, dx=\frac {a x^2}{2}+\frac {b x^4}{4}+\frac {c x^6}{6} \]

[In]

Integrate[a*x + b*x^3 + c*x^5,x]

[Out]

(a*x^2)/2 + (b*x^4)/4 + (c*x^6)/6

Maple [A] (verified)

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

method	result	size
default	\(\frac {1}{2} a \,x^{2}+\frac {1}{4} b \,x^{4}+\frac {1}{6} c \,x^{6}\)	\(20\)
norman	\(\frac {1}{2} a \,x^{2}+\frac {1}{4} b \,x^{4}+\frac {1}{6} c \,x^{6}\)	\(20\)
risch	\(\frac {1}{2} a \,x^{2}+\frac {1}{4} b \,x^{4}+\frac {1}{6} c \,x^{6}\)	\(20\)
parallelrisch	\(\frac {1}{2} a \,x^{2}+\frac {1}{4} b \,x^{4}+\frac {1}{6} c \,x^{6}\)	\(20\)
parts	\(\frac {1}{2} a \,x^{2}+\frac {1}{4} b \,x^{4}+\frac {1}{6} c \,x^{6}\)	\(20\)
gosper	\(\frac {x^{2} \left (2 c \,x^{4}+3 b \,x^{2}+6 a \right )}{12}\)	\(22\)

[In]

int(c*x^5+b*x^3+a*x,x,method=_RETURNVERBOSE)

[Out]

1/2*a*x^2+1/4*b*x^4+1/6*c*x^6

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \left (a x+b x^3+c x^5\right ) \, dx=\frac {1}{6} \, c x^{6} + \frac {1}{4} \, b x^{4} + \frac {1}{2} \, a x^{2} \]

[In]

integrate(c*x^5+b*x^3+a*x,x, algorithm="fricas")

[Out]

1/6*c*x^6 + 1/4*b*x^4 + 1/2*a*x^2

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \left (a x+b x^3+c x^5\right ) \, dx=\frac {a x^{2}}{2} + \frac {b x^{4}}{4} + \frac {c x^{6}}{6} \]

[In]

integrate(c*x**5+b*x**3+a*x,x)

[Out]

a*x**2/2 + b*x**4/4 + c*x**6/6

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \left (a x+b x^3+c x^5\right ) \, dx=\frac {1}{6} \, c x^{6} + \frac {1}{4} \, b x^{4} + \frac {1}{2} \, a x^{2} \]

[In]

integrate(c*x^5+b*x^3+a*x,x, algorithm="maxima")

[Out]

1/6*c*x^6 + 1/4*b*x^4 + 1/2*a*x^2

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \left (a x+b x^3+c x^5\right ) \, dx=\frac {1}{6} \, c x^{6} + \frac {1}{4} \, b x^{4} + \frac {1}{2} \, a x^{2} \]

[In]

integrate(c*x^5+b*x^3+a*x,x, algorithm="giac")

[Out]

1/6*c*x^6 + 1/4*b*x^4 + 1/2*a*x^2

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \left (a x+b x^3+c x^5\right ) \, dx=\frac {c\,x^6}{6}+\frac {b\,x^4}{4}+\frac {a\,x^2}{2} \]

[In]

int(a*x + b*x^3 + c*x^5,x)

[Out]

(a*x^2)/2 + (b*x^4)/4 + (c*x^6)/6