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

Optimal. Leaf size=25 \[ \frac {a x^2}{2}+\frac {b x^4}{4}+\frac {c x^6}{6} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \frac {a x^2}{2}+\frac {b x^4}{4}+\frac {c x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[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*} \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}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} \frac {a x^2}{2}+\frac {b x^4}{4}+\frac {c x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a*x + b*x^3 + c*x^5,x]

[Out]

(a*x^2)/2 + (b*x^4)/4 + (c*x^6)/6

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Maple [A]
time = 0.02, size = 20, normalized size = 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	\(\frac {1}{2} a \,x^{2}+\frac {1}{4} b \,x^{4}+\frac {1}{6} c \,x^{6}\)	\(20\)
gosper	\(\frac {x^{2} \left (2 c \,x^{4}+3 b \,x^{2}+6 a \right )}{12}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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Maxima [A]
time = 0.26, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, c x^{6} + \frac {1}{4} \, b x^{4} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]
time = 0.33, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, c x^{6} + \frac {1}{4} \, b x^{4} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]
time = 0.01, size = 19, normalized size = 0.76 \begin {gather*} \frac {a x^{2}}{2} + \frac {b x^{4}}{4} + \frac {c x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**2/2 + b*x**4/4 + c*x**6/6

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Giac [A]
time = 3.66, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, c x^{6} + \frac {1}{4} \, b x^{4} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Mupad [B]
time = 0.03, size = 19, normalized size = 0.76 \begin {gather*} \frac {c\,x^6}{6}+\frac {b\,x^4}{4}+\frac {a\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a*x + b*x^3 + c*x^5,x)

[Out]

(a*x^2)/2 + (b*x^4)/4 + (c*x^6)/6

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