3.2.37 \(\int \frac {b x^2+c x^4}{x} \, dx\) [137]

Optimal. Leaf size=17 \[ \frac {b x^2}{2}+\frac {c x^4}{4} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14} \begin {gather*} \frac {b x^2}{2}+\frac {c x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)/x,x]

[Out]

(b*x^2)/2 + (c*x^4)/4

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*} \int \frac {b x^2+c x^4}{x} \, dx &=\int \left (b x+c x^3\right ) \, dx\\ &=\frac {b x^2}{2}+\frac {c x^4}{4}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {b x^2}{2}+\frac {c x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)/x,x]

[Out]

(b*x^2)/2 + (c*x^4)/4

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Maple [A]
time = 0.03, size = 15, normalized size = 0.88

method	result	size
norman	\(\frac {1}{2} b \,x^{2}+\frac {1}{4} c \,x^{4}\)	\(14\)
gosper	\(\frac {x^{2} \left (c \,x^{2}+2 b \right )}{4}\)	\(15\)
default	\(\frac {\left (c \,x^{2}+b \right )^{2}}{4 c}\)	\(15\)
risch	\(\frac {c \,x^{4}}{4}+\frac {b \,x^{2}}{2}+\frac {b^{2}}{4 c}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c*x^2+b)^2/c

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Maxima [A]
time = 0.28, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, c x^{4} + \frac {1}{2} \, b x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c*x^4 + 1/2*b*x^2

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Fricas [A]
time = 0.31, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, c x^{4} + \frac {1}{2} \, b x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c*x^4 + 1/2*b*x^2

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Sympy [A]
time = 0.01, size = 12, normalized size = 0.71 \begin {gather*} \frac {b x^{2}}{2} + \frac {c x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x**2/2 + c*x**4/4

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Giac [A]
time = 5.37, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, c x^{4} + \frac {1}{2} \, b x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c*x^4 + 1/2*b*x^2

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Mupad [B]
time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {c\,x^4}{4}+\frac {b\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)/x,x)

[Out]

(b*x^2)/2 + (c*x^4)/4

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