3.9.17 \(\int \frac {a+b x^2+c x^4}{x} \, dx\) [817]

Optimal. Leaf size=21 \[ \frac {b x^2}{2}+\frac {c x^4}{4}+a \log (x) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14} \begin {gather*} a \log (x)+\frac {b x^2}{2}+\frac {c x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x,x]

[Out]

(b*x^2)/2 + (c*x^4)/4 + a*Log[x]

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

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Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} \frac {b x^2}{2}+\frac {c x^4}{4}+a \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x,x]

[Out]

(b*x^2)/2 + (c*x^4)/4 + a*Log[x]

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Maple [A]
time = 0.02, size = 18, normalized size = 0.86

method	result	size
default	\(\frac {b \,x^{2}}{2}+\frac {c \,x^{4}}{4}+a \ln \left (x \right )\)	\(18\)
norman	\(\frac {b \,x^{2}}{2}+\frac {c \,x^{4}}{4}+a \ln \left (x \right )\)	\(18\)
risch	\(\frac {c \,x^{4}}{4}+\frac {b \,x^{2}}{2}+\frac {b^{2}}{4 c}+a \ln \left (x \right )\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 20, normalized size = 0.95 \begin {gather*} \frac {1}{4} \, c x^{4} + \frac {1}{2} \, b x^{2} + \frac {1}{2} \, a \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.35, size = 17, normalized size = 0.81 \begin {gather*} \frac {1}{4} \, c x^{4} + \frac {1}{2} \, b x^{2} + a \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c*x^4 + 1/2*b*x^2 + a*log(x)

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Sympy [A]
time = 0.02, size = 17, normalized size = 0.81 \begin {gather*} a \log {\left (x \right )} + \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+a)/x,x)

[Out]

a*log(x) + b*x**2/2 + c*x**4/4

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Giac [A]
time = 3.99, size = 20, normalized size = 0.95 \begin {gather*} \frac {1}{4} \, c x^{4} + \frac {1}{2} \, b x^{2} + \frac {1}{2} \, a \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.02, size = 17, normalized size = 0.81 \begin {gather*} \frac {b\,x^2}{2}+\frac {c\,x^4}{4}+a\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)/x,x)

[Out]

(b*x^2)/2 + (c*x^4)/4 + a*log(x)

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