\(\int \frac {a+b x^2+c x^4}{x^3} \, dx\) [819]

Optimal result

Integrand size = 16, antiderivative size = 21 \[ \int \frac {a+b x^2+c x^4}{x^3} \, dx=-\frac {a}{2 x^2}+\frac {c x^2}{2}+b \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14} \[ \int \frac {a+b x^2+c x^4}{x^3} \, dx=-\frac {a}{2 x^2}+b \log (x)+\frac {c x^2}{2} \]

[In]

Int[(a + b*x^2 + c*x^4)/x^3,x]

[Out]

-1/2*a/x^2 + (c*x^2)/2 + b*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*} \text {integral}& = \int \left (\frac {a}{x^3}+\frac {b}{x}+c x\right ) \, dx \\ & = -\frac {a}{2 x^2}+\frac {c x^2}{2}+b \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^2+c x^4}{x^3} \, dx=-\frac {a}{2 x^2}+\frac {c x^2}{2}+b \log (x) \]

[In]

Integrate[(a + b*x^2 + c*x^4)/x^3,x]

[Out]

-1/2*a/x^2 + (c*x^2)/2 + b*Log[x]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86

method	result	size
default	\(-\frac {a}{2 x^{2}}+\frac {c \,x^{2}}{2}+b \ln \left (x \right )\)	\(18\)
risch	\(-\frac {a}{2 x^{2}}+\frac {c \,x^{2}}{2}+b \ln \left (x \right )\)	\(18\)
norman	\(\frac {-\frac {a}{2}+\frac {c \,x^{4}}{2}}{x^{2}}+b \ln \left (x \right )\)	\(20\)
parallelrisch	\(\frac {c \,x^{4}+2 \ln \left (x \right ) x^{2} b -a}{2 x^{2}}\)	\(23\)

[In]

int((c*x^4+b*x^2+a)/x^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {a+b x^2+c x^4}{x^3} \, dx=\frac {c x^{4} + 2 \, b x^{2} \log \left (x\right ) - a}{2 \, x^{2}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^3,x, algorithm="fricas")

[Out]

1/2*(c*x^4 + 2*b*x^2*log(x) - a)/x^2

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {a+b x^2+c x^4}{x^3} \, dx=- \frac {a}{2 x^{2}} + b \log {\left (x \right )} + \frac {c x^{2}}{2} \]

[In]

integrate((c*x**4+b*x**2+a)/x**3,x)

[Out]

-a/(2*x**2) + b*log(x) + c*x**2/2

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {a+b x^2+c x^4}{x^3} \, dx=\frac {1}{2} \, c x^{2} + \frac {1}{2} \, b \log \left (x^{2}\right ) - \frac {a}{2 \, x^{2}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^3,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {a+b x^2+c x^4}{x^3} \, dx=\frac {1}{2} \, c x^{2} + \frac {1}{2} \, b \log \left (x^{2}\right ) - \frac {b x^{2} + a}{2 \, x^{2}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {a+b x^2+c x^4}{x^3} \, dx=\frac {c\,x^2}{2}-\frac {a}{2\,x^2}+b\,\ln \left (x\right ) \]

[In]

int((a + b*x^2 + c*x^4)/x^3,x)

[Out]

(c*x^2)/2 - a/(2*x^2) + b*log(x)