\(\int \frac {b x^2+c x^4}{x^3} \, dx\) [139]

Optimal result

Integrand size = 15, antiderivative size = 13 \[ \int \frac {b x^2+c x^4}{x^3} \, dx=\frac {c x^2}{2}+b \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14} \[ \int \frac {b x^2+c x^4}{x^3} \, dx=b \log (x)+\frac {c x^2}{2} \]

[In]

Int[(b*x^2 + c*x^4)/x^3,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {b x^2+c x^4}{x^3} \, dx=\frac {c x^2}{2}+b \log (x) \]

[In]

Integrate[(b*x^2 + c*x^4)/x^3,x]

[Out]

(c*x^2)/2 + b*Log[x]

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {c \,x^{2}}{2}+b \ln \left (x \right )\)	\(12\)
norman	\(\frac {c \,x^{2}}{2}+b \ln \left (x \right )\)	\(12\)
risch	\(\frac {c \,x^{2}}{2}+b \ln \left (x \right )\)	\(12\)
parallelrisch	\(\frac {c \,x^{2}}{2}+b \ln \left (x \right )\)	\(12\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {b x^2+c x^4}{x^3} \, dx=\frac {1}{2} \, c x^{2} + b \log \left (x\right ) \]

[In]

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

[Out]

1/2*c*x^2 + b*log(x)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {b x^2+c x^4}{x^3} \, dx=b \log {\left (x \right )} + \frac {c x^{2}}{2} \]

[In]

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

[Out]

b*log(x) + c*x**2/2

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {b x^2+c x^4}{x^3} \, dx=\frac {1}{2} \, c x^{2} + \frac {1}{2} \, b \log \left (x^{2}\right ) \]

[In]

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

[Out]

1/2*c*x^2 + 1/2*b*log(x^2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {b x^2+c x^4}{x^3} \, dx=\frac {1}{2} \, c x^{2} + \frac {1}{2} \, b \log \left (x^{2}\right ) \]

[In]

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

[Out]

1/2*c*x^2 + 1/2*b*log(x^2)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {b x^2+c x^4}{x^3} \, dx=\frac {c\,x^2}{2}+b\,\ln \left (x\right ) \]

[In]

int((b*x^2 + c*x^4)/x^3,x)

[Out]

(c*x^2)/2 + b*log(x)