\(\int (a+\frac {d}{x^3}+\frac {c}{x^2}+\frac {b}{x}) \, dx\) [1902]

Optimal result

Integrand size = 17, antiderivative size = 22 \[ \int \left (a+\frac {d}{x^3}+\frac {c}{x^2}+\frac {b}{x}\right ) \, dx=-\frac {d}{2 x^2}-\frac {c}{x}+a x+b \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (a+\frac {d}{x^3}+\frac {c}{x^2}+\frac {b}{x}\right ) \, dx=a x+b \log (x)-\frac {c}{x}-\frac {d}{2 x^2} \]

[In]

Int[a + d/x^3 + c/x^2 + b/x,x]

[Out]

-1/2*d/x^2 - c/x + a*x + b*Log[x]

Rubi steps \begin{align*} \text {integral}& = -\frac {d}{2 x^2}-\frac {c}{x}+a x+b \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {d}{x^3}+\frac {c}{x^2}+\frac {b}{x}\right ) \, dx=-\frac {d}{2 x^2}-\frac {c}{x}+a x+b \log (x) \]

[In]

Integrate[a + d/x^3 + c/x^2 + b/x,x]

[Out]

-1/2*d/x^2 - c/x + a*x + b*Log[x]

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95

method	result	size
default	\(-\frac {d}{2 x^{2}}-\frac {c}{x}+a x +b \ln \left (x \right )\)	\(21\)
risch	\(-\frac {d}{2 x^{2}}-\frac {c}{x}+a x +b \ln \left (x \right )\)	\(21\)
norman	\(\frac {a \,x^{3}-\frac {1}{2} d -c x}{x^{2}}+b \ln \left (x \right )\)	\(23\)
parallelrisch	\(\frac {2 b \ln \left (x \right ) x^{2}-2 c x -d}{2 x^{2}}+a x\)	\(26\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \left (a+\frac {d}{x^3}+\frac {c}{x^2}+\frac {b}{x}\right ) \, dx=\frac {2 \, a x^{3} + 2 \, b x^{2} \log \left (x\right ) - 2 \, c x - d}{2 \, x^{2}} \]

[In]

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

[Out]

1/2*(2*a*x^3 + 2*b*x^2*log(x) - 2*c*x - d)/x^2

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \left (a+\frac {d}{x^3}+\frac {c}{x^2}+\frac {b}{x}\right ) \, dx=a x + b \log {\left (x \right )} + \frac {- 2 c x - d}{2 x^{2}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \left (a+\frac {d}{x^3}+\frac {c}{x^2}+\frac {b}{x}\right ) \, dx=a x + b \log \left (x\right ) - \frac {c}{x} - \frac {d}{2 \, x^{2}} \]

[In]

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

[Out]

a*x + b*log(x) - c/x - 1/2*d/x^2

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \left (a+\frac {d}{x^3}+\frac {c}{x^2}+\frac {b}{x}\right ) \, dx=a x + b \log \left ({\left | x \right |}\right ) - \frac {c}{x} - \frac {d}{2 \, x^{2}} \]

[In]

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

[Out]

a*x + b*log(abs(x)) - c/x - 1/2*d/x^2

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \left (a+\frac {d}{x^3}+\frac {c}{x^2}+\frac {b}{x}\right ) \, dx=a\,x-\frac {\frac {d}{2}+c\,x}{x^2}+b\,\ln \left (x\right ) \]

[In]

int(a + b/x + c/x^2 + d/x^3,x)

[Out]

a*x - (d/2 + c*x)/x^2 + b*log(x)