3.1902 \(\int (a+\frac{d}{x^3}+\frac{c}{x^2}+\frac{b}{x}) \, dx\)

Optimal. Leaf size=22 \[ a x+b \log (x)-\frac{c}{x}-\frac{d}{2 x^2} \]

[Out]

-d/(2*x^2) - c/x + a*x + b*Log[x]

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Rubi [A]  time = 0.0039866, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ a x+b \log (x)-\frac{c}{x}-\frac{d}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[a + d/x^3 + c/x^2 + b/x,x]

[Out]

-d/(2*x^2) - c/x + a*x + b*Log[x]

Rubi steps

\begin{align*} \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)\\ \end{align*}

Mathematica [A]  time = 0.0063733, size = 22, normalized size = 1. \[ a x+b \log (x)-\frac{c}{x}-\frac{d}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[a + d/x^3 + c/x^2 + b/x,x]

[Out]

-d/(2*x^2) - c/x + a*x + b*Log[x]

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Maple [A]  time = 0.001, size = 21, normalized size = 1. \begin{align*} -{\frac{d}{2\,{x}^{2}}}-{\frac{c}{x}}+ax+b\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.983506, size = 27, normalized size = 1.23 \begin{align*} a x + b \log \left (x\right ) - \frac{c}{x} - \frac{d}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 2.0407, size = 65, normalized size = 2.95 \begin{align*} \frac{2 \, a x^{3} + 2 \, b x^{2} \log \left (x\right ) - 2 \, c x - d}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 0.305357, size = 19, normalized size = 0.86 \begin{align*} a x + b \log{\left (x \right )} - \frac{2 c x + d}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.07752, size = 28, normalized size = 1.27 \begin{align*} a x + b \log \left ({\left | x \right |}\right ) - \frac{c}{x} - \frac{d}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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