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

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

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Rubi [A]  time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} a x+b \log (x)-\frac {c}{x}-\frac {d}{2 x^2} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} a x+b \log (x)-\frac {c}{x}-\frac {d}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[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]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+\frac {d}{x^3}+\frac {c}{x^2}+\frac {b}{x}\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[a + d/x^3 + c/x^2 + b/x,x]

[Out]

IntegrateAlgebraic[a + d/x^3 + c/x^2 + b/x, x]

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fricas [A]  time = 1.20, size = 27, normalized size = 1.23 \begin {gather*} \frac {2 \, a x^{3} + 2 \, b x^{2} \log \relax (x) - 2 \, c x - d}{2 \, x^{2}} \end {gather*}

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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giac [A]  time = 0.85, size = 21, normalized size = 0.95 \begin {gather*} a x + b \log \left ({\left | x \right |}\right ) - \frac {c}{x} - \frac {d}{2 \, x^{2}} \end {gather*}

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

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maple [A]  time = 0.00, size = 21, normalized size = 0.95 \begin {gather*} a x +b \ln \relax (x )-\frac {c}{x}-\frac {d}{2 x^{2}} \end {gather*}

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 = 1.03, size = 20, normalized size = 0.91 \begin {gather*} a x + b \log \relax (x) - \frac {c}{x} - \frac {d}{2 \, x^{2}} \end {gather*}

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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mupad [B]  time = 0.04, size = 20, normalized size = 0.91 \begin {gather*} a\,x-\frac {\frac {d}{2}+c\,x}{x^2}+b\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b/x + c/x^2 + d/x^3,x)

[Out]

a*x - (d/2 + c*x)/x^2 + b*log(x)

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sympy [A]  time = 0.16, size = 20, normalized size = 0.91 \begin {gather*} a x + b \log {\relax (x )} + \frac {- 2 c x - d}{2 x^{2}} \end {gather*}

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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