3.50.73 \(\int \frac {-3125+6250 x-1250 x^2+9225 x^3-4 x^4}{3125 x} \, dx\)

Optimal. Leaf size=29 \[ -4+2 x+x^3-\frac {1}{5} \left (x+\frac {x^2}{25}\right )^2-\log (2 x) \]

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Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {12, 14} \begin {gather*} -\frac {x^4}{3125}+\frac {123 x^3}{125}-\frac {x^2}{5}+2 x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3125 + 6250*x - 1250*x^2 + 9225*x^3 - 4*x^4)/(3125*x),x]

[Out]

2*x - x^2/5 + (123*x^3)/125 - x^4/3125 - Log[x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-3125+6250 x-1250 x^2+9225 x^3-4 x^4}{x} \, dx}{3125}\\ &=\frac {\int \left (6250-\frac {3125}{x}-1250 x+9225 x^2-4 x^3\right ) \, dx}{3125}\\ &=2 x-\frac {x^2}{5}+\frac {123 x^3}{125}-\frac {x^4}{3125}-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 29, normalized size = 1.00 \begin {gather*} 2 x-\frac {x^2}{5}+\frac {123 x^3}{125}-\frac {x^4}{3125}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3125 + 6250*x - 1250*x^2 + 9225*x^3 - 4*x^4)/(3125*x),x]

[Out]

2*x - x^2/5 + (123*x^3)/125 - x^4/3125 - Log[x]

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fricas [A]  time = 0.68, size = 23, normalized size = 0.79 \begin {gather*} -\frac {1}{3125} \, x^{4} + \frac {123}{125} \, x^{3} - \frac {1}{5} \, x^{2} + 2 \, x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3125*(-4*x^4+9225*x^3-1250*x^2+6250*x-3125)/x,x, algorithm="fricas")

[Out]

-1/3125*x^4 + 123/125*x^3 - 1/5*x^2 + 2*x - log(x)

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giac [A]  time = 0.14, size = 24, normalized size = 0.83 \begin {gather*} -\frac {1}{3125} \, x^{4} + \frac {123}{125} \, x^{3} - \frac {1}{5} \, x^{2} + 2 \, x - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3125*(-4*x^4+9225*x^3-1250*x^2+6250*x-3125)/x,x, algorithm="giac")

[Out]

-1/3125*x^4 + 123/125*x^3 - 1/5*x^2 + 2*x - log(abs(x))

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maple [A]  time = 0.02, size = 24, normalized size = 0.83

method	result	size
default	\(-\frac {x^{4}}{3125}+\frac {123 x^{3}}{125}-\frac {x^{2}}{5}+2 x -\ln \relax (x )\)	\(24\)
norman	\(-\frac {x^{4}}{3125}+\frac {123 x^{3}}{125}-\frac {x^{2}}{5}+2 x -\ln \relax (x )\)	\(24\)
risch	\(-\frac {x^{4}}{3125}+\frac {123 x^{3}}{125}-\frac {x^{2}}{5}+2 x -\ln \relax (x )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3125*(-4*x^4+9225*x^3-1250*x^2+6250*x-3125)/x,x,method=_RETURNVERBOSE)

[Out]

-1/3125*x^4+123/125*x^3-1/5*x^2+2*x-ln(x)

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maxima [A]  time = 0.35, size = 23, normalized size = 0.79 \begin {gather*} -\frac {1}{3125} \, x^{4} + \frac {123}{125} \, x^{3} - \frac {1}{5} \, x^{2} + 2 \, x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3125*(-4*x^4+9225*x^3-1250*x^2+6250*x-3125)/x,x, algorithm="maxima")

[Out]

-1/3125*x^4 + 123/125*x^3 - 1/5*x^2 + 2*x - log(x)

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mupad [B]  time = 0.03, size = 23, normalized size = 0.79 \begin {gather*} 2\,x-\ln \relax (x)-\frac {x^2}{5}+\frac {123\,x^3}{125}-\frac {x^4}{3125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x^2)/5 - 2*x - (369*x^3)/125 + (4*x^4)/3125 + 1)/x,x)

[Out]

2*x - log(x) - x^2/5 + (123*x^3)/125 - x^4/3125

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sympy [A]  time = 0.07, size = 22, normalized size = 0.76 \begin {gather*} - \frac {x^{4}}{3125} + \frac {123 x^{3}}{125} - \frac {x^{2}}{5} + 2 x - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3125*(-4*x**4+9225*x**3-1250*x**2+6250*x-3125)/x,x)

[Out]

-x**4/3125 + 123*x**3/125 - x**2/5 + 2*x - log(x)

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