3.27.5 \(\int \frac {36-9 x+288 x^2-108 x^3}{4 x} \, dx\)

Optimal. Leaf size=25 \[ 9 \left (-x \left (\frac {1}{4}+(-4+x) x\right )+\log \left (e^{e^{e^5}} x\right )\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 20, normalized size of antiderivative = 0.80, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {12, 14} \begin {gather*} -9 x^3+36 x^2-\frac {9 x}{4}+9 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(36 - 9*x + 288*x^2 - 108*x^3)/(4*x),x]

[Out]

(-9*x)/4 + 36*x^2 - 9*x^3 + 9*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 {1}{4} \int \frac {36-9 x+288 x^2-108 x^3}{x} \, dx\\ &=\frac {1}{4} \int \left (-9+\frac {36}{x}+288 x-108 x^2\right ) \, dx\\ &=-\frac {9 x}{4}+36 x^2-9 x^3+9 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 0.80 \begin {gather*} -\frac {9 x}{4}+36 x^2-9 x^3+9 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36 - 9*x + 288*x^2 - 108*x^3)/(4*x),x]

[Out]

(-9*x)/4 + 36*x^2 - 9*x^3 + 9*Log[x]

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fricas [A]  time = 0.59, size = 18, normalized size = 0.72 \begin {gather*} -9 \, x^{3} + 36 \, x^{2} - \frac {9}{4} \, x + 9 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-108*x^3+288*x^2-9*x+36)/x,x, algorithm="fricas")

[Out]

-9*x^3 + 36*x^2 - 9/4*x + 9*log(x)

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giac [A]  time = 0.17, size = 19, normalized size = 0.76 \begin {gather*} -9 \, x^{3} + 36 \, x^{2} - \frac {9}{4} \, x + 9 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-108*x^3+288*x^2-9*x+36)/x,x, algorithm="giac")

[Out]

-9*x^3 + 36*x^2 - 9/4*x + 9*log(abs(x))

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maple [A]  time = 0.01, size = 19, normalized size = 0.76

method	result	size
default	\(-9 x^{3}+36 x^{2}-\frac {9 x}{4}+9 \ln \relax (x )\)	\(19\)
norman	\(-9 x^{3}+36 x^{2}-\frac {9 x}{4}+9 \ln \relax (x )\)	\(19\)
risch	\(-9 x^{3}+36 x^{2}-\frac {9 x}{4}+9 \ln \relax (x )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(-108*x^3+288*x^2-9*x+36)/x,x,method=_RETURNVERBOSE)

[Out]

-9*x^3+36*x^2-9/4*x+9*ln(x)

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maxima [A]  time = 0.47, size = 18, normalized size = 0.72 \begin {gather*} -9 \, x^{3} + 36 \, x^{2} - \frac {9}{4} \, x + 9 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-108*x^3+288*x^2-9*x+36)/x,x, algorithm="maxima")

[Out]

-9*x^3 + 36*x^2 - 9/4*x + 9*log(x)

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mupad [B]  time = 0.03, size = 18, normalized size = 0.72 \begin {gather*} 9\,\ln \relax (x)-\frac {9\,x}{4}+36\,x^2-9\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((9*x)/4 - 72*x^2 + 27*x^3 - 9)/x,x)

[Out]

9*log(x) - (9*x)/4 + 36*x^2 - 9*x^3

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sympy [A]  time = 0.07, size = 19, normalized size = 0.76 \begin {gather*} - 9 x^{3} + 36 x^{2} - \frac {9 x}{4} + 9 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-108*x**3+288*x**2-9*x+36)/x,x)

[Out]

-9*x**3 + 36*x**2 - 9*x/4 + 9*log(x)

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