3.33.42 \(\int \frac {-1260 x+143 x^2-4 x^3}{648-72 x+2 x^2} \, dx\)

Optimal. Leaf size=25 \[ -x^2+\frac {x}{\log \left (e^{\frac {4 \left (9-\frac {x}{2}\right )}{x}}\right )} \]

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Rubi [A]  time = 0.03, antiderivative size = 20, normalized size of antiderivative = 0.80, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {27, 12, 1594, 771} \begin {gather*} -x^2-\frac {x}{2}+\frac {162}{18-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1260*x + 143*x^2 - 4*x^3)/(648 - 72*x + 2*x^2),x]

[Out]

162/(18 - x) - x/2 - x^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1260 x+143 x^2-4 x^3}{2 (-18+x)^2} \, dx\\ &=\frac {1}{2} \int \frac {-1260 x+143 x^2-4 x^3}{(-18+x)^2} \, dx\\ &=\frac {1}{2} \int \frac {x \left (-1260+143 x-4 x^2\right )}{(-18+x)^2} \, dx\\ &=\frac {1}{2} \int \left (-1+\frac {324}{(-18+x)^2}-4 x\right ) \, dx\\ &=\frac {162}{18-x}-\frac {x}{2}-x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{2} \left (666-\frac {324}{-18+x}-x-2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1260*x + 143*x^2 - 4*x^3)/(648 - 72*x + 2*x^2),x]

[Out]

(666 - 324/(-18 + x) - x - 2*x^2)/2

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fricas [A]  time = 0.62, size = 22, normalized size = 0.88 \begin {gather*} -\frac {2 \, x^{3} - 35 \, x^{2} - 18 \, x + 324}{2 \, {\left (x - 18\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^3+143*x^2-1260*x)/(2*x^2-72*x+648),x, algorithm="fricas")

[Out]

-1/2*(2*x^3 - 35*x^2 - 18*x + 324)/(x - 18)

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giac [A]  time = 0.28, size = 16, normalized size = 0.64 \begin {gather*} -x^{2} - \frac {1}{2} \, x - \frac {162}{x - 18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^3+143*x^2-1260*x)/(2*x^2-72*x+648),x, algorithm="giac")

[Out]

-x^2 - 1/2*x - 162/(x - 18)

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maple [A]  time = 0.61, size = 16, normalized size = 0.64

method	result	size
gosper	\(-\frac {x^{2} \left (2 x -35\right )}{2 \left (-18+x \right )}\)	\(16\)
default	\(-x^{2}-\frac {x}{2}-\frac {162}{-18+x}\)	\(17\)
risch	\(-x^{2}-\frac {x}{2}-\frac {162}{-18+x}\)	\(17\)
norman	\(\frac {\frac {35}{2} x^{2}-x^{3}}{-18+x}\)	\(18\)
meijerg	\(-\frac {9 x \left (-\frac {1}{162} x^{2}-\frac {1}{3} x +12\right )}{1-\frac {x}{18}}+\frac {143 x \left (-\frac {x}{6}+6\right )}{6 \left (1-\frac {x}{18}\right )}-\frac {35 x}{1-\frac {x}{18}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^3+143*x^2-1260*x)/(2*x^2-72*x+648),x,method=_RETURNVERBOSE)

[Out]

-1/2*x^2*(2*x-35)/(-18+x)

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maxima [A]  time = 0.39, size = 16, normalized size = 0.64 \begin {gather*} -x^{2} - \frac {1}{2} \, x - \frac {162}{x - 18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^3+143*x^2-1260*x)/(2*x^2-72*x+648),x, algorithm="maxima")

[Out]

-x^2 - 1/2*x - 162/(x - 18)

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mupad [B]  time = 1.85, size = 16, normalized size = 0.64 \begin {gather*} -\frac {x}{2}-\frac {162}{x-18}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(1260*x - 143*x^2 + 4*x^3)/(2*x^2 - 72*x + 648),x)

[Out]

- x/2 - 162/(x - 18) - x^2

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sympy [A]  time = 0.08, size = 12, normalized size = 0.48 \begin {gather*} - x^{2} - \frac {x}{2} - \frac {162}{x - 18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**3+143*x**2-1260*x)/(2*x**2-72*x+648),x)

[Out]

-x**2 - x/2 - 162/(x - 18)

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