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

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

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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} $$\begin{array}{c}-x^2-\frac{x}{2}+\frac{162}{18-x}\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-1260x+143x^2-4x^3}{2(-18+x{)}^2}dx\\ & =\frac{1}{2}\int \frac{-1260x+143x^2-4x^3}{(-18+x{)}^2}dx\\ & =\frac{1}{2}\int \frac{x(-1260+143x-4x^2)}{(-18+x{)}^2}dx\\ & =\frac{1}{2}\int (-1+\frac{324}{(-18+x{)}^2}-4x)dx\\ & =\frac{162}{18-x}-\frac{x}{2}-x^2\end{array}\end{array}$$

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

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 $$\begin{array}{c}-\frac{2x^3-35x^2-18x+324}{2(x-18)}\end{array}$$

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 $$\begin{array}{c}-x^2-\frac{1}{2}x-\frac{162}{x-18}\end{array}$$

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	$-\frac{x^2(2x-35)}{2(-18+x)}$	$16$
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	$-\frac{9x(-\frac{1}{162}{x}^2-\frac{1}{3}x+12)}{1-\frac{x}{18}}+\frac{143x(-\frac{x}{6}+6)}{6(1-\frac{x}{18})}-\frac{35x}{1-\frac{x}{18}}$	$47$

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 $$\begin{array}{c}-x^2-\frac{1}{2}x-\frac{162}{x-18}\end{array}$$

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 $$\begin{array}{c}-\frac{x}{2}-\frac{162}{x-18}-x^2\end{array}$$

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 $$\begin{array}{c}-x^2-\frac{x}{2}-\frac{162}{x-18}\end{array}$$

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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