3.85.89 \(\int \frac {-2112 x+3504 x^2-1756 x^3+3 x^4}{-64+288 x-432 x^2+216 x^3} \, dx\)

Optimal. Leaf size=26 \[ -3+e^2-8 x+\frac {(4-x)^4}{16 (-2+3 x)^2} \]

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Rubi [A]  time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.35, number of steps used = 2, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {2074} \begin {gather*} \frac {x^2}{144}-\frac {875 x}{108}+\frac {250}{81 (2-3 x)}+\frac {625}{81 (2-3 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2112*x + 3504*x^2 - 1756*x^3 + 3*x^4)/(-64 + 288*x - 432*x^2 + 216*x^3),x]

[Out]

625/(81*(2 - 3*x)^2) + 250/(81*(2 - 3*x)) - (875*x)/108 + x^2/144

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {875}{108}+\frac {x}{72}-\frac {1250}{27 (-2+3 x)^3}+\frac {250}{27 (-2+3 x)^2}\right ) \, dx\\ &=\frac {625}{81 (2-3 x)^2}+\frac {250}{81 (2-3 x)}-\frac {875 x}{108}+\frac {x^2}{144}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 1.19 \begin {gather*} \frac {15328-45984 x+63000 x^2-31536 x^3+27 x^4}{432 (2-3 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2112*x + 3504*x^2 - 1756*x^3 + 3*x^4)/(-64 + 288*x - 432*x^2 + 216*x^3),x]

[Out]

(15328 - 45984*x + 63000*x^2 - 31536*x^3 + 27*x^4)/(432*(2 - 3*x)^2)

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fricas [A]  time = 0.54, size = 34, normalized size = 1.31 \begin {gather*} \frac {9 \, x^{4} - 10512 \, x^{3} + 14004 \, x^{2} - 6000 \, x + 2000}{144 \, {\left (9 \, x^{2} - 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4-1756*x^3+3504*x^2-2112*x)/(216*x^3-432*x^2+288*x-64),x, algorithm="fricas")

[Out]

1/144*(9*x^4 - 10512*x^3 + 14004*x^2 - 6000*x + 2000)/(9*x^2 - 12*x + 4)

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giac [A]  time = 0.12, size = 23, normalized size = 0.88 \begin {gather*} \frac {1}{144} \, x^{2} - \frac {875}{108} \, x - \frac {125 \, {\left (2 \, x - 3\right )}}{27 \, {\left (3 \, x - 2\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4-1756*x^3+3504*x^2-2112*x)/(216*x^3-432*x^2+288*x-64),x, algorithm="giac")

[Out]

1/144*x^2 - 875/108*x - 125/27*(2*x - 3)/(3*x - 2)^2

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maple [A]  time = 0.03, size = 25, normalized size = 0.96

method	result	size
norman	\(\frac {66 x^{2}-73 x^{3}+\frac {1}{16} x^{4}}{\left (3 x -2\right )^{2}}\)	\(25\)
gosper	\(\frac {x^{2} \left (x^{2}-1168 x +1056\right )}{144 x^{2}-192 x +64}\)	\(26\)
risch	\(\frac {x^{2}}{144}-\frac {875 x}{108}+\frac {-\frac {250 x}{243}+\frac {125}{81}}{x^{2}-\frac {4}{3} x +\frac {4}{9}}\)	\(26\)
default	\(\frac {x^{2}}{144}-\frac {875 x}{108}-\frac {250}{81 \left (3 x -2\right )}+\frac {625}{81 \left (3 x -2\right )^{2}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^4-1756*x^3+3504*x^2-2112*x)/(216*x^3-432*x^2+288*x-64),x,method=_RETURNVERBOSE)

[Out]

(66*x^2-73*x^3+1/16*x^4)/(3*x-2)^2

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maxima [A]  time = 0.36, size = 28, normalized size = 1.08 \begin {gather*} \frac {1}{144} \, x^{2} - \frac {875}{108} \, x - \frac {125 \, {\left (2 \, x - 3\right )}}{27 \, {\left (9 \, x^{2} - 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4-1756*x^3+3504*x^2-2112*x)/(216*x^3-432*x^2+288*x-64),x, algorithm="maxima")

[Out]

1/144*x^2 - 875/108*x - 125/27*(2*x - 3)/(9*x^2 - 12*x + 4)

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mupad [B]  time = 5.38, size = 23, normalized size = 0.88 \begin {gather*} \frac {x^2}{144}-\frac {\frac {250\,x}{27}-\frac {125}{9}}{{\left (3\,x-2\right )}^2}-\frac {875\,x}{108} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2112*x - 3504*x^2 + 1756*x^3 - 3*x^4)/(288*x - 432*x^2 + 216*x^3 - 64),x)

[Out]

x^2/144 - ((250*x)/27 - 125/9)/(3*x - 2)^2 - (875*x)/108

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sympy [A]  time = 0.09, size = 24, normalized size = 0.92 \begin {gather*} \frac {x^{2}}{144} - \frac {875 x}{108} + \frac {375 - 250 x}{243 x^{2} - 324 x + 108} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**4-1756*x**3+3504*x**2-2112*x)/(216*x**3-432*x**2+288*x-64),x)

[Out]

x**2/144 - 875*x/108 + (375 - 250*x)/(243*x**2 - 324*x + 108)

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