3.22.15 \(\int \frac {-9-16 e^4+46 x-33 x^2-72 x^3}{1-6 x+9 x^2} \, dx\)

Optimal. Leaf size=27 \[ e^3+x \left (3+4 \left (-3-x+\frac {4 e^4}{-1+3 x}\right )\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 23, normalized size of antiderivative = 0.85, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {27, 1850} \begin {gather*} -4 x^2-9 x-\frac {16 e^4}{3 (1-3 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9 - 16*E^4 + 46*x - 33*x^2 - 72*x^3)/(1 - 6*x + 9*x^2),x]

[Out]

(-16*E^4)/(3*(1 - 3*x)) - 9*x - 4*x^2

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 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-9-16 e^4+46 x-33 x^2-72 x^3}{(-1+3 x)^2} \, dx\\ &=\int \left (-9-8 x-\frac {16 e^4}{(-1+3 x)^2}\right ) \, dx\\ &=-\frac {16 e^4}{3 (1-3 x)}-9 x-4 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.85 \begin {gather*} -\frac {16 e^4}{3 (1-3 x)}-9 x-4 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 - 16*E^4 + 46*x - 33*x^2 - 72*x^3)/(1 - 6*x + 9*x^2),x]

[Out]

(-16*E^4)/(3*(1 - 3*x)) - 9*x - 4*x^2

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fricas [A]  time = 0.50, size = 27, normalized size = 1.00 \begin {gather*} -\frac {36 \, x^{3} + 69 \, x^{2} - 27 \, x - 16 \, e^{4}}{3 \, {\left (3 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*exp(4)-72*x^3-33*x^2+46*x-9)/(9*x^2-6*x+1),x, algorithm="fricas")

[Out]

-1/3*(36*x^3 + 69*x^2 - 27*x - 16*e^4)/(3*x - 1)

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giac [A]  time = 0.21, size = 20, normalized size = 0.74 \begin {gather*} -4 \, x^{2} - 9 \, x + \frac {16 \, e^{4}}{3 \, {\left (3 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*exp(4)-72*x^3-33*x^2+46*x-9)/(9*x^2-6*x+1),x, algorithm="giac")

[Out]

-4*x^2 - 9*x + 16/3*e^4/(3*x - 1)

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maple [A]  time = 0.41, size = 19, normalized size = 0.70

method	result	size
risch	\(-4 x^{2}-9 x +\frac {16 \,{\mathrm e}^{4}}{9 \left (x -\frac {1}{3}\right )}\)	\(19\)
default	\(-4 x^{2}-9 x +\frac {16 \,{\mathrm e}^{4}}{3 \left (3 x -1\right )}\)	\(21\)
norman	\(\frac {-23 x^{2}-12 x^{3}+3+\frac {16 \,{\mathrm e}^{4}}{3}}{3 x -1}\)	\(25\)
gosper	\(\frac {-36 x^{3}-69 x^{2}+16 \,{\mathrm e}^{4}+9}{9 x -3}\)	\(26\)
meijerg	\(-\frac {16 \,{\mathrm e}^{4} x}{-3 x +1}-\frac {2 x \left (-18 x^{2}-18 x +12\right )}{3 \left (-3 x +1\right )}-\frac {11 x \left (-9 x +6\right )}{9 \left (-3 x +1\right )}+\frac {19 x}{3 \left (-3 x +1\right )}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-16*exp(4)-72*x^3-33*x^2+46*x-9)/(9*x^2-6*x+1),x,method=_RETURNVERBOSE)

[Out]

-4*x^2-9*x+16/9*exp(4)/(x-1/3)

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maxima [A]  time = 0.38, size = 20, normalized size = 0.74 \begin {gather*} -4 \, x^{2} - 9 \, x + \frac {16 \, e^{4}}{3 \, {\left (3 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*exp(4)-72*x^3-33*x^2+46*x-9)/(9*x^2-6*x+1),x, algorithm="maxima")

[Out]

-4*x^2 - 9*x + 16/3*e^4/(3*x - 1)

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mupad [B]  time = 0.06, size = 20, normalized size = 0.74 \begin {gather*} \frac {16\,{\mathrm {e}}^4}{3\,\left (3\,x-1\right )}-9\,x-4\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*exp(4) - 46*x + 33*x^2 + 72*x^3 + 9)/(9*x^2 - 6*x + 1),x)

[Out]

(16*exp(4))/(3*(3*x - 1)) - 9*x - 4*x^2

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sympy [A]  time = 0.10, size = 17, normalized size = 0.63 \begin {gather*} - 4 x^{2} - 9 x + \frac {16 e^{4}}{9 x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*exp(4)-72*x**3-33*x**2+46*x-9)/(9*x**2-6*x+1),x)

[Out]

-4*x**2 - 9*x + 16*exp(4)/(9*x - 3)

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