3.57.67 \(\int \frac {12150-450 e^3+2025 x-27 x^3}{729 x^3+e^6 x^3+162 x^4+9 x^5+e^3 (-54 x^3-6 x^4)} \, dx\)

Optimal. Leaf size=27 \[ \frac {3 (5-x) (5+x)}{\left (-9+\frac {e^3}{3}-x\right ) x^2} \]

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Rubi [B]  time = 0.10, antiderivative size = 62, normalized size of antiderivative = 2.30, number of steps used = 3, number of rules used = 2, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {6, 2074} \begin {gather*} -\frac {225}{\left (27-e^3\right ) x^2}+\frac {9 \left (504-54 e^3+e^6\right )}{\left (27-e^3\right )^2 \left (3 x-e^3+27\right )}+\frac {675}{\left (27-e^3\right )^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(12150 - 450*E^3 + 2025*x - 27*x^3)/(729*x^3 + E^6*x^3 + 162*x^4 + 9*x^5 + E^3*(-54*x^3 - 6*x^4)),x]

[Out]

-225/((27 - E^3)*x^2) + 675/((27 - E^3)^2*x) + (9*(504 - 54*E^3 + E^6))/((27 - E^3)^2*(27 - E^3 + 3*x))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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 \frac {12150-450 e^3+2025 x-27 x^3}{\left (729+e^6\right ) x^3+162 x^4+9 x^5+e^3 \left (-54 x^3-6 x^4\right )} \, dx\\ &=\int \left (-\frac {27 \left (504-54 e^3+e^6\right )}{(-3+e)^2 \left (9+3 e+e^2\right )^2 \left (-27+e^3-3 x\right )^2}-\frac {450}{\left (-27+e^3\right ) x^3}-\frac {675}{\left (-27+e^3\right )^2 x^2}\right ) \, dx\\ &=-\frac {225}{\left (27-e^3\right ) x^2}+\frac {675}{\left (27-e^3\right )^2 x}+\frac {9 \left (504-54 e^3+e^6\right )}{\left (27-e^3\right )^2 \left (27-e^3+3 x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 21, normalized size = 0.78 \begin {gather*} -\frac {9 \left (-25+x^2\right )}{x^2 \left (e^3-3 (9+x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12150 - 450*E^3 + 2025*x - 27*x^3)/(729*x^3 + E^6*x^3 + 162*x^4 + 9*x^5 + E^3*(-54*x^3 - 6*x^4)),x]

[Out]

(-9*(-25 + x^2))/(x^2*(E^3 - 3*(9 + x)))

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fricas [A]  time = 0.55, size = 27, normalized size = 1.00 \begin {gather*} \frac {9 \, {\left (x^{2} - 25\right )}}{3 \, x^{3} - x^{2} e^{3} + 27 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-450*exp(3)-27*x^3+2025*x+12150)/(x^3*exp(3)^2+(-6*x^4-54*x^3)*exp(3)+9*x^5+162*x^4+729*x^3),x, alg
orithm="fricas")

[Out]

9*(x^2 - 25)/(3*x^3 - x^2*e^3 + 27*x^2)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-450*exp(3)-27*x^3+2025*x+12150)/(x^3*exp(3)^2+(-6*x^4-54*x^3)*exp(3)+9*x^5+162*x^4+729*x^3),x, alg
orithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -9*((900*exp(6)*exp(3)-24300*exp(6)-900*
exp(3)^3+24300*exp(3)^2)/(exp(6)^3-162*exp(6)^2*exp(3)+2187*exp(6)^2+8748*exp(6)*exp(3)^2-236196*exp(6)*exp(3)
+1594323*exp(6)-15746

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maple [A]  time = 7.33, size = 20, normalized size = 0.74

method	result	size
gosper	\(-\frac {9 \left (x^{2}-25\right )}{x^{2} \left (-27+{\mathrm e}^{3}-3 x \right )}\)	\(20\)
norman	\(\frac {-9 x^{2}+225}{x^{2} \left (-27+{\mathrm e}^{3}-3 x \right )}\)	\(21\)
risch	\(\frac {-9 x^{2}+225}{x^{2} \left (-27+{\mathrm e}^{3}-3 x \right )}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-450*exp(3)-27*x^3+2025*x+12150)/(x^3*exp(3)^2+(-6*x^4-54*x^3)*exp(3)+9*x^5+162*x^4+729*x^3),x,method=_RE
TURNVERBOSE)

[Out]

-9/x^2*(x^2-25)/(-27+exp(3)-3*x)

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maxima [A]  time = 0.36, size = 24, normalized size = 0.89 \begin {gather*} \frac {9 \, {\left (x^{2} - 25\right )}}{3 \, x^{3} - x^{2} {\left (e^{3} - 27\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-450*exp(3)-27*x^3+2025*x+12150)/(x^3*exp(3)^2+(-6*x^4-54*x^3)*exp(3)+9*x^5+162*x^4+729*x^3),x, alg
orithm="maxima")

[Out]

9*(x^2 - 25)/(3*x^3 - x^2*(e^3 - 27))

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mupad [B]  time = 3.59, size = 25, normalized size = 0.93 \begin {gather*} \frac {9\,x^2-225}{3\,x^3-x^2\,\left ({\mathrm {e}}^3-27\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2025*x - 450*exp(3) - 27*x^3 + 12150)/(x^3*exp(6) - exp(3)*(54*x^3 + 6*x^4) + 729*x^3 + 162*x^4 + 9*x^5),
x)

[Out]

(9*x^2 - 225)/(3*x^3 - x^2*(exp(3) - 27))

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sympy [A]  time = 0.62, size = 20, normalized size = 0.74 \begin {gather*} - \frac {225 - 9 x^{2}}{3 x^{3} + x^{2} \left (27 - e^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-450*exp(3)-27*x**3+2025*x+12150)/(x**3*exp(3)**2+(-6*x**4-54*x**3)*exp(3)+9*x**5+162*x**4+729*x**3
),x)

[Out]

-(225 - 9*x**2)/(3*x**3 + x**2*(27 - exp(3)))

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