3.50.72 \(\int -\frac {440}{576+4 e^{16}-2640 x+3025 x^2+e^8 (-96+220 x)} \, dx\)

Optimal. Leaf size=20 \[ \frac {4}{e^8-3 \left (4+\frac {x}{2}\right )+29 x} \]

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 1981, 27, 32} \begin {gather*} -\frac {8}{2 \left (12-e^8\right )-55 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-440/(576 + 4*E^16 - 2640*x + 3025*x^2 + E^8*(-96 + 220*x)),x]

[Out]

-8/(2*(12 - E^8) - 55*x)

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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 1981

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && QuadraticQ[u, x] &&  !QuadraticMatch
Q[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (440 \int \frac {1}{576+4 e^{16}-2640 x+3025 x^2+e^8 (-96+220 x)} \, dx\right )\\ &=-\left (440 \int \frac {1}{4 \left (12-e^8\right )^2-220 \left (12-e^8\right ) x+3025 x^2} \, dx\right )\\ &=-\left (440 \int \frac {1}{\left (-24+2 e^8+55 x\right )^2} \, dx\right )\\ &=-\frac {8}{2 \left (12-e^8\right )-55 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 0.70 \begin {gather*} \frac {8}{-24+2 e^8+55 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-440/(576 + 4*E^16 - 2640*x + 3025*x^2 + E^8*(-96 + 220*x)),x]

[Out]

8/(-24 + 2*E^8 + 55*x)

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fricas [A]  time = 0.67, size = 13, normalized size = 0.65 \begin {gather*} \frac {8}{55 \, x + 2 \, e^{8} - 24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-440/(4*exp(8)^2+(220*x-96)*exp(8)+3025*x^2-2640*x+576),x, algorithm="fricas")

[Out]

8/(55*x + 2*e^8 - 24)

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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(-440/(4*exp(8)^2+(220*x-96)*exp(8)+3025*x^2-2640*x+576),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -880*1/220/sqrt(-exp(8)^2+exp(16))*atan(
(55*sageVARx+2*exp(8)-24)*1/2/sqrt(-exp(8)^2+exp(16)))

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maple [A]  time = 0.18, size = 12, normalized size = 0.60

method	result	size
risch	\(\frac {4}{\frac {55 x}{2}+{\mathrm e}^{8}-12}\)	\(12\)
gosper	\(\frac {8}{2 \,{\mathrm e}^{8}+55 x -24}\)	\(14\)
norman	\(\frac {8}{2 \,{\mathrm e}^{8}+55 x -24}\)	\(14\)
meijerg	\(-\frac {4 x}{\left (\frac {2 \,{\mathrm e}^{8}}{55}-\frac {24}{55}\right ) \left ({\mathrm e}^{8}-12\right ) \left (1+\frac {55 x}{2 \left ({\mathrm e}^{8}-12\right )}\right )}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-440/(4*exp(8)^2+(220*x-96)*exp(8)+3025*x^2-2640*x+576),x,method=_RETURNVERBOSE)

[Out]

4/(55/2*x+exp(8)-12)

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maxima [A]  time = 0.35, size = 13, normalized size = 0.65 \begin {gather*} \frac {8}{55 \, x + 2 \, e^{8} - 24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-440/(4*exp(8)^2+(220*x-96)*exp(8)+3025*x^2-2640*x+576),x, algorithm="maxima")

[Out]

8/(55*x + 2*e^8 - 24)

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mupad [B]  time = 3.24, size = 13, normalized size = 0.65 \begin {gather*} \frac {8}{55\,x+2\,{\mathrm {e}}^8-24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-440/(4*exp(16) - 2640*x + 3025*x^2 + exp(8)*(220*x - 96) + 576),x)

[Out]

8/(55*x + 2*exp(8) - 24)

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sympy [A]  time = 0.16, size = 10, normalized size = 0.50 \begin {gather*} \frac {440}{3025 x - 1320 + 110 e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-440/(4*exp(8)**2+(220*x-96)*exp(8)+3025*x**2-2640*x+576),x)

[Out]

440/(3025*x - 1320 + 110*exp(8))

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