3.95.97 \(\int \frac {144 e^{34}}{16 x^2+e^{16} (-24 e^2 x-24 x^2)+e^{32} (9 e^4+18 e^2 x+9 x^2)} \, dx\)

Optimal. Leaf size=18 \[ \frac {16 x}{e^2+x-\frac {4 x}{3 e^{16}}} \]

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Rubi [A]  time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.78, number of steps used = 5, number of rules used = 5, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 1981, 27, 6, 32} \begin {gather*} \frac {144 e^{34}}{\left (4-3 e^{16}\right ) \left (3 e^{18}-\left (4-3 e^{16}\right ) x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(144*E^34)/(16*x^2 + E^16*(-24*E^2*x - 24*x^2) + E^32*(9*E^4 + 18*E^2*x + 9*x^2)),x]

[Out]

(144*E^34)/((4 - 3*E^16)*(3*E^18 - (4 - 3*E^16)*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 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 (144 e^{34}\right ) \int \frac {1}{16 x^2+e^{16} \left (-24 e^2 x-24 x^2\right )+e^{32} \left (9 e^4+18 e^2 x+9 x^2\right )} \, dx\\ &=\left (144 e^{34}\right ) \int \frac {1}{9 e^{36}-6 e^{18} \left (4-3 e^{16}\right ) x+\left (4-3 e^{16}\right )^2 x^2} \, dx\\ &=\left (144 e^{34}\right ) \int \frac {1}{\left (3 e^{18}-4 x+3 e^{16} x\right )^2} \, dx\\ &=\left (144 e^{34}\right ) \int \frac {1}{\left (3 e^{18}+\left (-4+3 e^{16}\right ) x\right )^2} \, dx\\ &=\frac {144 e^{34}}{\left (4-3 e^{16}\right ) \left (3 e^{18}-\left (4-3 e^{16}\right ) x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 1.72 \begin {gather*} -\frac {144 e^{34}}{\left (-4+3 e^{16}\right ) \left (3 e^{18}-4 x+3 e^{16} x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(144*E^34)/(16*x^2 + E^16*(-24*E^2*x - 24*x^2) + E^32*(9*E^4 + 18*E^2*x + 9*x^2)),x]

[Out]

(-144*E^34)/((-4 + 3*E^16)*(3*E^18 - 4*x + 3*E^16*x))

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fricas [A]  time = 0.55, size = 28, normalized size = 1.56 \begin {gather*} -\frac {144 \, e^{34}}{9 \, x e^{32} - 24 \, x e^{16} + 16 \, x + 9 \, e^{34} - 12 \, e^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(144*exp(2)*exp(16)^2/((9*exp(2)^2+18*exp(2)*x+9*x^2)*exp(16)^2+(-24*exp(2)*x-24*x^2)*exp(16)+16*x^2)
,x, algorithm="fricas")

[Out]

-144*e^34/(9*x*e^32 - 24*x*e^16 + 16*x + 9*e^34 - 12*e^18)

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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(144*exp(2)*exp(16)^2/((9*exp(2)^2+18*exp(2)*x+9*x^2)*exp(16)^2+(-24*exp(2)*x-24*x^2)*exp(16)+16*x^2)
,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 144*exp(34)*2*1/6/sqrt(-16*exp(16)^2*exp
(2)^2+24*exp(16)*exp(2)^2*exp(32)-24*exp(16)*exp(32)*exp(4)-9*exp(2)^2*exp(32)^2+9*exp(32)^2*exp(4)+16*exp(32)
*exp(4))*atan((-24*sa

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maple [A]  time = 1.83, size = 23, normalized size = 1.28

method	result	size
norman	\(\frac {48 \,{\mathrm e}^{16} x}{3 \,{\mathrm e}^{16} {\mathrm e}^{2}+3 x \,{\mathrm e}^{16}-4 x}\)	\(23\)
risch	\(-\frac {48 \,{\mathrm e}^{34}}{\left (3 \,{\mathrm e}^{16}-4\right ) \left ({\mathrm e}^{18}+x \,{\mathrm e}^{16}-\frac {4 x}{3}\right )}\)	\(25\)
gosper	\(-\frac {144 \,{\mathrm e}^{32} {\mathrm e}^{2}}{\left (3 \,{\mathrm e}^{16} {\mathrm e}^{2}+3 x \,{\mathrm e}^{16}-4 x \right ) \left (3 \,{\mathrm e}^{16}-4\right )}\)	\(34\)
meijerg	\(\frac {16 \left (3 \,{\mathrm e}^{16}-4\right )^{2} {\mathrm e}^{-2} x}{\left (9 \,{\mathrm e}^{32}-24 \,{\mathrm e}^{16}+16\right ) \left (1+\frac {x \,{\mathrm e}^{-18} \left (3 \,{\mathrm e}^{16}-4\right )}{3}\right )}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(144*exp(2)*exp(16)^2/((9*exp(2)^2+18*exp(2)*x+9*x^2)*exp(16)^2+(-24*exp(2)*x-24*x^2)*exp(16)+16*x^2),x,met
hod=_RETURNVERBOSE)

[Out]

48*exp(16)*x/(3*exp(16)*exp(2)+3*x*exp(16)-4*x)

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maxima [A]  time = 0.35, size = 27, normalized size = 1.50 \begin {gather*} -\frac {144 \, e^{34}}{x {\left (9 \, e^{32} - 24 \, e^{16} + 16\right )} + 9 \, e^{34} - 12 \, e^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(144*exp(2)*exp(16)^2/((9*exp(2)^2+18*exp(2)*x+9*x^2)*exp(16)^2+(-24*exp(2)*x-24*x^2)*exp(16)+16*x^2)
,x, algorithm="maxima")

[Out]

-144*e^34/(x*(9*e^32 - 24*e^16 + 16) + 9*e^34 - 12*e^18)

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mupad [B]  time = 6.53, size = 27, normalized size = 1.50 \begin {gather*} -\frac {144\,{\mathrm {e}}^{34}}{\left (3\,{\mathrm {e}}^{18}+x\,\left (3\,{\mathrm {e}}^{16}-4\right )\right )\,\left (3\,{\mathrm {e}}^{16}-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((144*exp(34))/(exp(32)*(9*exp(4) + 18*x*exp(2) + 9*x^2) - exp(16)*(24*x*exp(2) + 24*x^2) + 16*x^2),x)

[Out]

-(144*exp(34))/((3*exp(18) + x*(3*exp(16) - 4))*(3*exp(16) - 4))

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sympy [A]  time = 0.23, size = 29, normalized size = 1.61 \begin {gather*} - \frac {144 e^{34}}{x \left (- 24 e^{16} + 16 + 9 e^{32}\right ) - 12 e^{18} + 9 e^{34}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(144*exp(2)*exp(16)**2/((9*exp(2)**2+18*exp(2)*x+9*x**2)*exp(16)**2+(-24*exp(2)*x-24*x**2)*exp(16)+16
*x**2),x)

[Out]

-144*exp(34)/(x*(-24*exp(16) + 16 + 9*exp(32)) - 12*exp(18) + 9*exp(34))

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