3.71.61 \(\int \frac {384 x+128 e^{32} x+256 x^2}{9+e^{64}+24 x+16 x^2+e^{32} (6+8 x)} \, dx\)

Optimal. Leaf size=18 \[ \frac {16 x^2}{\frac {1}{4} \left (3+e^{32}\right )+x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 22, normalized size of antiderivative = 1.22, number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6, 1593, 1983, 27, 74} \begin {gather*} \frac {16 \left (2 x+e^{32}+3\right )^2}{4 x+e^{32}+3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(384*x + 128*E^32*x + 256*x^2)/(9 + E^64 + 24*x + 16*x^2 + E^32*(6 + 8*x)),x]

[Out]

(16*(3 + E^32 + 2*x)^2)/(3 + E^32 + 4*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 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 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1983

Int[(u_)^(m_.)*(v_)^(n_.)*(w_)^(p_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^n*ExpandToSum[w,
x]^p, x] /; FreeQ[{m, n, p}, x] && LinearQ[{u, v}, x] && QuadraticQ[w, x] &&  !(LinearMatchQ[{u, v}, x] && Qua
draticMatchQ[w, x])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (384+128 e^{32}\right ) x+256 x^2}{9+e^{64}+24 x+16 x^2+e^{32} (6+8 x)} \, dx\\ &=\int \frac {x \left (384+128 e^{32}+256 x\right )}{9+e^{64}+24 x+16 x^2+e^{32} (6+8 x)} \, dx\\ &=\int \frac {x \left (128 \left (3+e^{32}\right )+256 x\right )}{\left (3+e^{32}\right )^2+8 \left (3+e^{32}\right ) x+16 x^2} \, dx\\ &=\int \frac {x \left (128 \left (3+e^{32}\right )+256 x\right )}{\left (3+e^{32}+4 x\right )^2} \, dx\\ &=\frac {16 \left (3+e^{32}+2 x\right )^2}{3+e^{32}+4 x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 36, normalized size = 2.00 \begin {gather*} 128 \left (\frac {\left (3+e^{32}\right )^2}{32 \left (3+e^{32}+4 x\right )}+\frac {1}{32} \left (3+e^{32}+4 x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(384*x + 128*E^32*x + 256*x^2)/(9 + E^64 + 24*x + 16*x^2 + E^32*(6 + 8*x)),x]

[Out]

128*((3 + E^32)^2/(32*(3 + E^32 + 4*x)) + (3 + E^32 + 4*x)/32)

________________________________________________________________________________________

fricas [B]  time = 0.66, size = 32, normalized size = 1.78 \begin {gather*} \frac {4 \, {\left (16 \, x^{2} + 2 \, {\left (2 \, x + 3\right )} e^{32} + 12 \, x + e^{64} + 9\right )}}{4 \, x + e^{32} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((128*x*exp(16)^2+256*x^2+384*x)/(exp(16)^4+(8*x+6)*exp(16)^2+16*x^2+24*x+9),x, algorithm="fricas")

[Out]

4*(16*x^2 + 2*(2*x + 3)*e^32 + 12*x + e^64 + 9)/(4*x + e^32 + 3)

________________________________________________________________________________________

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((128*x*exp(16)^2+256*x^2+384*x)/(exp(16)^4+(8*x+6)*exp(16)^2+16*x^2+24*x+9),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 128*(2*sageVARx/16+(-6*exp(32)-exp(64)-9
)*1/32/sqrt(-exp(32)^2+exp(64))*atan((4*sageVARx+exp(32)+3)/sqrt(-exp(32)^2+exp(64))))

________________________________________________________________________________________

maple [A]  time = 0.12, size = 17, normalized size = 0.94

method	result	size
gosper	\(\frac {64 x^{2}}{{\mathrm e}^{32}+4 x +3}\)	\(17\)
norman	\(\frac {64 x^{2}}{{\mathrm e}^{32}+4 x +3}\)	\(17\)
risch	\(16 x +\frac {4 \,{\mathrm e}^{64}}{{\mathrm e}^{32}+4 x +3}+\frac {24 \,{\mathrm e}^{32}}{{\mathrm e}^{32}+4 x +3}+\frac {36}{{\mathrm e}^{32}+4 x +3}\)	\(42\)
meijerg	\(\frac {\left (128 \,{\mathrm e}^{32}+384\right ) \left (-\frac {4 x}{\left ({\mathrm e}^{32}+3\right ) \left (1+\frac {4 x}{{\mathrm e}^{32}+3}\right )}+\ln \left (1+\frac {4 x}{{\mathrm e}^{32}+3}\right )\right )}{16}+\frac {\left ({\mathrm e}^{32}+3\right )^{2} \left (\frac {4 x \left (\frac {12 x}{{\mathrm e}^{32}+3}+6\right )}{3 \left ({\mathrm e}^{32}+3\right ) \left (1+\frac {4 x}{{\mathrm e}^{32}+3}\right )}-2 \ln \left (1+\frac {4 x}{{\mathrm e}^{32}+3}\right )\right )}{\frac {{\mathrm e}^{32}}{4}+\frac {3}{4}}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((128*x*exp(16)^2+256*x^2+384*x)/(exp(16)^4+(8*x+6)*exp(16)^2+16*x^2+24*x+9),x,method=_RETURNVERBOSE)

[Out]

64*x^2/(exp(16)^2+4*x+3)

________________________________________________________________________________________

maxima [A]  time = 0.38, size = 23, normalized size = 1.28 \begin {gather*} 16 \, x + \frac {4 \, {\left (e^{64} + 6 \, e^{32} + 9\right )}}{4 \, x + e^{32} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((128*x*exp(16)^2+256*x^2+384*x)/(exp(16)^4+(8*x+6)*exp(16)^2+16*x^2+24*x+9),x, algorithm="maxima")

[Out]

16*x + 4*(e^64 + 6*e^32 + 9)/(4*x + e^32 + 3)

________________________________________________________________________________________

mupad [B]  time = 4.18, size = 24, normalized size = 1.33 \begin {gather*} 16\,x+\frac {24\,{\mathrm {e}}^{32}+4\,{\mathrm {e}}^{64}+36}{4\,x+{\mathrm {e}}^{32}+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((384*x + 128*x*exp(32) + 256*x^2)/(24*x + exp(64) + 16*x^2 + exp(32)*(8*x + 6) + 9),x)

[Out]

16*x + (24*exp(32) + 4*exp(64) + 36)/(4*x + exp(32) + 3)

________________________________________________________________________________________

sympy [A]  time = 0.22, size = 22, normalized size = 1.22 \begin {gather*} 16 x + \frac {36 + 24 e^{32} + 4 e^{64}}{4 x + 3 + e^{32}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((128*x*exp(16)**2+256*x**2+384*x)/(exp(16)**4+(8*x+6)*exp(16)**2+16*x**2+24*x+9),x)

[Out]

16*x + (36 + 24*exp(32) + 4*exp(64))/(4*x + 3 + exp(32))

________________________________________________________________________________________