3.4.61 \(\int \frac {64 x-144 x^2-1188 x^3+459 x^4+e^{4 x} (-128 x+32 x^2+360 x^3-378 x^4+108 x^5)+e^{2 x} (864 x^2-504 x^3-540 x^4+324 x^5)}{-64+144 x-108 x^2+27 x^3} \, dx\)

Optimal. Leaf size=30 \[ -\frac {x^2}{2}+x^2 \left (e^{2 x}+\frac {9 x}{-4+3 x}\right )^2 \]

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Rubi [B]  time = 0.95, antiderivative size = 97, normalized size of antiderivative = 3.23, number of steps used = 28, number of rules used = 10, integrand size = 94, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {6688, 6742, 2196, 2176, 2194, 37, 43, 2199, 2177, 2178} \begin {gather*} 6 e^{2 x} x^2+e^{4 x} x^2-\frac {8 x^2}{(4-3 x)^2}+\frac {17 x^2}{2}+8 e^{2 x} x+24 x+\frac {32 e^{2 x}}{3}-\frac {128 e^{2 x}}{3 (4-3 x)}-\frac {2368}{9 (4-3 x)}+\frac {2432}{9 (4-3 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(64*x - 144*x^2 - 1188*x^3 + 459*x^4 + E^(4*x)*(-128*x + 32*x^2 + 360*x^3 - 378*x^4 + 108*x^5) + E^(2*x)*(
864*x^2 - 504*x^3 - 540*x^4 + 324*x^5))/(-64 + 144*x - 108*x^2 + 27*x^3),x]

[Out]

(32*E^(2*x))/3 + 2432/(9*(4 - 3*x)^2) - 2368/(9*(4 - 3*x)) - (128*E^(2*x))/(3*(4 - 3*x)) + 24*x + 8*E^(2*x)*x
+ (17*x^2)/2 + 6*E^(2*x)*x^2 + E^(4*x)*x^2 - (8*x^2)/(4 - 3*x)^2

Rule 37

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (-64+144 x+1188 x^2-459 x^3-2 e^{4 x} (1+2 x) (-4+3 x)^3-36 e^{2 x} x \left (24-14 x-15 x^2+9 x^3\right )\right )}{(4-3 x)^3} \, dx\\ &=\int \left (2 e^{4 x} x (1+2 x)+\frac {64 x}{(-4+3 x)^3}-\frac {144 x^2}{(-4+3 x)^3}-\frac {1188 x^3}{(-4+3 x)^3}+\frac {459 x^4}{(-4+3 x)^3}+\frac {36 e^{2 x} x^2 \left (-6-x+3 x^2\right )}{(-4+3 x)^2}\right ) \, dx\\ &=2 \int e^{4 x} x (1+2 x) \, dx+36 \int \frac {e^{2 x} x^2 \left (-6-x+3 x^2\right )}{(-4+3 x)^2} \, dx+64 \int \frac {x}{(-4+3 x)^3} \, dx-144 \int \frac {x^2}{(-4+3 x)^3} \, dx+459 \int \frac {x^4}{(-4+3 x)^3} \, dx-1188 \int \frac {x^3}{(-4+3 x)^3} \, dx\\ &=-\frac {8 x^2}{(4-3 x)^2}+2 \int \left (e^{4 x} x+2 e^{4 x} x^2\right ) \, dx+36 \int \left (\frac {22 e^{2 x}}{27}+\frac {7}{9} e^{2 x} x+\frac {1}{3} e^{2 x} x^2-\frac {32 e^{2 x}}{9 (-4+3 x)^2}+\frac {64 e^{2 x}}{27 (-4+3 x)}\right ) \, dx-144 \int \left (\frac {16}{9 (-4+3 x)^3}+\frac {8}{9 (-4+3 x)^2}+\frac {1}{9 (-4+3 x)}\right ) \, dx+459 \int \left (\frac {4}{27}+\frac {x}{27}+\frac {256}{81 (-4+3 x)^3}+\frac {256}{81 (-4+3 x)^2}+\frac {32}{27 (-4+3 x)}\right ) \, dx-1188 \int \left (\frac {1}{27}+\frac {64}{27 (-4+3 x)^3}+\frac {16}{9 (-4+3 x)^2}+\frac {4}{9 (-4+3 x)}\right ) \, dx\\ &=\frac {2432}{9 (4-3 x)^2}-\frac {2368}{9 (4-3 x)}+24 x+\frac {17 x^2}{2}-\frac {8 x^2}{(4-3 x)^2}+2 \int e^{4 x} x \, dx+4 \int e^{4 x} x^2 \, dx+12 \int e^{2 x} x^2 \, dx+28 \int e^{2 x} x \, dx+\frac {88}{3} \int e^{2 x} \, dx+\frac {256}{3} \int \frac {e^{2 x}}{-4+3 x} \, dx-128 \int \frac {e^{2 x}}{(-4+3 x)^2} \, dx\\ &=\frac {44 e^{2 x}}{3}+\frac {2432}{9 (4-3 x)^2}-\frac {2368}{9 (4-3 x)}-\frac {128 e^{2 x}}{3 (4-3 x)}+24 x+14 e^{2 x} x+\frac {1}{2} e^{4 x} x+\frac {17 x^2}{2}+6 e^{2 x} x^2+e^{4 x} x^2-\frac {8 x^2}{(4-3 x)^2}+\frac {256}{9} e^{8/3} \text {Ei}\left (-\frac {2}{3} (4-3 x)\right )-\frac {1}{2} \int e^{4 x} \, dx-2 \int e^{4 x} x \, dx-12 \int e^{2 x} x \, dx-14 \int e^{2 x} \, dx-\frac {256}{3} \int \frac {e^{2 x}}{-4+3 x} \, dx\\ &=\frac {23 e^{2 x}}{3}-\frac {e^{4 x}}{8}+\frac {2432}{9 (4-3 x)^2}-\frac {2368}{9 (4-3 x)}-\frac {128 e^{2 x}}{3 (4-3 x)}+24 x+8 e^{2 x} x+\frac {17 x^2}{2}+6 e^{2 x} x^2+e^{4 x} x^2-\frac {8 x^2}{(4-3 x)^2}+\frac {1}{2} \int e^{4 x} \, dx+6 \int e^{2 x} \, dx\\ &=\frac {32 e^{2 x}}{3}+\frac {2432}{9 (4-3 x)^2}-\frac {2368}{9 (4-3 x)}-\frac {128 e^{2 x}}{3 (4-3 x)}+24 x+8 e^{2 x} x+\frac {17 x^2}{2}+6 e^{2 x} x^2+e^{4 x} x^2-\frac {8 x^2}{(4-3 x)^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 46, normalized size = 1.53 \begin {gather*} \frac {768 (-1+x)}{(4-3 x)^2}+24 x+\left (\frac {17}{2}+e^{4 x}\right ) x^2+\frac {18 e^{2 x} x^3}{-4+3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(64*x - 144*x^2 - 1188*x^3 + 459*x^4 + E^(4*x)*(-128*x + 32*x^2 + 360*x^3 - 378*x^4 + 108*x^5) + E^(
2*x)*(864*x^2 - 504*x^3 - 540*x^4 + 324*x^5))/(-64 + 144*x - 108*x^2 + 27*x^3),x]

[Out]

(768*(-1 + x))/(4 - 3*x)^2 + 24*x + (17/2 + E^(4*x))*x^2 + (18*E^(2*x)*x^3)/(-4 + 3*x)

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fricas [B]  time = 2.36, size = 73, normalized size = 2.43 \begin {gather*} \frac {153 \, x^{4} + 24 \, x^{3} - 880 \, x^{2} + 2 \, {\left (9 \, x^{4} - 24 \, x^{3} + 16 \, x^{2}\right )} e^{\left (4 \, x\right )} + 36 \, {\left (3 \, x^{4} - 4 \, x^{3}\right )} e^{\left (2 \, x\right )} + 2304 \, x - 1536}{2 \, {\left (9 \, x^{2} - 24 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((108*x^5-378*x^4+360*x^3+32*x^2-128*x)*exp(2*x)^2+(324*x^5-540*x^4-504*x^3+864*x^2)*exp(2*x)+459*x^
4-1188*x^3-144*x^2+64*x)/(27*x^3-108*x^2+144*x-64),x, algorithm="fricas")

[Out]

1/2*(153*x^4 + 24*x^3 - 880*x^2 + 2*(9*x^4 - 24*x^3 + 16*x^2)*e^(4*x) + 36*(3*x^4 - 4*x^3)*e^(2*x) + 2304*x -
1536)/(9*x^2 - 24*x + 16)

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giac [B]  time = 0.52, size = 79, normalized size = 2.63 \begin {gather*} \frac {18 \, x^{4} e^{\left (4 \, x\right )} + 108 \, x^{4} e^{\left (2 \, x\right )} + 153 \, x^{4} - 48 \, x^{3} e^{\left (4 \, x\right )} - 144 \, x^{3} e^{\left (2 \, x\right )} + 24 \, x^{3} + 32 \, x^{2} e^{\left (4 \, x\right )} - 880 \, x^{2} + 2304 \, x - 1536}{2 \, {\left (9 \, x^{2} - 24 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((108*x^5-378*x^4+360*x^3+32*x^2-128*x)*exp(2*x)^2+(324*x^5-540*x^4-504*x^3+864*x^2)*exp(2*x)+459*x^
4-1188*x^3-144*x^2+64*x)/(27*x^3-108*x^2+144*x-64),x, algorithm="giac")

[Out]

1/2*(18*x^4*e^(4*x) + 108*x^4*e^(2*x) + 153*x^4 - 48*x^3*e^(4*x) - 144*x^3*e^(2*x) + 24*x^3 + 32*x^2*e^(4*x) -
880*x^2 + 2304*x - 1536)/(9*x^2 - 24*x + 16)

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maple [A]  time = 0.13, size = 50, normalized size = 1.67

method	result	size
risch	\(\frac {17 x^{2}}{2}+24 x +\frac {\frac {256 x}{3}-\frac {256}{3}}{x^{2}-\frac {8}{3} x +\frac {16}{9}}+x^{2} {\mathrm e}^{4 x}+\frac {18 x^{3} {\mathrm e}^{2 x}}{3 x -4}\)	\(50\)
derivativedivides	\(\frac {1024}{\left (6 x -8\right )^{2}}+\frac {512}{6 x -8}+24 x +\frac {17 x^{2}}{2}+\frac {256 \,{\mathrm e}^{2 x}}{9 \left (2 x -\frac {8}{3}\right )}+\frac {32 \,{\mathrm e}^{2 x}}{3}+8 x \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{2 x} x^{2}+x^{2} {\mathrm e}^{4 x}\)	\(73\)
default	\(\frac {1024}{\left (6 x -8\right )^{2}}+\frac {512}{6 x -8}+24 x +\frac {17 x^{2}}{2}+\frac {256 \,{\mathrm e}^{2 x}}{9 \left (2 x -\frac {8}{3}\right )}+\frac {32 \,{\mathrm e}^{2 x}}{3}+8 x \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{2 x} x^{2}+x^{2} {\mathrm e}^{4 x}\)	\(73\)
norman	\(\frac {-\frac {64 x}{3}+12 x^{3}+\frac {153 x^{4}}{2}+16 x^{2} {\mathrm e}^{4 x}-24 x^{3} {\mathrm e}^{4 x}+9 x^{4} {\mathrm e}^{4 x}-72 \,{\mathrm e}^{2 x} x^{3}+54 \,{\mathrm e}^{2 x} x^{4}+\frac {128}{9}}{\left (3 x -4\right )^{2}}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((108*x^5-378*x^4+360*x^3+32*x^2-128*x)*exp(2*x)^2+(324*x^5-540*x^4-504*x^3+864*x^2)*exp(2*x)+459*x^4-1188
*x^3-144*x^2+64*x)/(27*x^3-108*x^2+144*x-64),x,method=_RETURNVERBOSE)

[Out]

17/2*x^2+24*x+(256/3*x-256/3)/(x^2-8/3*x+16/9)+x^2*exp(4*x)+18*x^3/(3*x-4)*exp(2*x)

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maxima [B]  time = 0.46, size = 117, normalized size = 3.90 \begin {gather*} \frac {17}{2} \, x^{2} + 24 \, x + \frac {704 \, {\left (9 \, x - 10\right )}}{3 \, {\left (9 \, x^{2} - 24 \, x + 16\right )}} - \frac {2176 \, {\left (6 \, x - 7\right )}}{9 \, {\left (9 \, x^{2} - 24 \, x + 16\right )}} - \frac {64 \, {\left (3 \, x - 2\right )}}{9 \, {\left (9 \, x^{2} - 24 \, x + 16\right )}} + \frac {18 \, x^{3} e^{\left (2 \, x\right )} + {\left (3 \, x^{3} - 4 \, x^{2}\right )} e^{\left (4 \, x\right )}}{3 \, x - 4} + \frac {128 \, {\left (x - 1\right )}}{9 \, x^{2} - 24 \, x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((108*x^5-378*x^4+360*x^3+32*x^2-128*x)*exp(2*x)^2+(324*x^5-540*x^4-504*x^3+864*x^2)*exp(2*x)+459*x^
4-1188*x^3-144*x^2+64*x)/(27*x^3-108*x^2+144*x-64),x, algorithm="maxima")

[Out]

17/2*x^2 + 24*x + 704/3*(9*x - 10)/(9*x^2 - 24*x + 16) - 2176/9*(6*x - 7)/(9*x^2 - 24*x + 16) - 64/9*(3*x - 2)
/(9*x^2 - 24*x + 16) + (18*x^3*e^(2*x) + (3*x^3 - 4*x^2)*e^(4*x))/(3*x - 4) + 128*(x - 1)/(9*x^2 - 24*x + 16)

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mupad [B]  time = 0.14, size = 59, normalized size = 1.97 \begin {gather*} \frac {x^2\,\left (12\,x+16\,{\mathrm {e}}^{4\,x}-72\,x\,{\mathrm {e}}^{2\,x}-24\,x\,{\mathrm {e}}^{4\,x}+54\,x^2\,{\mathrm {e}}^{2\,x}+9\,x^2\,{\mathrm {e}}^{4\,x}+\frac {153\,x^2}{2}-8\right )}{{\left (3\,x-4\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((64*x + exp(4*x)*(32*x^2 - 128*x + 360*x^3 - 378*x^4 + 108*x^5) + exp(2*x)*(864*x^2 - 504*x^3 - 540*x^4 +
324*x^5) - 144*x^2 - 1188*x^3 + 459*x^4)/(144*x - 108*x^2 + 27*x^3 - 64),x)

[Out]

(x^2*(12*x + 16*exp(4*x) - 72*x*exp(2*x) - 24*x*exp(4*x) + 54*x^2*exp(2*x) + 9*x^2*exp(4*x) + (153*x^2)/2 - 8)
)/(3*x - 4)^2

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sympy [B]  time = 0.20, size = 54, normalized size = 1.80 \begin {gather*} \frac {17 x^{2}}{2} + 24 x + \frac {768 x - 768}{9 x^{2} - 24 x + 16} + \frac {18 x^{3} e^{2 x} + \left (3 x^{3} - 4 x^{2}\right ) e^{4 x}}{3 x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((108*x**5-378*x**4+360*x**3+32*x**2-128*x)*exp(2*x)**2+(324*x**5-540*x**4-504*x**3+864*x**2)*exp(2*
x)+459*x**4-1188*x**3-144*x**2+64*x)/(27*x**3-108*x**2+144*x-64),x)

[Out]

17*x**2/2 + 24*x + (768*x - 768)/(9*x**2 - 24*x + 16) + (18*x**3*exp(2*x) + (3*x**3 - 4*x**2)*exp(4*x))/(3*x -
4)

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