3.4.11 \(\int \frac {40 x^4+96 x^5+32 x^6-40 x^7-6 x^8+4 x^9+e^{x/3} (18+22 x-17 x^2+x^3)}{8 x^4+16 x^5-8 x^7+2 x^8} \, dx\)

Optimal. Leaf size=29 \[ x (5+x)+\frac {3 e^{x/3}}{2 x^3 (-2+(-2+x) x)} \]

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Rubi [C]  time = 1.84, antiderivative size = 445, normalized size of antiderivative = 15.34, number of steps used = 37, number of rules used = 5, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.066, Rules used = {6688, 6742, 2177, 2178, 6728} \begin {gather*} \frac {5}{48} \left (1+\sqrt {3}\right ) e^{\frac {1}{3}+\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (x-\sqrt {3}-1\right )\right )+\frac {1}{48} \left (9-14 \sqrt {3}\right ) e^{\frac {1}{3}+\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (x-\sqrt {3}-1\right )\right )+\frac {3}{16} \sqrt {3} e^{\frac {1}{3}+\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (x-\sqrt {3}-1\right )\right )-\frac {7}{24} e^{\frac {1}{3}+\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (x-\sqrt {3}-1\right )\right )+\frac {1}{48} \left (9+14 \sqrt {3}\right ) e^{\frac {1}{3}-\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (x+\sqrt {3}-1\right )\right )+\frac {5}{48} \left (1-\sqrt {3}\right ) e^{\frac {1}{3}-\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (x+\sqrt {3}-1\right )\right )-\frac {3}{16} \sqrt {3} e^{\frac {1}{3}-\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (x+\sqrt {3}-1\right )\right )-\frac {7}{24} e^{\frac {1}{3}-\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (x+\sqrt {3}-1\right )\right )-\frac {3 e^{x/3}}{4 x^3}+x^2+\frac {3 e^{x/3}}{4 x^2}+5 x+\frac {5 \left (1-\sqrt {3}\right ) e^{x/3}}{16 \left (-x-\sqrt {3}+1\right )}-\frac {7 e^{x/3}}{8 \left (-x-\sqrt {3}+1\right )}+\frac {5 \left (1+\sqrt {3}\right ) e^{x/3}}{16 \left (-x+\sqrt {3}+1\right )}-\frac {7 e^{x/3}}{8 \left (-x+\sqrt {3}+1\right )}-\frac {9 e^{x/3}}{8 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(40*x^4 + 96*x^5 + 32*x^6 - 40*x^7 - 6*x^8 + 4*x^9 + E^(x/3)*(18 + 22*x - 17*x^2 + x^3))/(8*x^4 + 16*x^5 -
8*x^7 + 2*x^8),x]

[Out]

(-7*E^(x/3))/(8*(1 - Sqrt[3] - x)) + (5*(1 - Sqrt[3])*E^(x/3))/(16*(1 - Sqrt[3] - x)) - (7*E^(x/3))/(8*(1 + Sq
rt[3] - x)) + (5*(1 + Sqrt[3])*E^(x/3))/(16*(1 + Sqrt[3] - x)) - (3*E^(x/3))/(4*x^3) + (3*E^(x/3))/(4*x^2) - (
9*E^(x/3))/(8*x) + 5*x + x^2 - (7*E^(1/3 + 1/Sqrt[3])*ExpIntegralEi[(-1 - Sqrt[3] + x)/3])/24 + (3*Sqrt[3]*E^(
1/3 + 1/Sqrt[3])*ExpIntegralEi[(-1 - Sqrt[3] + x)/3])/16 + ((9 - 14*Sqrt[3])*E^(1/3 + 1/Sqrt[3])*ExpIntegralEi
[(-1 - Sqrt[3] + x)/3])/48 + (5*(1 + Sqrt[3])*E^(1/3 + 1/Sqrt[3])*ExpIntegralEi[(-1 - Sqrt[3] + x)/3])/48 - (7
*E^(1/3 - 1/Sqrt[3])*ExpIntegralEi[(-1 + Sqrt[3] + x)/3])/24 - (3*Sqrt[3]*E^(1/3 - 1/Sqrt[3])*ExpIntegralEi[(-
1 + Sqrt[3] + x)/3])/16 + (5*(1 - Sqrt[3])*E^(1/3 - 1/Sqrt[3])*ExpIntegralEi[(-1 + Sqrt[3] + x)/3])/48 + ((9 +
14*Sqrt[3])*E^(1/3 - 1/Sqrt[3])*ExpIntegralEi[(-1 + Sqrt[3] + x)/3])/48

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 6688

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

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

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 \left (5+2 x+\frac {e^{x/3} \left (18+22 x-17 x^2+x^3\right )}{2 x^4 \left (-2-2 x+x^2\right )^2}\right ) \, dx\\ &=5 x+x^2+\frac {1}{2} \int \frac {e^{x/3} \left (18+22 x-17 x^2+x^3\right )}{x^4 \left (-2-2 x+x^2\right )^2} \, dx\\ &=5 x+x^2+\frac {1}{2} \int \left (\frac {9 e^{x/3}}{2 x^4}-\frac {7 e^{x/3}}{2 x^3}+\frac {11 e^{x/3}}{4 x^2}-\frac {3 e^{x/3}}{4 x}+\frac {3 e^{x/3} (-14+5 x)}{2 \left (-2-2 x+x^2\right )^2}+\frac {e^{x/3} (-17+3 x)}{4 \left (-2-2 x+x^2\right )}\right ) \, dx\\ &=5 x+x^2+\frac {1}{8} \int \frac {e^{x/3} (-17+3 x)}{-2-2 x+x^2} \, dx-\frac {3}{8} \int \frac {e^{x/3}}{x} \, dx+\frac {3}{4} \int \frac {e^{x/3} (-14+5 x)}{\left (-2-2 x+x^2\right )^2} \, dx+\frac {11}{8} \int \frac {e^{x/3}}{x^2} \, dx-\frac {7}{4} \int \frac {e^{x/3}}{x^3} \, dx+\frac {9}{4} \int \frac {e^{x/3}}{x^4} \, dx\\ &=-\frac {3 e^{x/3}}{4 x^3}+\frac {7 e^{x/3}}{8 x^2}-\frac {11 e^{x/3}}{8 x}+5 x+x^2-\frac {3 \text {Ei}\left (\frac {x}{3}\right )}{8}+\frac {1}{8} \int \left (\frac {\left (3-\frac {14}{\sqrt {3}}\right ) e^{x/3}}{-2-2 \sqrt {3}+2 x}+\frac {\left (3+\frac {14}{\sqrt {3}}\right ) e^{x/3}}{-2+2 \sqrt {3}+2 x}\right ) \, dx+\frac {1}{4} \int \frac {e^{x/3}}{x^3} \, dx-\frac {7}{24} \int \frac {e^{x/3}}{x^2} \, dx+\frac {11}{24} \int \frac {e^{x/3}}{x} \, dx+\frac {3}{4} \int \left (-\frac {14 e^{x/3}}{\left (-2-2 x+x^2\right )^2}+\frac {5 e^{x/3} x}{\left (-2-2 x+x^2\right )^2}\right ) \, dx\\ &=-\frac {3 e^{x/3}}{4 x^3}+\frac {3 e^{x/3}}{4 x^2}-\frac {13 e^{x/3}}{12 x}+5 x+x^2+\frac {\text {Ei}\left (\frac {x}{3}\right )}{12}+\frac {1}{24} \int \frac {e^{x/3}}{x^2} \, dx-\frac {7}{72} \int \frac {e^{x/3}}{x} \, dx+\frac {15}{4} \int \frac {e^{x/3} x}{\left (-2-2 x+x^2\right )^2} \, dx-\frac {21}{2} \int \frac {e^{x/3}}{\left (-2-2 x+x^2\right )^2} \, dx+\frac {1}{24} \left (9-14 \sqrt {3}\right ) \int \frac {e^{x/3}}{-2-2 \sqrt {3}+2 x} \, dx+\frac {1}{24} \left (9+14 \sqrt {3}\right ) \int \frac {e^{x/3}}{-2+2 \sqrt {3}+2 x} \, dx\\ &=-\frac {3 e^{x/3}}{4 x^3}+\frac {3 e^{x/3}}{4 x^2}-\frac {9 e^{x/3}}{8 x}+5 x+x^2-\frac {\text {Ei}\left (\frac {x}{3}\right )}{72}+\frac {1}{48} \left (9-14 \sqrt {3}\right ) e^{\frac {1}{3}+\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1-\sqrt {3}+x\right )\right )+\frac {1}{48} \left (9+14 \sqrt {3}\right ) e^{\frac {1}{3}-\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1+\sqrt {3}+x\right )\right )+\frac {1}{72} \int \frac {e^{x/3}}{x} \, dx+\frac {15}{4} \int \left (\frac {\left (2+2 \sqrt {3}\right ) e^{x/3}}{6 \left (2+2 \sqrt {3}-2 x\right )^2}+\frac {e^{x/3}}{6 \sqrt {3} \left (2+2 \sqrt {3}-2 x\right )}+\frac {\left (2-2 \sqrt {3}\right ) e^{x/3}}{6 \left (-2+2 \sqrt {3}+2 x\right )^2}+\frac {e^{x/3}}{6 \sqrt {3} \left (-2+2 \sqrt {3}+2 x\right )}\right ) \, dx-\frac {63}{2} \operatorname {Subst}\left (\int \frac {e^x}{\left (-2-6 x+9 x^2\right )^2} \, dx,x,\frac {x}{3}\right )\\ &=-\frac {3 e^{x/3}}{4 x^3}+\frac {3 e^{x/3}}{4 x^2}-\frac {9 e^{x/3}}{8 x}+5 x+x^2+\frac {1}{48} \left (9-14 \sqrt {3}\right ) e^{\frac {1}{3}+\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1-\sqrt {3}+x\right )\right )+\frac {1}{48} \left (9+14 \sqrt {3}\right ) e^{\frac {1}{3}-\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1+\sqrt {3}+x\right )\right )-\frac {63}{2} \operatorname {Subst}\left (\int \left (\frac {3 e^x}{\left (6+6 \sqrt {3}-18 x\right )^2}+\frac {e^x}{2 \sqrt {3} \left (6+6 \sqrt {3}-18 x\right )}+\frac {3 e^x}{\left (-6+6 \sqrt {3}+18 x\right )^2}+\frac {e^x}{2 \sqrt {3} \left (-6+6 \sqrt {3}+18 x\right )}\right ) \, dx,x,\frac {x}{3}\right )+\frac {5 \int \frac {e^{x/3}}{2+2 \sqrt {3}-2 x} \, dx}{8 \sqrt {3}}+\frac {5 \int \frac {e^{x/3}}{-2+2 \sqrt {3}+2 x} \, dx}{8 \sqrt {3}}+\frac {1}{4} \left (5 \left (1-\sqrt {3}\right )\right ) \int \frac {e^{x/3}}{\left (-2+2 \sqrt {3}+2 x\right )^2} \, dx+\frac {1}{4} \left (5 \left (1+\sqrt {3}\right )\right ) \int \frac {e^{x/3}}{\left (2+2 \sqrt {3}-2 x\right )^2} \, dx\\ &=\frac {5 \left (1-\sqrt {3}\right ) e^{x/3}}{16 \left (1-\sqrt {3}-x\right )}+\frac {5 \left (1+\sqrt {3}\right ) e^{x/3}}{16 \left (1+\sqrt {3}-x\right )}-\frac {3 e^{x/3}}{4 x^3}+\frac {3 e^{x/3}}{4 x^2}-\frac {9 e^{x/3}}{8 x}+5 x+x^2-\frac {5 e^{\frac {1}{3}+\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1-\sqrt {3}+x\right )\right )}{16 \sqrt {3}}+\frac {1}{48} \left (9-14 \sqrt {3}\right ) e^{\frac {1}{3}+\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1-\sqrt {3}+x\right )\right )+\frac {5 e^{\frac {1}{3}-\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1+\sqrt {3}+x\right )\right )}{16 \sqrt {3}}+\frac {1}{48} \left (9+14 \sqrt {3}\right ) e^{\frac {1}{3}-\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1+\sqrt {3}+x\right )\right )-\frac {189}{2} \operatorname {Subst}\left (\int \frac {e^x}{\left (6+6 \sqrt {3}-18 x\right )^2} \, dx,x,\frac {x}{3}\right )-\frac {189}{2} \operatorname {Subst}\left (\int \frac {e^x}{\left (-6+6 \sqrt {3}+18 x\right )^2} \, dx,x,\frac {x}{3}\right )-\frac {1}{4} \left (21 \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {e^x}{6+6 \sqrt {3}-18 x} \, dx,x,\frac {x}{3}\right )-\frac {1}{4} \left (21 \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {e^x}{-6+6 \sqrt {3}+18 x} \, dx,x,\frac {x}{3}\right )+\frac {1}{24} \left (5 \left (1-\sqrt {3}\right )\right ) \int \frac {e^{x/3}}{-2+2 \sqrt {3}+2 x} \, dx-\frac {1}{24} \left (5 \left (1+\sqrt {3}\right )\right ) \int \frac {e^{x/3}}{2+2 \sqrt {3}-2 x} \, dx\\ &=-\frac {7 e^{x/3}}{8 \left (1-\sqrt {3}-x\right )}+\frac {5 \left (1-\sqrt {3}\right ) e^{x/3}}{16 \left (1-\sqrt {3}-x\right )}-\frac {7 e^{x/3}}{8 \left (1+\sqrt {3}-x\right )}+\frac {5 \left (1+\sqrt {3}\right ) e^{x/3}}{16 \left (1+\sqrt {3}-x\right )}-\frac {3 e^{x/3}}{4 x^3}+\frac {3 e^{x/3}}{4 x^2}-\frac {9 e^{x/3}}{8 x}+5 x+x^2+\frac {3}{16} \sqrt {3} e^{\frac {1}{3}+\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1-\sqrt {3}+x\right )\right )+\frac {1}{48} \left (9-14 \sqrt {3}\right ) e^{\frac {1}{3}+\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1-\sqrt {3}+x\right )\right )+\frac {5}{48} \left (1+\sqrt {3}\right ) e^{\frac {1}{3}+\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1-\sqrt {3}+x\right )\right )-\frac {3}{16} \sqrt {3} e^{\frac {1}{3}-\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1+\sqrt {3}+x\right )\right )+\frac {5}{48} \left (1-\sqrt {3}\right ) e^{\frac {1}{3}-\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1+\sqrt {3}+x\right )\right )+\frac {1}{48} \left (9+14 \sqrt {3}\right ) e^{\frac {1}{3}-\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1+\sqrt {3}+x\right )\right )+\frac {21}{4} \operatorname {Subst}\left (\int \frac {e^x}{6+6 \sqrt {3}-18 x} \, dx,x,\frac {x}{3}\right )-\frac {21}{4} \operatorname {Subst}\left (\int \frac {e^x}{-6+6 \sqrt {3}+18 x} \, dx,x,\frac {x}{3}\right )\\ &=-\frac {7 e^{x/3}}{8 \left (1-\sqrt {3}-x\right )}+\frac {5 \left (1-\sqrt {3}\right ) e^{x/3}}{16 \left (1-\sqrt {3}-x\right )}-\frac {7 e^{x/3}}{8 \left (1+\sqrt {3}-x\right )}+\frac {5 \left (1+\sqrt {3}\right ) e^{x/3}}{16 \left (1+\sqrt {3}-x\right )}-\frac {3 e^{x/3}}{4 x^3}+\frac {3 e^{x/3}}{4 x^2}-\frac {9 e^{x/3}}{8 x}+5 x+x^2-\frac {7}{24} e^{\frac {1}{3}+\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1-\sqrt {3}+x\right )\right )+\frac {3}{16} \sqrt {3} e^{\frac {1}{3}+\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1-\sqrt {3}+x\right )\right )+\frac {1}{48} \left (9-14 \sqrt {3}\right ) e^{\frac {1}{3}+\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1-\sqrt {3}+x\right )\right )+\frac {5}{48} \left (1+\sqrt {3}\right ) e^{\frac {1}{3}+\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1-\sqrt {3}+x\right )\right )-\frac {7}{24} e^{\frac {1}{3}-\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1+\sqrt {3}+x\right )\right )-\frac {3}{16} \sqrt {3} e^{\frac {1}{3}-\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1+\sqrt {3}+x\right )\right )+\frac {5}{48} \left (1-\sqrt {3}\right ) e^{\frac {1}{3}-\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1+\sqrt {3}+x\right )\right )+\frac {1}{48} \left (9+14 \sqrt {3}\right ) e^{\frac {1}{3}-\frac {1}{\sqrt {3}}} \text {Ei}\left (\frac {1}{3} \left (-1+\sqrt {3}+x\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.36, size = 59, normalized size = 2.03 \begin {gather*} 5 x+x^2+\frac {1}{2} e^{x/3} \left (-\frac {3}{2 x^3}+\frac {3}{2 x^2}-\frac {9}{4 x}+\frac {3 (-8+3 x)}{4 \left (-2-2 x+x^2\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(40*x^4 + 96*x^5 + 32*x^6 - 40*x^7 - 6*x^8 + 4*x^9 + E^(x/3)*(18 + 22*x - 17*x^2 + x^3))/(8*x^4 + 16
*x^5 - 8*x^7 + 2*x^8),x]

[Out]

5*x + x^2 + (E^(x/3)*(-3/(2*x^3) + 3/(2*x^2) - 9/(4*x) + (3*(-8 + 3*x))/(4*(-2 - 2*x + x^2))))/2

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fricas [A]  time = 0.72, size = 45, normalized size = 1.55 \begin {gather*} \frac {2 \, x^{7} + 6 \, x^{6} - 24 \, x^{5} - 20 \, x^{4} + 3 \, e^{\left (\frac {1}{3} \, x\right )}}{2 \, {\left (x^{5} - 2 \, x^{4} - 2 \, x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-17*x^2+22*x+18)*exp(1/3*x)+4*x^9-6*x^8-40*x^7+32*x^6+96*x^5+40*x^4)/(2*x^8-8*x^7+16*x^5+8*x^4)
,x, algorithm="fricas")

[Out]

1/2*(2*x^7 + 6*x^6 - 24*x^5 - 20*x^4 + 3*e^(1/3*x))/(x^5 - 2*x^4 - 2*x^3)

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giac [A]  time = 0.38, size = 45, normalized size = 1.55 \begin {gather*} \frac {2 \, x^{7} + 6 \, x^{6} - 24 \, x^{5} - 20 \, x^{4} + 3 \, e^{\left (\frac {1}{3} \, x\right )}}{2 \, {\left (x^{5} - 2 \, x^{4} - 2 \, x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-17*x^2+22*x+18)*exp(1/3*x)+4*x^9-6*x^8-40*x^7+32*x^6+96*x^5+40*x^4)/(2*x^8-8*x^7+16*x^5+8*x^4)
,x, algorithm="giac")

[Out]

1/2*(2*x^7 + 6*x^6 - 24*x^5 - 20*x^4 + 3*e^(1/3*x))/(x^5 - 2*x^4 - 2*x^3)

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maple [A]  time = 0.19, size = 27, normalized size = 0.93

method	result	size
risch	\(x^{2}+5 x +\frac {3 \,{\mathrm e}^{\frac {x}{3}}}{2 x^{3} \left (x^{2}-2 x -2\right )}\)	\(27\)
norman	\(\frac {x^{7}-34 x^{4}-24 x^{3}+3 x^{6}+\frac {3 \,{\mathrm e}^{\frac {x}{3}}}{2}}{x^{3} \left (x^{2}-2 x -2\right )}\)	\(40\)
derivativedivides	\(-\frac {5 \left (6 x -6\right )}{9 \left (x^{2}-2 x -2\right )}-\frac {4 \left (2 x +4\right )}{x^{2}-2 x -2}+\frac {-\frac {32 x}{27}-\frac {16}{27}}{\frac {1}{9} x^{2}-\frac {2}{9} x -\frac {2}{9}}-\frac {1620 \left (-\frac {5 x}{2187}-\frac {4}{2187}\right )}{\frac {1}{9} x^{2}-\frac {2}{9} x -\frac {2}{9}}+5 x -\frac {36 \left (-\frac {7 x}{162}-\frac {5}{162}\right )}{\frac {1}{9} x^{2}-\frac {2}{9} x -\frac {2}{9}}+x^{2}+\frac {-\frac {76 x}{27}-\frac {56}{27}}{\frac {1}{9} x^{2}-\frac {2}{9} x -\frac {2}{9}}-\frac {3 \,{\mathrm e}^{\frac {x}{3}} \left (\frac {131}{9} x^{4}-\frac {310}{9} x^{3}-\frac {139}{9} x^{2}+\frac {11}{3} x -2\right )}{4 x^{3} \left (x^{2}-2 x -2\right )}+\frac {11 \,{\mathrm e}^{\frac {x}{3}} \left (\frac {16}{3} x^{3}-13 x^{2}-\frac {16}{3} x +2\right )}{8 x^{2} \left (x^{2}-2 x -2\right )}+\frac {17 \,{\mathrm e}^{\frac {x}{3}} \left (\frac {5}{3} x^{2}-\frac {11}{3} x -2\right )}{8 \left (x^{2}-2 x -2\right ) x}+\frac {{\mathrm e}^{\frac {x}{3}} \left (x -4\right )}{24 x^{2}-48 x -48}\)	\(239\)
default	\(-\frac {5 \left (6 x -6\right )}{9 \left (x^{2}-2 x -2\right )}-\frac {4 \left (2 x +4\right )}{x^{2}-2 x -2}+\frac {-\frac {32 x}{27}-\frac {16}{27}}{\frac {1}{9} x^{2}-\frac {2}{9} x -\frac {2}{9}}-\frac {1620 \left (-\frac {5 x}{2187}-\frac {4}{2187}\right )}{\frac {1}{9} x^{2}-\frac {2}{9} x -\frac {2}{9}}+5 x -\frac {36 \left (-\frac {7 x}{162}-\frac {5}{162}\right )}{\frac {1}{9} x^{2}-\frac {2}{9} x -\frac {2}{9}}+x^{2}+\frac {-\frac {76 x}{27}-\frac {56}{27}}{\frac {1}{9} x^{2}-\frac {2}{9} x -\frac {2}{9}}-\frac {3 \,{\mathrm e}^{\frac {x}{3}} \left (\frac {131}{9} x^{4}-\frac {310}{9} x^{3}-\frac {139}{9} x^{2}+\frac {11}{3} x -2\right )}{4 x^{3} \left (x^{2}-2 x -2\right )}+\frac {11 \,{\mathrm e}^{\frac {x}{3}} \left (\frac {16}{3} x^{3}-13 x^{2}-\frac {16}{3} x +2\right )}{8 x^{2} \left (x^{2}-2 x -2\right )}+\frac {17 \,{\mathrm e}^{\frac {x}{3}} \left (\frac {5}{3} x^{2}-\frac {11}{3} x -2\right )}{8 \left (x^{2}-2 x -2\right ) x}+\frac {{\mathrm e}^{\frac {x}{3}} \left (x -4\right )}{24 x^{2}-48 x -48}\)	\(239\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3-17*x^2+22*x+18)*exp(1/3*x)+4*x^9-6*x^8-40*x^7+32*x^6+96*x^5+40*x^4)/(2*x^8-8*x^7+16*x^5+8*x^4),x,met
hod=_RETURNVERBOSE)

[Out]

x^2+5*x+3/2/x^3/(x^2-2*x-2)*exp(1/3*x)

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maxima [B]  time = 0.67, size = 127, normalized size = 4.38 \begin {gather*} x^{2} + 5 \, x - \frac {4 \, {\left (19 \, x + 14\right )}}{3 \, {\left (x^{2} - 2 \, x - 2\right )}} + \frac {2 \, {\left (7 \, x + 5\right )}}{x^{2} - 2 \, x - 2} + \frac {20 \, {\left (5 \, x + 4\right )}}{3 \, {\left (x^{2} - 2 \, x - 2\right )}} - \frac {16 \, {\left (2 \, x + 1\right )}}{3 \, {\left (x^{2} - 2 \, x - 2\right )}} - \frac {8 \, {\left (x + 2\right )}}{x^{2} - 2 \, x - 2} - \frac {10 \, {\left (x - 1\right )}}{3 \, {\left (x^{2} - 2 \, x - 2\right )}} + \frac {3 \, e^{\left (\frac {1}{3} \, x\right )}}{2 \, {\left (x^{5} - 2 \, x^{4} - 2 \, x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-17*x^2+22*x+18)*exp(1/3*x)+4*x^9-6*x^8-40*x^7+32*x^6+96*x^5+40*x^4)/(2*x^8-8*x^7+16*x^5+8*x^4)
,x, algorithm="maxima")

[Out]

x^2 + 5*x - 4/3*(19*x + 14)/(x^2 - 2*x - 2) + 2*(7*x + 5)/(x^2 - 2*x - 2) + 20/3*(5*x + 4)/(x^2 - 2*x - 2) - 1
6/3*(2*x + 1)/(x^2 - 2*x - 2) - 8*(x + 2)/(x^2 - 2*x - 2) - 10/3*(x - 1)/(x^2 - 2*x - 2) + 3/2*e^(1/3*x)/(x^5
- 2*x^4 - 2*x^3)

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mupad [B]  time = 0.48, size = 28, normalized size = 0.97 \begin {gather*} 5\,x+x^2-\frac {3\,{\mathrm {e}}^{x/3}}{2\,x^3\,\left (-x^2+2\,x+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((40*x^4 + 96*x^5 + 32*x^6 - 40*x^7 - 6*x^8 + 4*x^9 + exp(x/3)*(22*x - 17*x^2 + x^3 + 18))/(8*x^4 + 16*x^5
- 8*x^7 + 2*x^8),x)

[Out]

5*x + x^2 - (3*exp(x/3))/(2*x^3*(2*x - x^2 + 2))

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sympy [A]  time = 0.13, size = 27, normalized size = 0.93 \begin {gather*} x^{2} + 5 x + \frac {3 e^{\frac {x}{3}}}{2 x^{5} - 4 x^{4} - 4 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3-17*x**2+22*x+18)*exp(1/3*x)+4*x**9-6*x**8-40*x**7+32*x**6+96*x**5+40*x**4)/(2*x**8-8*x**7+16*
x**5+8*x**4),x)

[Out]

x**2 + 5*x + 3*exp(x/3)/(2*x**5 - 4*x**4 - 4*x**3)

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