3.28.40 \(\int \frac {-225 x^5+162 x^6+e^{3 x} (12 x-12 x^2-108 x^5)+e^x (-68 x^3+68 x^4-36 x^5)+e^{2 x} (60 x^2-78 x^3+36 x^4-306 x^5-288 x^6+324 x^7)+e^{4 x} (-4+4 x-18 x^3+72 x^4+162 x^6+324 x^7)}{81 x^5} \, dx\)

Optimal. Leaf size=31 \[ 3+x+\left (-2+x+e^{2 x} x+\frac {\left (-e^x+x\right )^2}{9 x^2}\right )^2 \]

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Rubi [B]  time = 0.68, antiderivative size = 127, normalized size of antiderivative = 4.10, number of steps used = 52, number of rules used = 7, integrand size = 126, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {12, 14, 2199, 2194, 2177, 2178, 2176} \begin {gather*} \frac {e^{4 x}}{81 x^4}-\frac {4 e^{3 x}}{81 x^3}+2 e^{2 x} x^2+e^{4 x} x^2-\frac {10 e^{2 x}}{27 x^2}+\frac {1}{324} (25-18 x)^2-\frac {4 e^x}{9}-\frac {4 e^{3 x}}{9}-\frac {34}{9} e^{2 x} x+\frac {68 e^x}{81 x}+\frac {2 e^{2 x}}{9 x}+\frac {2 e^{4 x}}{9 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-225*x^5 + 162*x^6 + E^(3*x)*(12*x - 12*x^2 - 108*x^5) + E^x*(-68*x^3 + 68*x^4 - 36*x^5) + E^(2*x)*(60*x^
2 - 78*x^3 + 36*x^4 - 306*x^5 - 288*x^6 + 324*x^7) + E^(4*x)*(-4 + 4*x - 18*x^3 + 72*x^4 + 162*x^6 + 324*x^7))
/(81*x^5),x]

[Out]

(-4*E^x)/9 - (4*E^(3*x))/9 + (25 - 18*x)^2/324 + E^(4*x)/(81*x^4) - (4*E^(3*x))/(81*x^3) - (10*E^(2*x))/(27*x^
2) + (68*E^x)/(81*x) + (2*E^(2*x))/(9*x) + (2*E^(4*x))/(9*x) - (34*E^(2*x)*x)/9 + 2*E^(2*x)*x^2 + E^(4*x)*x^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{81} \int \frac {-225 x^5+162 x^6+e^{3 x} \left (12 x-12 x^2-108 x^5\right )+e^x \left (-68 x^3+68 x^4-36 x^5\right )+e^{2 x} \left (60 x^2-78 x^3+36 x^4-306 x^5-288 x^6+324 x^7\right )+e^{4 x} \left (-4+4 x-18 x^3+72 x^4+162 x^6+324 x^7\right )}{x^5} \, dx\\ &=\frac {1}{81} \int \left (9 (-25+18 x)-\frac {4 e^x \left (17-17 x+9 x^2\right )}{x^2}-\frac {12 e^{3 x} \left (-1+x+9 x^4\right )}{x^4}+\frac {2 e^{4 x} \left (1+9 x^3\right ) \left (-2+2 x+9 x^3+18 x^4\right )}{x^5}+\frac {6 e^{2 x} \left (10-13 x+6 x^2-51 x^3-48 x^4+54 x^5\right )}{x^3}\right ) \, dx\\ &=\frac {1}{324} (25-18 x)^2+\frac {2}{81} \int \frac {e^{4 x} \left (1+9 x^3\right ) \left (-2+2 x+9 x^3+18 x^4\right )}{x^5} \, dx-\frac {4}{81} \int \frac {e^x \left (17-17 x+9 x^2\right )}{x^2} \, dx+\frac {2}{27} \int \frac {e^{2 x} \left (10-13 x+6 x^2-51 x^3-48 x^4+54 x^5\right )}{x^3} \, dx-\frac {4}{27} \int \frac {e^{3 x} \left (-1+x+9 x^4\right )}{x^4} \, dx\\ &=\frac {1}{324} (25-18 x)^2+\frac {2}{81} \int \left (-\frac {2 e^{4 x}}{x^5}+\frac {2 e^{4 x}}{x^4}-\frac {9 e^{4 x}}{x^2}+\frac {36 e^{4 x}}{x}+81 e^{4 x} x+162 e^{4 x} x^2\right ) \, dx-\frac {4}{81} \int \left (9 e^x+\frac {17 e^x}{x^2}-\frac {17 e^x}{x}\right ) \, dx+\frac {2}{27} \int \left (-51 e^{2 x}+\frac {10 e^{2 x}}{x^3}-\frac {13 e^{2 x}}{x^2}+\frac {6 e^{2 x}}{x}-48 e^{2 x} x+54 e^{2 x} x^2\right ) \, dx-\frac {4}{27} \int \left (9 e^{3 x}-\frac {e^{3 x}}{x^4}+\frac {e^{3 x}}{x^3}\right ) \, dx\\ &=\frac {1}{324} (25-18 x)^2-\frac {4}{81} \int \frac {e^{4 x}}{x^5} \, dx+\frac {4}{81} \int \frac {e^{4 x}}{x^4} \, dx+\frac {4}{27} \int \frac {e^{3 x}}{x^4} \, dx-\frac {4}{27} \int \frac {e^{3 x}}{x^3} \, dx-\frac {2}{9} \int \frac {e^{4 x}}{x^2} \, dx-\frac {4 \int e^x \, dx}{9}+\frac {4}{9} \int \frac {e^{2 x}}{x} \, dx+\frac {20}{27} \int \frac {e^{2 x}}{x^3} \, dx-\frac {68}{81} \int \frac {e^x}{x^2} \, dx+\frac {68}{81} \int \frac {e^x}{x} \, dx+\frac {8}{9} \int \frac {e^{4 x}}{x} \, dx-\frac {26}{27} \int \frac {e^{2 x}}{x^2} \, dx-\frac {4}{3} \int e^{3 x} \, dx+2 \int e^{4 x} x \, dx-\frac {32}{9} \int e^{2 x} x \, dx-\frac {34}{9} \int e^{2 x} \, dx+4 \int e^{2 x} x^2 \, dx+4 \int e^{4 x} x^2 \, dx\\ &=-\frac {4 e^x}{9}-\frac {17 e^{2 x}}{9}-\frac {4 e^{3 x}}{9}+\frac {1}{324} (25-18 x)^2+\frac {e^{4 x}}{81 x^4}-\frac {4 e^{3 x}}{81 x^3}-\frac {4 e^{4 x}}{243 x^3}-\frac {10 e^{2 x}}{27 x^2}+\frac {2 e^{3 x}}{27 x^2}+\frac {68 e^x}{81 x}+\frac {26 e^{2 x}}{27 x}+\frac {2 e^{4 x}}{9 x}-\frac {16}{9} e^{2 x} x+\frac {1}{2} e^{4 x} x+2 e^{2 x} x^2+e^{4 x} x^2+\frac {68 \text {Ei}(x)}{81}+\frac {4 \text {Ei}(2 x)}{9}+\frac {8 \text {Ei}(4 x)}{9}-\frac {4}{81} \int \frac {e^{4 x}}{x^4} \, dx+\frac {16}{243} \int \frac {e^{4 x}}{x^3} \, dx+\frac {4}{27} \int \frac {e^{3 x}}{x^3} \, dx-\frac {2}{9} \int \frac {e^{3 x}}{x^2} \, dx-\frac {1}{2} \int e^{4 x} \, dx+\frac {20}{27} \int \frac {e^{2 x}}{x^2} \, dx-\frac {68}{81} \int \frac {e^x}{x} \, dx-\frac {8}{9} \int \frac {e^{4 x}}{x} \, dx+\frac {16}{9} \int e^{2 x} \, dx-\frac {52}{27} \int \frac {e^{2 x}}{x} \, dx-2 \int e^{4 x} x \, dx-4 \int e^{2 x} x \, dx\\ &=-\frac {4 e^x}{9}-e^{2 x}-\frac {4 e^{3 x}}{9}-\frac {e^{4 x}}{8}+\frac {1}{324} (25-18 x)^2+\frac {e^{4 x}}{81 x^4}-\frac {4 e^{3 x}}{81 x^3}-\frac {10 e^{2 x}}{27 x^2}-\frac {8 e^{4 x}}{243 x^2}+\frac {68 e^x}{81 x}+\frac {2 e^{2 x}}{9 x}+\frac {2 e^{3 x}}{9 x}+\frac {2 e^{4 x}}{9 x}-\frac {34}{9} e^{2 x} x+2 e^{2 x} x^2+e^{4 x} x^2-\frac {40 \text {Ei}(2 x)}{27}-\frac {16}{243} \int \frac {e^{4 x}}{x^3} \, dx+\frac {32}{243} \int \frac {e^{4 x}}{x^2} \, dx+\frac {2}{9} \int \frac {e^{3 x}}{x^2} \, dx+\frac {1}{2} \int e^{4 x} \, dx-\frac {2}{3} \int \frac {e^{3 x}}{x} \, dx+\frac {40}{27} \int \frac {e^{2 x}}{x} \, dx+2 \int e^{2 x} \, dx\\ &=-\frac {4 e^x}{9}-\frac {4 e^{3 x}}{9}+\frac {1}{324} (25-18 x)^2+\frac {e^{4 x}}{81 x^4}-\frac {4 e^{3 x}}{81 x^3}-\frac {10 e^{2 x}}{27 x^2}+\frac {68 e^x}{81 x}+\frac {2 e^{2 x}}{9 x}+\frac {22 e^{4 x}}{243 x}-\frac {34}{9} e^{2 x} x+2 e^{2 x} x^2+e^{4 x} x^2-\frac {2 \text {Ei}(3 x)}{3}-\frac {32}{243} \int \frac {e^{4 x}}{x^2} \, dx+\frac {128}{243} \int \frac {e^{4 x}}{x} \, dx+\frac {2}{3} \int \frac {e^{3 x}}{x} \, dx\\ &=-\frac {4 e^x}{9}-\frac {4 e^{3 x}}{9}+\frac {1}{324} (25-18 x)^2+\frac {e^{4 x}}{81 x^4}-\frac {4 e^{3 x}}{81 x^3}-\frac {10 e^{2 x}}{27 x^2}+\frac {68 e^x}{81 x}+\frac {2 e^{2 x}}{9 x}+\frac {2 e^{4 x}}{9 x}-\frac {34}{9} e^{2 x} x+2 e^{2 x} x^2+e^{4 x} x^2+\frac {128 \text {Ei}(4 x)}{243}-\frac {128}{243} \int \frac {e^{4 x}}{x} \, dx\\ &=-\frac {4 e^x}{9}-\frac {4 e^{3 x}}{9}+\frac {1}{324} (25-18 x)^2+\frac {e^{4 x}}{81 x^4}-\frac {4 e^{3 x}}{81 x^3}-\frac {10 e^{2 x}}{27 x^2}+\frac {68 e^x}{81 x}+\frac {2 e^{2 x}}{9 x}+\frac {2 e^{4 x}}{9 x}-\frac {34}{9} e^{2 x} x+2 e^{2 x} x^2+e^{4 x} x^2\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.06, size = 82, normalized size = 2.65 \begin {gather*} \frac {1}{81} \left (e^{3 x} \left (-36-\frac {4}{x^3}\right )+e^x \left (-36+\frac {68}{x}\right )-225 x+81 x^2+e^{4 x} \left (\frac {1}{x^4}+\frac {18}{x}+81 x^2\right )+e^{2 x} \left (-\frac {30}{x^2}+\frac {18}{x}-306 x+162 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-225*x^5 + 162*x^6 + E^(3*x)*(12*x - 12*x^2 - 108*x^5) + E^x*(-68*x^3 + 68*x^4 - 36*x^5) + E^(2*x)*
(60*x^2 - 78*x^3 + 36*x^4 - 306*x^5 - 288*x^6 + 324*x^7) + E^(4*x)*(-4 + 4*x - 18*x^3 + 72*x^4 + 162*x^6 + 324
*x^7))/(81*x^5),x]

[Out]

(E^(3*x)*(-36 - 4/x^3) + E^x*(-36 + 68/x) - 225*x + 81*x^2 + E^(4*x)*(x^(-4) + 18/x + 81*x^2) + E^(2*x)*(-30/x
^2 + 18/x - 306*x + 162*x^2))/81

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fricas [B]  time = 0.49, size = 88, normalized size = 2.84 \begin {gather*} \frac {81 \, x^{6} - 225 \, x^{5} + {\left (81 \, x^{6} + 18 \, x^{3} + 1\right )} e^{\left (4 \, x\right )} - 4 \, {\left (9 \, x^{4} + x\right )} e^{\left (3 \, x\right )} + 6 \, {\left (27 \, x^{6} - 51 \, x^{5} + 3 \, x^{3} - 5 \, x^{2}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (9 \, x^{4} - 17 \, x^{3}\right )} e^{x}}{81 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*((324*x^7+162*x^6+72*x^4-18*x^3+4*x-4)*exp(x)^4+(-108*x^5-12*x^2+12*x)*exp(x)^3+(324*x^7-288*x^
6-306*x^5+36*x^4-78*x^3+60*x^2)*exp(x)^2+(-36*x^5+68*x^4-68*x^3)*exp(x)+162*x^6-225*x^5)/x^5,x, algorithm="fri
cas")

[Out]

1/81*(81*x^6 - 225*x^5 + (81*x^6 + 18*x^3 + 1)*e^(4*x) - 4*(9*x^4 + x)*e^(3*x) + 6*(27*x^6 - 51*x^5 + 3*x^3 -
5*x^2)*e^(2*x) - 4*(9*x^4 - 17*x^3)*e^x)/x^4

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giac [B]  time = 0.25, size = 104, normalized size = 3.35 \begin {gather*} \frac {81 \, x^{6} e^{\left (4 \, x\right )} + 162 \, x^{6} e^{\left (2 \, x\right )} + 81 \, x^{6} - 306 \, x^{5} e^{\left (2 \, x\right )} - 225 \, x^{5} - 36 \, x^{4} e^{\left (3 \, x\right )} - 36 \, x^{4} e^{x} + 18 \, x^{3} e^{\left (4 \, x\right )} + 18 \, x^{3} e^{\left (2 \, x\right )} + 68 \, x^{3} e^{x} - 30 \, x^{2} e^{\left (2 \, x\right )} - 4 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}}{81 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*((324*x^7+162*x^6+72*x^4-18*x^3+4*x-4)*exp(x)^4+(-108*x^5-12*x^2+12*x)*exp(x)^3+(324*x^7-288*x^
6-306*x^5+36*x^4-78*x^3+60*x^2)*exp(x)^2+(-36*x^5+68*x^4-68*x^3)*exp(x)+162*x^6-225*x^5)/x^5,x, algorithm="gia
c")

[Out]

1/81*(81*x^6*e^(4*x) + 162*x^6*e^(2*x) + 81*x^6 - 306*x^5*e^(2*x) - 225*x^5 - 36*x^4*e^(3*x) - 36*x^4*e^x + 18
*x^3*e^(4*x) + 18*x^3*e^(2*x) + 68*x^3*e^x - 30*x^2*e^(2*x) - 4*x*e^(3*x) + e^(4*x))/x^4

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maple [B]  time = 0.15, size = 81, normalized size = 2.61

method	result	size
risch	\(x^{2}-\frac {25 x}{9}+\frac {\left (81 x^{6}+18 x^{3}+1\right ) {\mathrm e}^{4 x}}{81 x^{4}}-\frac {4 \left (9 x^{3}+1\right ) {\mathrm e}^{3 x}}{81 x^{3}}+\frac {2 \left (27 x^{4}-51 x^{3}+3 x -5\right ) {\mathrm e}^{2 x}}{27 x^{2}}-\frac {4 \left (9 x -17\right ) {\mathrm e}^{x}}{81 x}\)	\(81\)
default	\(x^{2}-\frac {25 x}{9}-\frac {4 \,{\mathrm e}^{3 x}}{9}+\frac {{\mathrm e}^{4 x}}{81 x^{4}}-\frac {4 \,{\mathrm e}^{3 x}}{81 x^{3}}-\frac {10 \,{\mathrm e}^{2 x}}{27 x^{2}}+\frac {2 \,{\mathrm e}^{2 x}}{9 x}-\frac {34 x \,{\mathrm e}^{2 x}}{9}+x^{2} {\mathrm e}^{4 x}+\frac {68 \,{\mathrm e}^{x}}{81 x}+2 \,{\mathrm e}^{2 x} x^{2}+\frac {2 \,{\mathrm e}^{4 x}}{9 x}-\frac {4 \,{\mathrm e}^{x}}{9}\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/81*((324*x^7+162*x^6+72*x^4-18*x^3+4*x-4)*exp(x)^4+(-108*x^5-12*x^2+12*x)*exp(x)^3+(324*x^7-288*x^6-306*
x^5+36*x^4-78*x^3+60*x^2)*exp(x)^2+(-36*x^5+68*x^4-68*x^3)*exp(x)+162*x^6-225*x^5)/x^5,x,method=_RETURNVERBOSE
)

[Out]

x^2-25/9*x+1/81*(81*x^6+18*x^3+1)/x^4*exp(4*x)-4/81*(9*x^3+1)/x^3*exp(3*x)+2/27*(27*x^4-51*x^3+3*x-5)/x^2*exp(
2*x)-4/81*(9*x-17)/x*exp(x)

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maxima [C]  time = 0.45, size = 148, normalized size = 4.77 \begin {gather*} x^{2} + \frac {1}{8} \, {\left (8 \, x^{2} - 4 \, x + 1\right )} e^{\left (4 \, x\right )} + \frac {1}{8} \, {\left (4 \, x - 1\right )} e^{\left (4 \, x\right )} + {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} - \frac {8}{9} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - \frac {25}{9} \, x + \frac {8}{9} \, {\rm Ei}\left (4 \, x\right ) + \frac {4}{9} \, {\rm Ei}\left (2 \, x\right ) + \frac {68}{81} \, {\rm Ei}\relax (x) - \frac {4}{9} \, e^{\left (3 \, x\right )} - \frac {17}{9} \, e^{\left (2 \, x\right )} - \frac {4}{9} \, e^{x} - \frac {68}{81} \, \Gamma \left (-1, -x\right ) - \frac {52}{27} \, \Gamma \left (-1, -2 \, x\right ) - \frac {8}{9} \, \Gamma \left (-1, -4 \, x\right ) - \frac {80}{27} \, \Gamma \left (-2, -2 \, x\right ) + \frac {4}{3} \, \Gamma \left (-2, -3 \, x\right ) + 4 \, \Gamma \left (-3, -3 \, x\right ) + \frac {256}{81} \, \Gamma \left (-3, -4 \, x\right ) + \frac {1024}{81} \, \Gamma \left (-4, -4 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*((324*x^7+162*x^6+72*x^4-18*x^3+4*x-4)*exp(x)^4+(-108*x^5-12*x^2+12*x)*exp(x)^3+(324*x^7-288*x^
6-306*x^5+36*x^4-78*x^3+60*x^2)*exp(x)^2+(-36*x^5+68*x^4-68*x^3)*exp(x)+162*x^6-225*x^5)/x^5,x, algorithm="max
ima")

[Out]

x^2 + 1/8*(8*x^2 - 4*x + 1)*e^(4*x) + 1/8*(4*x - 1)*e^(4*x) + (2*x^2 - 2*x + 1)*e^(2*x) - 8/9*(2*x - 1)*e^(2*x
) - 25/9*x + 8/9*Ei(4*x) + 4/9*Ei(2*x) + 68/81*Ei(x) - 4/9*e^(3*x) - 17/9*e^(2*x) - 4/9*e^x - 68/81*gamma(-1,
-x) - 52/27*gamma(-1, -2*x) - 8/9*gamma(-1, -4*x) - 80/27*gamma(-2, -2*x) + 4/3*gamma(-2, -3*x) + 4*gamma(-3,
-3*x) + 256/81*gamma(-3, -4*x) + 1024/81*gamma(-4, -4*x)

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mupad [B]  time = 1.81, size = 86, normalized size = 2.77 \begin {gather*} x^2\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )-\frac {4\,{\mathrm {e}}^x}{9}-\frac {4\,{\mathrm {e}}^{3\,x}}{9}+\frac {\frac {{\mathrm {e}}^{4\,x}}{81}+x^3\,\left (\frac {2\,{\mathrm {e}}^{2\,x}}{9}+\frac {2\,{\mathrm {e}}^{4\,x}}{9}+\frac {68\,{\mathrm {e}}^x}{81}\right )-\frac {4\,x\,{\mathrm {e}}^{3\,x}}{81}-\frac {10\,x^2\,{\mathrm {e}}^{2\,x}}{27}}{x^4}-x\,\left (\frac {34\,{\mathrm {e}}^{2\,x}}{9}+\frac {25}{9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((exp(3*x)*(12*x^2 - 12*x + 108*x^5))/81 + (exp(x)*(68*x^3 - 68*x^4 + 36*x^5))/81 - (exp(2*x)*(60*x^2 - 7
8*x^3 + 36*x^4 - 306*x^5 - 288*x^6 + 324*x^7))/81 - (exp(4*x)*(4*x - 18*x^3 + 72*x^4 + 162*x^6 + 324*x^7 - 4))
/81 + (25*x^5)/9 - 2*x^6)/x^5,x)

[Out]

x^2*(2*exp(2*x) + exp(4*x) + 1) - (4*exp(x))/9 - (4*exp(3*x))/9 + (exp(4*x)/81 + x^3*((2*exp(2*x))/9 + (2*exp(
4*x))/9 + (68*exp(x))/81) - (4*x*exp(3*x))/81 - (10*x^2*exp(2*x))/27)/x^4 - x*((34*exp(2*x))/9 + 25/9)

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sympy [B]  time = 0.24, size = 88, normalized size = 2.84 \begin {gather*} x^{2} - \frac {25 x}{9} + \frac {\left (- 6377292 x^{10} - 708588 x^{7}\right ) e^{3 x} + \left (- 6377292 x^{10} + 12045996 x^{9}\right ) e^{x} + \left (14348907 x^{12} + 3188646 x^{9} + 177147 x^{6}\right ) e^{4 x} + \left (28697814 x^{12} - 54206982 x^{11} + 3188646 x^{9} - 5314410 x^{8}\right ) e^{2 x}}{14348907 x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*((324*x**7+162*x**6+72*x**4-18*x**3+4*x-4)*exp(x)**4+(-108*x**5-12*x**2+12*x)*exp(x)**3+(324*x*
*7-288*x**6-306*x**5+36*x**4-78*x**3+60*x**2)*exp(x)**2+(-36*x**5+68*x**4-68*x**3)*exp(x)+162*x**6-225*x**5)/x
**5,x)

[Out]

x**2 - 25*x/9 + ((-6377292*x**10 - 708588*x**7)*exp(3*x) + (-6377292*x**10 + 12045996*x**9)*exp(x) + (14348907
*x**12 + 3188646*x**9 + 177147*x**6)*exp(4*x) + (28697814*x**12 - 54206982*x**11 + 3188646*x**9 - 5314410*x**8
)*exp(2*x))/(14348907*x**10)

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