3.84.8 \(\int \frac {-6+12 x^3-80 x^4+2 x^5-71 x^7+808 x^8-132 x^9+164 x^{10}-10 x^{11}+6 x^{12}+e^{4 x} (32 x^8+64 x^9)+e^{3 x} (-96 x^7+96 x^9+96 x^{10})+e^{2 x} (-48 x^4+48 x^5-48 x^7+224 x^8+296 x^9+80 x^{10}+48 x^{11})+e^x (32 x^2-8 x^3-24 x^5+24 x^6-152 x^7-40 x^8+456 x^9+128 x^{10}+32 x^{11}+8 x^{12})}{9 x^7} \, dx\)

Optimal. Leaf size=30 \[ x+\frac {1}{9} \left (20+\left (-2 e^x+\frac {1}{x^2}-x\right )^2-x\right )^2 x^2 \]

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Rubi [B]  time = 0.79, antiderivative size = 235, normalized size of antiderivative = 7.83, number of steps used = 79, number of rules used = 8, integrand size = 190, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {12, 14, 2196, 2176, 2194, 2199, 2177, 2178} \begin {gather*} \frac {x^6}{9}+\frac {1}{9 x^6}+\frac {8 e^x x^5}{9}-\frac {2 x^5}{9}-\frac {8 e^x x^4}{9}+\frac {8}{3} e^{2 x} x^4+\frac {41 x^4}{9}-\frac {8 e^x}{9 x^4}+\frac {160 e^x x^3}{9}-\frac {8}{9} e^{2 x} x^3+\frac {32}{9} e^{3 x} x^3-\frac {44 x^3}{9}-\frac {4}{9 x^3}-\frac {8 e^x x^2}{3}+\frac {160}{9} e^{2 x} x^2+\frac {16}{9} e^{4 x} x^2+\frac {404 x^2}{9}+\frac {8 e^{2 x}}{3 x^2}+\frac {40}{9 x^2}+\frac {8 e^x x}{9}-\frac {16}{3} e^{2 x} x-\frac {71 x}{9}-\frac {160 e^x}{9}-\frac {32 e^{3 x}}{9}+\frac {8 e^x}{3 x}-\frac {2}{9 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6 + 12*x^3 - 80*x^4 + 2*x^5 - 71*x^7 + 808*x^8 - 132*x^9 + 164*x^10 - 10*x^11 + 6*x^12 + E^(4*x)*(32*x^8
+ 64*x^9) + E^(3*x)*(-96*x^7 + 96*x^9 + 96*x^10) + E^(2*x)*(-48*x^4 + 48*x^5 - 48*x^7 + 224*x^8 + 296*x^9 + 8
0*x^10 + 48*x^11) + E^x*(32*x^2 - 8*x^3 - 24*x^5 + 24*x^6 - 152*x^7 - 40*x^8 + 456*x^9 + 128*x^10 + 32*x^11 +
8*x^12))/(9*x^7),x]

[Out]

(-160*E^x)/9 - (32*E^(3*x))/9 + 1/(9*x^6) - (8*E^x)/(9*x^4) - 4/(9*x^3) + 40/(9*x^2) + (8*E^(2*x))/(3*x^2) - 2
/(9*x) + (8*E^x)/(3*x) - (71*x)/9 + (8*E^x*x)/9 - (16*E^(2*x)*x)/3 + (404*x^2)/9 - (8*E^x*x^2)/3 + (160*E^(2*x
)*x^2)/9 + (16*E^(4*x)*x^2)/9 - (44*x^3)/9 + (160*E^x*x^3)/9 - (8*E^(2*x)*x^3)/9 + (32*E^(3*x)*x^3)/9 + (41*x^
4)/9 - (8*E^x*x^4)/9 + (8*E^(2*x)*x^4)/3 - (2*x^5)/9 + (8*E^x*x^5)/9 + x^6/9

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 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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \frac {-6+12 x^3-80 x^4+2 x^5-71 x^7+808 x^8-132 x^9+164 x^{10}-10 x^{11}+6 x^{12}+e^{4 x} \left (32 x^8+64 x^9\right )+e^{3 x} \left (-96 x^7+96 x^9+96 x^{10}\right )+e^{2 x} \left (-48 x^4+48 x^5-48 x^7+224 x^8+296 x^9+80 x^{10}+48 x^{11}\right )+e^x \left (32 x^2-8 x^3-24 x^5+24 x^6-152 x^7-40 x^8+456 x^9+128 x^{10}+32 x^{11}+8 x^{12}\right )}{x^7} \, dx\\ &=\frac {1}{9} \int \left (32 e^{4 x} x (1+2 x)+96 e^{3 x} \left (-1+x^2+x^3\right )+\frac {8 e^{2 x} \left (-6+6 x-6 x^3+28 x^4+37 x^5+10 x^6+6 x^7\right )}{x^3}+\frac {8 e^x \left (4-x-3 x^3+3 x^4-19 x^5-5 x^6+57 x^7+16 x^8+4 x^9+x^{10}\right )}{x^5}+\frac {-6+12 x^3-80 x^4+2 x^5-71 x^7+808 x^8-132 x^9+164 x^{10}-10 x^{11}+6 x^{12}}{x^7}\right ) \, dx\\ &=\frac {1}{9} \int \frac {-6+12 x^3-80 x^4+2 x^5-71 x^7+808 x^8-132 x^9+164 x^{10}-10 x^{11}+6 x^{12}}{x^7} \, dx+\frac {8}{9} \int \frac {e^{2 x} \left (-6+6 x-6 x^3+28 x^4+37 x^5+10 x^6+6 x^7\right )}{x^3} \, dx+\frac {8}{9} \int \frac {e^x \left (4-x-3 x^3+3 x^4-19 x^5-5 x^6+57 x^7+16 x^8+4 x^9+x^{10}\right )}{x^5} \, dx+\frac {32}{9} \int e^{4 x} x (1+2 x) \, dx+\frac {32}{3} \int e^{3 x} \left (-1+x^2+x^3\right ) \, dx\\ &=\frac {1}{9} \int \left (-71-\frac {6}{x^7}+\frac {12}{x^4}-\frac {80}{x^3}+\frac {2}{x^2}+808 x-132 x^2+164 x^3-10 x^4+6 x^5\right ) \, dx+\frac {8}{9} \int \left (-6 e^{2 x}-\frac {6 e^{2 x}}{x^3}+\frac {6 e^{2 x}}{x^2}+28 e^{2 x} x+37 e^{2 x} x^2+10 e^{2 x} x^3+6 e^{2 x} x^4\right ) \, dx+\frac {8}{9} \int \left (-19 e^x+\frac {4 e^x}{x^5}-\frac {e^x}{x^4}-\frac {3 e^x}{x^2}+\frac {3 e^x}{x}-5 e^x x+57 e^x x^2+16 e^x x^3+4 e^x x^4+e^x x^5\right ) \, dx+\frac {32}{9} \int \left (e^{4 x} x+2 e^{4 x} x^2\right ) \, dx+\frac {32}{3} \int \left (-e^{3 x}+e^{3 x} x^2+e^{3 x} x^3\right ) \, dx\\ &=\frac {1}{9 x^6}-\frac {4}{9 x^3}+\frac {40}{9 x^2}-\frac {2}{9 x}-\frac {71 x}{9}+\frac {404 x^2}{9}-\frac {44 x^3}{9}+\frac {41 x^4}{9}-\frac {2 x^5}{9}+\frac {x^6}{9}-\frac {8}{9} \int \frac {e^x}{x^4} \, dx+\frac {8}{9} \int e^x x^5 \, dx-\frac {8}{3} \int \frac {e^x}{x^2} \, dx+\frac {8}{3} \int \frac {e^x}{x} \, dx+\frac {32}{9} \int \frac {e^x}{x^5} \, dx+\frac {32}{9} \int e^{4 x} x \, dx+\frac {32}{9} \int e^x x^4 \, dx-\frac {40}{9} \int e^x x \, dx-\frac {16}{3} \int e^{2 x} \, dx-\frac {16}{3} \int \frac {e^{2 x}}{x^3} \, dx+\frac {16}{3} \int \frac {e^{2 x}}{x^2} \, dx+\frac {16}{3} \int e^{2 x} x^4 \, dx+\frac {64}{9} \int e^{4 x} x^2 \, dx+\frac {80}{9} \int e^{2 x} x^3 \, dx-\frac {32}{3} \int e^{3 x} \, dx+\frac {32}{3} \int e^{3 x} x^2 \, dx+\frac {32}{3} \int e^{3 x} x^3 \, dx+\frac {128}{9} \int e^x x^3 \, dx-\frac {152 \int e^x \, dx}{9}+\frac {224}{9} \int e^{2 x} x \, dx+\frac {296}{9} \int e^{2 x} x^2 \, dx+\frac {152}{3} \int e^x x^2 \, dx\\ &=-\frac {152 e^x}{9}-\frac {8 e^{2 x}}{3}-\frac {32 e^{3 x}}{9}+\frac {1}{9 x^6}-\frac {8 e^x}{9 x^4}-\frac {4}{9 x^3}+\frac {8 e^x}{27 x^3}+\frac {40}{9 x^2}+\frac {8 e^{2 x}}{3 x^2}-\frac {2}{9 x}+\frac {8 e^x}{3 x}-\frac {16 e^{2 x}}{3 x}-\frac {71 x}{9}-\frac {40 e^x x}{9}+\frac {112}{9} e^{2 x} x+\frac {8}{9} e^{4 x} x+\frac {404 x^2}{9}+\frac {152 e^x x^2}{3}+\frac {148}{9} e^{2 x} x^2+\frac {32}{9} e^{3 x} x^2+\frac {16}{9} e^{4 x} x^2-\frac {44 x^3}{9}+\frac {128 e^x x^3}{9}+\frac {40}{9} e^{2 x} x^3+\frac {32}{9} e^{3 x} x^3+\frac {41 x^4}{9}+\frac {32 e^x x^4}{9}+\frac {8}{3} e^{2 x} x^4-\frac {2 x^5}{9}+\frac {8 e^x x^5}{9}+\frac {x^6}{9}+\frac {8 \text {Ei}(x)}{3}-\frac {8}{27} \int \frac {e^x}{x^3} \, dx-\frac {8}{9} \int e^{4 x} \, dx+\frac {8}{9} \int \frac {e^x}{x^4} \, dx-\frac {8}{3} \int \frac {e^x}{x} \, dx-\frac {32}{9} \int e^{4 x} x \, dx+\frac {40 \int e^x \, dx}{9}-\frac {40}{9} \int e^x x^4 \, dx-\frac {16}{3} \int \frac {e^{2 x}}{x^2} \, dx-\frac {64}{9} \int e^{3 x} x \, dx+\frac {32}{3} \int \frac {e^{2 x}}{x} \, dx-\frac {32}{3} \int e^{3 x} x^2 \, dx-\frac {32}{3} \int e^{2 x} x^3 \, dx-\frac {112}{9} \int e^{2 x} \, dx-\frac {40}{3} \int e^{2 x} x^2 \, dx-\frac {128}{9} \int e^x x^3 \, dx-\frac {296}{9} \int e^{2 x} x \, dx-\frac {128}{3} \int e^x x^2 \, dx-\frac {304}{3} \int e^x x \, dx\\ &=-\frac {112 e^x}{9}-\frac {80 e^{2 x}}{9}-\frac {32 e^{3 x}}{9}-\frac {2 e^{4 x}}{9}+\frac {1}{9 x^6}-\frac {8 e^x}{9 x^4}-\frac {4}{9 x^3}+\frac {40}{9 x^2}+\frac {4 e^x}{27 x^2}+\frac {8 e^{2 x}}{3 x^2}-\frac {2}{9 x}+\frac {8 e^x}{3 x}-\frac {71 x}{9}-\frac {952 e^x x}{9}-4 e^{2 x} x-\frac {64}{27} e^{3 x} x+\frac {404 x^2}{9}+8 e^x x^2+\frac {88}{9} e^{2 x} x^2+\frac {16}{9} e^{4 x} x^2-\frac {44 x^3}{9}-\frac {8}{9} e^{2 x} x^3+\frac {32}{9} e^{3 x} x^3+\frac {41 x^4}{9}-\frac {8 e^x x^4}{9}+\frac {8}{3} e^{2 x} x^4-\frac {2 x^5}{9}+\frac {8 e^x x^5}{9}+\frac {x^6}{9}+\frac {32 \text {Ei}(2 x)}{3}-\frac {4}{27} \int \frac {e^x}{x^2} \, dx+\frac {8}{27} \int \frac {e^x}{x^3} \, dx+\frac {8}{9} \int e^{4 x} \, dx+\frac {64}{27} \int e^{3 x} \, dx+\frac {64}{9} \int e^{3 x} x \, dx-\frac {32}{3} \int \frac {e^{2 x}}{x} \, dx+\frac {40}{3} \int e^{2 x} x \, dx+16 \int e^{2 x} x^2 \, dx+\frac {148}{9} \int e^{2 x} \, dx+\frac {160}{9} \int e^x x^3 \, dx+\frac {128}{3} \int e^x x^2 \, dx+\frac {256}{3} \int e^x x \, dx+\frac {304 \int e^x \, dx}{3}\\ &=\frac {800 e^x}{9}-\frac {2 e^{2 x}}{3}-\frac {224 e^{3 x}}{81}+\frac {1}{9 x^6}-\frac {8 e^x}{9 x^4}-\frac {4}{9 x^3}+\frac {40}{9 x^2}+\frac {8 e^{2 x}}{3 x^2}-\frac {2}{9 x}+\frac {76 e^x}{27 x}-\frac {71 x}{9}-\frac {184 e^x x}{9}+\frac {8}{3} e^{2 x} x+\frac {404 x^2}{9}+\frac {152 e^x x^2}{3}+\frac {160}{9} e^{2 x} x^2+\frac {16}{9} e^{4 x} x^2-\frac {44 x^3}{9}+\frac {160 e^x x^3}{9}-\frac {8}{9} e^{2 x} x^3+\frac {32}{9} e^{3 x} x^3+\frac {41 x^4}{9}-\frac {8 e^x x^4}{9}+\frac {8}{3} e^{2 x} x^4-\frac {2 x^5}{9}+\frac {8 e^x x^5}{9}+\frac {x^6}{9}+\frac {4}{27} \int \frac {e^x}{x^2} \, dx-\frac {4}{27} \int \frac {e^x}{x} \, dx-\frac {64}{27} \int e^{3 x} \, dx-\frac {20}{3} \int e^{2 x} \, dx-16 \int e^{2 x} x \, dx-\frac {160}{3} \int e^x x^2 \, dx-\frac {256 \int e^x \, dx}{3}-\frac {256}{3} \int e^x x \, dx\\ &=\frac {32 e^x}{9}-4 e^{2 x}-\frac {32 e^{3 x}}{9}+\frac {1}{9 x^6}-\frac {8 e^x}{9 x^4}-\frac {4}{9 x^3}+\frac {40}{9 x^2}+\frac {8 e^{2 x}}{3 x^2}-\frac {2}{9 x}+\frac {8 e^x}{3 x}-\frac {71 x}{9}-\frac {952 e^x x}{9}-\frac {16}{3} e^{2 x} x+\frac {404 x^2}{9}-\frac {8 e^x x^2}{3}+\frac {160}{9} e^{2 x} x^2+\frac {16}{9} e^{4 x} x^2-\frac {44 x^3}{9}+\frac {160 e^x x^3}{9}-\frac {8}{9} e^{2 x} x^3+\frac {32}{9} e^{3 x} x^3+\frac {41 x^4}{9}-\frac {8 e^x x^4}{9}+\frac {8}{3} e^{2 x} x^4-\frac {2 x^5}{9}+\frac {8 e^x x^5}{9}+\frac {x^6}{9}-\frac {4 \text {Ei}(x)}{27}+\frac {4}{27} \int \frac {e^x}{x} \, dx+8 \int e^{2 x} \, dx+\frac {256 \int e^x \, dx}{3}+\frac {320}{3} \int e^x x \, dx\\ &=\frac {800 e^x}{9}-\frac {32 e^{3 x}}{9}+\frac {1}{9 x^6}-\frac {8 e^x}{9 x^4}-\frac {4}{9 x^3}+\frac {40}{9 x^2}+\frac {8 e^{2 x}}{3 x^2}-\frac {2}{9 x}+\frac {8 e^x}{3 x}-\frac {71 x}{9}+\frac {8 e^x x}{9}-\frac {16}{3} e^{2 x} x+\frac {404 x^2}{9}-\frac {8 e^x x^2}{3}+\frac {160}{9} e^{2 x} x^2+\frac {16}{9} e^{4 x} x^2-\frac {44 x^3}{9}+\frac {160 e^x x^3}{9}-\frac {8}{9} e^{2 x} x^3+\frac {32}{9} e^{3 x} x^3+\frac {41 x^4}{9}-\frac {8 e^x x^4}{9}+\frac {8}{3} e^{2 x} x^4-\frac {2 x^5}{9}+\frac {8 e^x x^5}{9}+\frac {x^6}{9}-\frac {320 \int e^x \, dx}{3}\\ &=-\frac {160 e^x}{9}-\frac {32 e^{3 x}}{9}+\frac {1}{9 x^6}-\frac {8 e^x}{9 x^4}-\frac {4}{9 x^3}+\frac {40}{9 x^2}+\frac {8 e^{2 x}}{3 x^2}-\frac {2}{9 x}+\frac {8 e^x}{3 x}-\frac {71 x}{9}+\frac {8 e^x x}{9}-\frac {16}{3} e^{2 x} x+\frac {404 x^2}{9}-\frac {8 e^x x^2}{3}+\frac {160}{9} e^{2 x} x^2+\frac {16}{9} e^{4 x} x^2-\frac {44 x^3}{9}+\frac {160 e^x x^3}{9}-\frac {8}{9} e^{2 x} x^3+\frac {32}{9} e^{3 x} x^3+\frac {41 x^4}{9}-\frac {8 e^x x^4}{9}+\frac {8}{3} e^{2 x} x^4-\frac {2 x^5}{9}+\frac {8 e^x x^5}{9}+\frac {x^6}{9}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.10, size = 141, normalized size = 4.70 \begin {gather*} \frac {1}{9} \left (\frac {1}{x^6}-\frac {4}{x^3}+\frac {40}{x^2}-\frac {2}{x}-71 x+404 x^2+16 e^{4 x} x^2-44 x^3+41 x^4-2 x^5+x^6+e^{3 x} \left (-32+32 x^3\right )+e^{2 x} \left (\frac {24}{x^2}-48 x+160 x^2-8 x^3+24 x^4\right )+e^x \left (-160-\frac {8}{x^4}+\frac {24}{x}+8 x-24 x^2+160 x^3-8 x^4+8 x^5\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 + 12*x^3 - 80*x^4 + 2*x^5 - 71*x^7 + 808*x^8 - 132*x^9 + 164*x^10 - 10*x^11 + 6*x^12 + E^(4*x)*(
32*x^8 + 64*x^9) + E^(3*x)*(-96*x^7 + 96*x^9 + 96*x^10) + E^(2*x)*(-48*x^4 + 48*x^5 - 48*x^7 + 224*x^8 + 296*x
^9 + 80*x^10 + 48*x^11) + E^x*(32*x^2 - 8*x^3 - 24*x^5 + 24*x^6 - 152*x^7 - 40*x^8 + 456*x^9 + 128*x^10 + 32*x
^11 + 8*x^12))/(9*x^7),x]

[Out]

(x^(-6) - 4/x^3 + 40/x^2 - 2/x - 71*x + 404*x^2 + 16*E^(4*x)*x^2 - 44*x^3 + 41*x^4 - 2*x^5 + x^6 + E^(3*x)*(-3
2 + 32*x^3) + E^(2*x)*(24/x^2 - 48*x + 160*x^2 - 8*x^3 + 24*x^4) + E^x*(-160 - 8/x^4 + 24/x + 8*x - 24*x^2 + 1
60*x^3 - 8*x^4 + 8*x^5))/9

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fricas [B]  time = 1.07, size = 147, normalized size = 4.90 \begin {gather*} \frac {x^{12} - 2 \, x^{11} + 41 \, x^{10} - 44 \, x^{9} + 16 \, x^{8} e^{\left (4 \, x\right )} + 404 \, x^{8} - 71 \, x^{7} - 2 \, x^{5} + 40 \, x^{4} - 4 \, x^{3} + 32 \, {\left (x^{9} - x^{6}\right )} e^{\left (3 \, x\right )} + 8 \, {\left (3 \, x^{10} - x^{9} + 20 \, x^{8} - 6 \, x^{7} + 3 \, x^{4}\right )} e^{\left (2 \, x\right )} + 8 \, {\left (x^{11} - x^{10} + 20 \, x^{9} - 3 \, x^{8} + x^{7} - 20 \, x^{6} + 3 \, x^{5} - x^{2}\right )} e^{x} + 1}{9 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((64*x^9+32*x^8)*exp(x)^4+(96*x^10+96*x^9-96*x^7)*exp(x)^3+(48*x^11+80*x^10+296*x^9+224*x^8-48*x
^7+48*x^5-48*x^4)*exp(x)^2+(8*x^12+32*x^11+128*x^10+456*x^9-40*x^8-152*x^7+24*x^6-24*x^5-8*x^3+32*x^2)*exp(x)+
6*x^12-10*x^11+164*x^10-132*x^9+808*x^8-71*x^7+2*x^5-80*x^4+12*x^3-6)/x^7,x, algorithm="fricas")

[Out]

1/9*(x^12 - 2*x^11 + 41*x^10 - 44*x^9 + 16*x^8*e^(4*x) + 404*x^8 - 71*x^7 - 2*x^5 + 40*x^4 - 4*x^3 + 32*(x^9 -
x^6)*e^(3*x) + 8*(3*x^10 - x^9 + 20*x^8 - 6*x^7 + 3*x^4)*e^(2*x) + 8*(x^11 - x^10 + 20*x^9 - 3*x^8 + x^7 - 20
*x^6 + 3*x^5 - x^2)*e^x + 1)/x^6

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giac [B]  time = 0.17, size = 178, normalized size = 5.93 \begin {gather*} \frac {x^{12} + 8 \, x^{11} e^{x} - 2 \, x^{11} + 24 \, x^{10} e^{\left (2 \, x\right )} - 8 \, x^{10} e^{x} + 41 \, x^{10} + 32 \, x^{9} e^{\left (3 \, x\right )} - 8 \, x^{9} e^{\left (2 \, x\right )} + 160 \, x^{9} e^{x} - 44 \, x^{9} + 16 \, x^{8} e^{\left (4 \, x\right )} + 160 \, x^{8} e^{\left (2 \, x\right )} - 24 \, x^{8} e^{x} + 404 \, x^{8} - 48 \, x^{7} e^{\left (2 \, x\right )} + 8 \, x^{7} e^{x} - 71 \, x^{7} - 32 \, x^{6} e^{\left (3 \, x\right )} - 160 \, x^{6} e^{x} + 24 \, x^{5} e^{x} - 2 \, x^{5} + 24 \, x^{4} e^{\left (2 \, x\right )} + 40 \, x^{4} - 4 \, x^{3} - 8 \, x^{2} e^{x} + 1}{9 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((64*x^9+32*x^8)*exp(x)^4+(96*x^10+96*x^9-96*x^7)*exp(x)^3+(48*x^11+80*x^10+296*x^9+224*x^8-48*x
^7+48*x^5-48*x^4)*exp(x)^2+(8*x^12+32*x^11+128*x^10+456*x^9-40*x^8-152*x^7+24*x^6-24*x^5-8*x^3+32*x^2)*exp(x)+
6*x^12-10*x^11+164*x^10-132*x^9+808*x^8-71*x^7+2*x^5-80*x^4+12*x^3-6)/x^7,x, algorithm="giac")

[Out]

1/9*(x^12 + 8*x^11*e^x - 2*x^11 + 24*x^10*e^(2*x) - 8*x^10*e^x + 41*x^10 + 32*x^9*e^(3*x) - 8*x^9*e^(2*x) + 16
0*x^9*e^x - 44*x^9 + 16*x^8*e^(4*x) + 160*x^8*e^(2*x) - 24*x^8*e^x + 404*x^8 - 48*x^7*e^(2*x) + 8*x^7*e^x - 71
*x^7 - 32*x^6*e^(3*x) - 160*x^6*e^x + 24*x^5*e^x - 2*x^5 + 24*x^4*e^(2*x) + 40*x^4 - 4*x^3 - 8*x^2*e^x + 1)/x^
6

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maple [B]  time = 0.13, size = 145, normalized size = 4.83

method	result	size
risch	\(\frac {x^{6}}{9}-\frac {2 x^{5}}{9}+\frac {41 x^{4}}{9}-\frac {44 x^{3}}{9}+\frac {404 x^{2}}{9}-\frac {71 x}{9}+\frac {-2 x^{5}+40 x^{4}-4 x^{3}+1}{9 x^{6}}+\frac {16 x^{2} {\mathrm e}^{4 x}}{9}+\frac {\left (32 x^{3}-32\right ) {\mathrm e}^{3 x}}{9}+\frac {8 \left (3 x^{6}-x^{5}+20 x^{4}-6 x^{3}+3\right ) {\mathrm e}^{2 x}}{9 x^{2}}+\frac {8 \left (x^{9}-x^{8}+20 x^{7}-3 x^{6}+x^{5}-20 x^{4}+3 x^{3}-1\right ) {\mathrm e}^{x}}{9 x^{4}}\)	\(145\)
default	\(-\frac {71 x}{9}+\frac {32 x^{3} {\mathrm e}^{3 x}}{9}+\frac {8 x^{5} {\mathrm e}^{x}}{9}-\frac {32 \,{\mathrm e}^{3 x}}{9}+\frac {x^{6}}{9}-\frac {2 x^{5}}{9}+\frac {41 x^{4}}{9}-\frac {44 x^{3}}{9}+\frac {404 x^{2}}{9}-\frac {160 \,{\mathrm e}^{x}}{9}-\frac {8 \,{\mathrm e}^{2 x} x^{3}}{9}+\frac {1}{9 x^{6}}+\frac {8 \,{\mathrm e}^{x}}{3 x}+\frac {160 \,{\mathrm e}^{2 x} x^{2}}{9}-\frac {16 x \,{\mathrm e}^{2 x}}{3}-\frac {8 \,{\mathrm e}^{x} x^{4}}{9}-\frac {8 \,{\mathrm e}^{x} x^{2}}{3}+\frac {160 \,{\mathrm e}^{x} x^{3}}{9}+\frac {8 \,{\mathrm e}^{x} x}{9}+\frac {40}{9 x^{2}}-\frac {2}{9 x}-\frac {8 \,{\mathrm e}^{x}}{9 x^{4}}-\frac {4}{9 x^{3}}+\frac {16 x^{2} {\mathrm e}^{4 x}}{9}+\frac {8 \,{\mathrm e}^{2 x}}{3 x^{2}}+\frac {8 \,{\mathrm e}^{2 x} x^{4}}{3}\)	\(168\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*((64*x^9+32*x^8)*exp(x)^4+(96*x^10+96*x^9-96*x^7)*exp(x)^3+(48*x^11+80*x^10+296*x^9+224*x^8-48*x^7+48*
x^5-48*x^4)*exp(x)^2+(8*x^12+32*x^11+128*x^10+456*x^9-40*x^8-152*x^7+24*x^6-24*x^5-8*x^3+32*x^2)*exp(x)+6*x^12
-10*x^11+164*x^10-132*x^9+808*x^8-71*x^7+2*x^5-80*x^4+12*x^3-6)/x^7,x,method=_RETURNVERBOSE)

[Out]

1/9*x^6-2/9*x^5+41/9*x^4-44/9*x^3+404/9*x^2-71/9*x+1/9*(-2*x^5+40*x^4-4*x^3+1)/x^6+16/9*x^2*exp(4*x)+1/9*(32*x
^3-32)*exp(3*x)+8/9*(3*x^6-x^5+20*x^4-6*x^3+3)/x^2*exp(2*x)+8/9*(x^9-x^8+20*x^7-3*x^6+x^5-20*x^4+3*x^3-1)/x^4*
exp(x)

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maxima [C]  time = 0.43, size = 327, normalized size = 10.90 \begin {gather*} \frac {1}{9} \, x^{6} - \frac {2}{9} \, x^{5} + \frac {41}{9} \, x^{4} - \frac {44}{9} \, x^{3} + \frac {404}{9} \, x^{2} + \frac {2}{9} \, {\left (8 \, x^{2} - 4 \, x + 1\right )} e^{\left (4 \, x\right )} + \frac {2}{9} \, {\left (4 \, x - 1\right )} e^{\left (4 \, x\right )} + \frac {32}{81} \, {\left (9 \, x^{3} - 9 \, x^{2} + 6 \, x - 2\right )} e^{\left (3 \, x\right )} + \frac {32}{81} \, {\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} + \frac {4}{3} \, {\left (2 \, x^{4} - 4 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} e^{\left (2 \, x\right )} + \frac {10}{9} \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} + \frac {74}{9} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {56}{9} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + \frac {8}{9} \, {\left (x^{5} - 5 \, x^{4} + 20 \, x^{3} - 60 \, x^{2} + 120 \, x - 120\right )} e^{x} + \frac {32}{9} \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} + \frac {128}{9} \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} + \frac {152}{3} \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} - \frac {40}{9} \, {\left (x - 1\right )} e^{x} - \frac {71}{9} \, x - \frac {2}{9 \, x} + \frac {40}{9 \, x^{2}} - \frac {4}{9 \, x^{3}} + \frac {1}{9 \, x^{6}} + \frac {8}{3} \, {\rm Ei}\relax (x) - \frac {32}{9} \, e^{\left (3 \, x\right )} - \frac {8}{3} \, e^{\left (2 \, x\right )} - \frac {152}{9} \, e^{x} - \frac {8}{3} \, \Gamma \left (-1, -x\right ) + \frac {32}{3} \, \Gamma \left (-1, -2 \, x\right ) + \frac {64}{3} \, \Gamma \left (-2, -2 \, x\right ) - \frac {8}{9} \, \Gamma \left (-3, -x\right ) - \frac {32}{9} \, \Gamma \left (-4, -x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((64*x^9+32*x^8)*exp(x)^4+(96*x^10+96*x^9-96*x^7)*exp(x)^3+(48*x^11+80*x^10+296*x^9+224*x^8-48*x
^7+48*x^5-48*x^4)*exp(x)^2+(8*x^12+32*x^11+128*x^10+456*x^9-40*x^8-152*x^7+24*x^6-24*x^5-8*x^3+32*x^2)*exp(x)+
6*x^12-10*x^11+164*x^10-132*x^9+808*x^8-71*x^7+2*x^5-80*x^4+12*x^3-6)/x^7,x, algorithm="maxima")

[Out]

1/9*x^6 - 2/9*x^5 + 41/9*x^4 - 44/9*x^3 + 404/9*x^2 + 2/9*(8*x^2 - 4*x + 1)*e^(4*x) + 2/9*(4*x - 1)*e^(4*x) +
32/81*(9*x^3 - 9*x^2 + 6*x - 2)*e^(3*x) + 32/81*(9*x^2 - 6*x + 2)*e^(3*x) + 4/3*(2*x^4 - 4*x^3 + 6*x^2 - 6*x +
3)*e^(2*x) + 10/9*(4*x^3 - 6*x^2 + 6*x - 3)*e^(2*x) + 74/9*(2*x^2 - 2*x + 1)*e^(2*x) + 56/9*(2*x - 1)*e^(2*x)
+ 8/9*(x^5 - 5*x^4 + 20*x^3 - 60*x^2 + 120*x - 120)*e^x + 32/9*(x^4 - 4*x^3 + 12*x^2 - 24*x + 24)*e^x + 128/9
*(x^3 - 3*x^2 + 6*x - 6)*e^x + 152/3*(x^2 - 2*x + 2)*e^x - 40/9*(x - 1)*e^x - 71/9*x - 2/9/x + 40/9/x^2 - 4/9/
x^3 + 1/9/x^6 + 8/3*Ei(x) - 32/9*e^(3*x) - 8/3*e^(2*x) - 152/9*e^x - 8/3*gamma(-1, -x) + 32/3*gamma(-1, -2*x)
+ 64/3*gamma(-2, -2*x) - 8/9*gamma(-3, -x) - 32/9*gamma(-4, -x)

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mupad [B]  time = 0.22, size = 142, normalized size = 4.73 \begin {gather*} x^2\,\left (\frac {160\,{\mathrm {e}}^{2\,x}}{9}+\frac {16\,{\mathrm {e}}^{4\,x}}{9}-\frac {8\,{\mathrm {e}}^x}{3}+\frac {404}{9}\right )-\frac {160\,{\mathrm {e}}^x}{9}-x^3\,\left (\frac {8\,{\mathrm {e}}^{2\,x}}{9}-\frac {32\,{\mathrm {e}}^{3\,x}}{9}-\frac {160\,{\mathrm {e}}^x}{9}+\frac {44}{9}\right )-\frac {32\,{\mathrm {e}}^{3\,x}}{9}-x\,\left (\frac {16\,{\mathrm {e}}^{2\,x}}{3}-\frac {8\,{\mathrm {e}}^x}{9}+\frac {71}{9}\right )+x^4\,\left (\frac {8\,{\mathrm {e}}^{2\,x}}{3}-\frac {8\,{\mathrm {e}}^x}{9}+\frac {41}{9}\right )+x^5\,\left (\frac {8\,{\mathrm {e}}^x}{9}-\frac {2}{9}\right )+\frac {x^6}{9}+\frac {x^4\,\left (\frac {8\,{\mathrm {e}}^{2\,x}}{3}+\frac {40}{9}\right )-\frac {8\,x^2\,{\mathrm {e}}^x}{9}+x^5\,\left (\frac {8\,{\mathrm {e}}^x}{3}-\frac {2}{9}\right )-\frac {4\,x^3}{9}+\frac {1}{9}}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(4*x)*(32*x^8 + 64*x^9))/9 + (exp(2*x)*(48*x^5 - 48*x^4 - 48*x^7 + 224*x^8 + 296*x^9 + 80*x^10 + 48*x
^11))/9 + (exp(3*x)*(96*x^9 - 96*x^7 + 96*x^10))/9 + (exp(x)*(32*x^2 - 8*x^3 - 24*x^5 + 24*x^6 - 152*x^7 - 40*
x^8 + 456*x^9 + 128*x^10 + 32*x^11 + 8*x^12))/9 + (4*x^3)/3 - (80*x^4)/9 + (2*x^5)/9 - (71*x^7)/9 + (808*x^8)/
9 - (44*x^9)/3 + (164*x^10)/9 - (10*x^11)/9 + (2*x^12)/3 - 2/3)/x^7,x)

[Out]

x^2*((160*exp(2*x))/9 + (16*exp(4*x))/9 - (8*exp(x))/3 + 404/9) - (160*exp(x))/9 - x^3*((8*exp(2*x))/9 - (32*e
xp(3*x))/9 - (160*exp(x))/9 + 44/9) - (32*exp(3*x))/9 - x*((16*exp(2*x))/3 - (8*exp(x))/9 + 71/9) + x^4*((8*ex
p(2*x))/3 - (8*exp(x))/9 + 41/9) + x^5*((8*exp(x))/9 - 2/9) + x^6/9 + (x^4*((8*exp(2*x))/3 + 40/9) - (8*x^2*ex
p(x))/9 + x^5*((8*exp(x))/3 - 2/9) - (4*x^3)/9 + 1/9)/x^6

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sympy [B]  time = 0.33, size = 163, normalized size = 5.43 \begin {gather*} \frac {x^{6}}{9} - \frac {2 x^{5}}{9} + \frac {41 x^{4}}{9} - \frac {44 x^{3}}{9} + \frac {404 x^{2}}{9} - \frac {71 x}{9} + \frac {- 2 x^{5} + 40 x^{4} - 4 x^{3} + 1}{9 x^{6}} + \frac {11664 x^{8} e^{4 x} + \left (23328 x^{9} - 23328 x^{6}\right ) e^{3 x} + \left (17496 x^{10} - 5832 x^{9} + 116640 x^{8} - 34992 x^{7} + 17496 x^{4}\right ) e^{2 x} + \left (5832 x^{11} - 5832 x^{10} + 116640 x^{9} - 17496 x^{8} + 5832 x^{7} - 116640 x^{6} + 17496 x^{5} - 5832 x^{2}\right ) e^{x}}{6561 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((64*x**9+32*x**8)*exp(x)**4+(96*x**10+96*x**9-96*x**7)*exp(x)**3+(48*x**11+80*x**10+296*x**9+22
4*x**8-48*x**7+48*x**5-48*x**4)*exp(x)**2+(8*x**12+32*x**11+128*x**10+456*x**9-40*x**8-152*x**7+24*x**6-24*x**
5-8*x**3+32*x**2)*exp(x)+6*x**12-10*x**11+164*x**10-132*x**9+808*x**8-71*x**7+2*x**5-80*x**4+12*x**3-6)/x**7,x
)

[Out]

x**6/9 - 2*x**5/9 + 41*x**4/9 - 44*x**3/9 + 404*x**2/9 - 71*x/9 + (-2*x**5 + 40*x**4 - 4*x**3 + 1)/(9*x**6) +
(11664*x**8*exp(4*x) + (23328*x**9 - 23328*x**6)*exp(3*x) + (17496*x**10 - 5832*x**9 + 116640*x**8 - 34992*x**
7 + 17496*x**4)*exp(2*x) + (5832*x**11 - 5832*x**10 + 116640*x**9 - 17496*x**8 + 5832*x**7 - 116640*x**6 + 174
96*x**5 - 5832*x**2)*exp(x))/(6561*x**6)

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