3.45.36 \(\int \frac {-3-96 x^2-24 x^3-720 x^4-288 x^5+2880 x^7+7488 x^8+21120 x^9+23040 x^{10}+7680 x^{11}+768 x^{12}+e^{2 x} (64 x^8+64 x^9)+e^x (16 x^4-8 x^5-256 x^7-1216 x^8-2880 x^9-1280 x^{10}-128 x^{11})}{32 x^7} \, dx\)

Optimal. Leaf size=25 \[ x^2 \left (9+e^x-2 \left (3+\frac {1}{4 x^2}+x\right )^2\right )^2 \]

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Rubi [B]  time = 0.35, antiderivative size = 116, normalized size of antiderivative = 4.64, number of steps used = 34, number of rules used = 8, integrand size = 116, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {12, 14, 2196, 2176, 2194, 2199, 2177, 2178} \begin {gather*} 4 x^6+\frac {1}{64 x^6}+48 x^5-4 e^x x^4+180 x^4+\frac {3}{4 x^4}-24 e^x x^3+220 x^3+\frac {1}{4 x^3}-18 e^x x^2+e^{2 x} x^2+117 x^2-\frac {e^x}{4 x^2}+\frac {45}{4 x^2}-2 e^x x+90 x-6 e^x+\frac {9}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 - 96*x^2 - 24*x^3 - 720*x^4 - 288*x^5 + 2880*x^7 + 7488*x^8 + 21120*x^9 + 23040*x^10 + 7680*x^11 + 768
*x^12 + E^(2*x)*(64*x^8 + 64*x^9) + E^x*(16*x^4 - 8*x^5 - 256*x^7 - 1216*x^8 - 2880*x^9 - 1280*x^10 - 128*x^11
))/(32*x^7),x]

[Out]

-6*E^x + 1/(64*x^6) + 3/(4*x^4) + 1/(4*x^3) + 45/(4*x^2) - E^x/(4*x^2) + 9/x + 90*x - 2*E^x*x + 117*x^2 - 18*E
^x*x^2 + E^(2*x)*x^2 + 220*x^3 - 24*E^x*x^3 + 180*x^4 - 4*E^x*x^4 + 48*x^5 + 4*x^6

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}{32} \int \frac {-3-96 x^2-24 x^3-720 x^4-288 x^5+2880 x^7+7488 x^8+21120 x^9+23040 x^{10}+7680 x^{11}+768 x^{12}+e^{2 x} \left (64 x^8+64 x^9\right )+e^x \left (16 x^4-8 x^5-256 x^7-1216 x^8-2880 x^9-1280 x^{10}-128 x^{11}\right )}{x^7} \, dx\\ &=\frac {1}{32} \int \left (64 e^{2 x} x (1+x)-\frac {8 e^x \left (-2+x+32 x^3+152 x^4+360 x^5+160 x^6+16 x^7\right )}{x^3}+\frac {3 \left (-1-32 x^2-8 x^3-240 x^4-96 x^5+960 x^7+2496 x^8+7040 x^9+7680 x^{10}+2560 x^{11}+256 x^{12}\right )}{x^7}\right ) \, dx\\ &=\frac {3}{32} \int \frac {-1-32 x^2-8 x^3-240 x^4-96 x^5+960 x^7+2496 x^8+7040 x^9+7680 x^{10}+2560 x^{11}+256 x^{12}}{x^7} \, dx-\frac {1}{4} \int \frac {e^x \left (-2+x+32 x^3+152 x^4+360 x^5+160 x^6+16 x^7\right )}{x^3} \, dx+2 \int e^{2 x} x (1+x) \, dx\\ &=\frac {3}{32} \int \left (960-\frac {1}{x^7}-\frac {32}{x^5}-\frac {8}{x^4}-\frac {240}{x^3}-\frac {96}{x^2}+2496 x+7040 x^2+7680 x^3+2560 x^4+256 x^5\right ) \, dx-\frac {1}{4} \int \left (32 e^x-\frac {2 e^x}{x^3}+\frac {e^x}{x^2}+152 e^x x+360 e^x x^2+160 e^x x^3+16 e^x x^4\right ) \, dx+2 \int \left (e^{2 x} x+e^{2 x} x^2\right ) \, dx\\ &=\frac {1}{64 x^6}+\frac {3}{4 x^4}+\frac {1}{4 x^3}+\frac {45}{4 x^2}+\frac {9}{x}+90 x+117 x^2+220 x^3+180 x^4+48 x^5+4 x^6-\frac {1}{4} \int \frac {e^x}{x^2} \, dx+\frac {1}{2} \int \frac {e^x}{x^3} \, dx+2 \int e^{2 x} x \, dx+2 \int e^{2 x} x^2 \, dx-4 \int e^x x^4 \, dx-8 \int e^x \, dx-38 \int e^x x \, dx-40 \int e^x x^3 \, dx-90 \int e^x x^2 \, dx\\ &=-8 e^x+\frac {1}{64 x^6}+\frac {3}{4 x^4}+\frac {1}{4 x^3}+\frac {45}{4 x^2}-\frac {e^x}{4 x^2}+\frac {9}{x}+\frac {e^x}{4 x}+90 x-38 e^x x+e^{2 x} x+117 x^2-90 e^x x^2+e^{2 x} x^2+220 x^3-40 e^x x^3+180 x^4-4 e^x x^4+48 x^5+4 x^6+\frac {1}{4} \int \frac {e^x}{x^2} \, dx-\frac {1}{4} \int \frac {e^x}{x} \, dx-2 \int e^{2 x} x \, dx+16 \int e^x x^3 \, dx+38 \int e^x \, dx+120 \int e^x x^2 \, dx+180 \int e^x x \, dx-\int e^{2 x} \, dx\\ &=30 e^x-\frac {e^{2 x}}{2}+\frac {1}{64 x^6}+\frac {3}{4 x^4}+\frac {1}{4 x^3}+\frac {45}{4 x^2}-\frac {e^x}{4 x^2}+\frac {9}{x}+90 x+142 e^x x+117 x^2+30 e^x x^2+e^{2 x} x^2+220 x^3-24 e^x x^3+180 x^4-4 e^x x^4+48 x^5+4 x^6-\frac {\text {Ei}(x)}{4}+\frac {1}{4} \int \frac {e^x}{x} \, dx-48 \int e^x x^2 \, dx-180 \int e^x \, dx-240 \int e^x x \, dx+\int e^{2 x} \, dx\\ &=-150 e^x+\frac {1}{64 x^6}+\frac {3}{4 x^4}+\frac {1}{4 x^3}+\frac {45}{4 x^2}-\frac {e^x}{4 x^2}+\frac {9}{x}+90 x-98 e^x x+117 x^2-18 e^x x^2+e^{2 x} x^2+220 x^3-24 e^x x^3+180 x^4-4 e^x x^4+48 x^5+4 x^6+96 \int e^x x \, dx+240 \int e^x \, dx\\ &=90 e^x+\frac {1}{64 x^6}+\frac {3}{4 x^4}+\frac {1}{4 x^3}+\frac {45}{4 x^2}-\frac {e^x}{4 x^2}+\frac {9}{x}+90 x-2 e^x x+117 x^2-18 e^x x^2+e^{2 x} x^2+220 x^3-24 e^x x^3+180 x^4-4 e^x x^4+48 x^5+4 x^6-96 \int e^x \, dx\\ &=-6 e^x+\frac {1}{64 x^6}+\frac {3}{4 x^4}+\frac {1}{4 x^3}+\frac {45}{4 x^2}-\frac {e^x}{4 x^2}+\frac {9}{x}+90 x-2 e^x x+117 x^2-18 e^x x^2+e^{2 x} x^2+220 x^3-24 e^x x^3+180 x^4-4 e^x x^4+48 x^5+4 x^6\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.23, size = 101, normalized size = 4.04 \begin {gather*} \frac {1}{64 x^6}+\frac {3}{4 x^4}+\frac {1}{4 x^3}+\frac {45}{4 x^2}+\frac {9}{x}+90 x+117 x^2+e^{2 x} x^2+220 x^3+180 x^4+48 x^5+4 x^6-\frac {1}{4} e^x \left (24+\frac {1}{x^2}+8 x+72 x^2+96 x^3+16 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 - 96*x^2 - 24*x^3 - 720*x^4 - 288*x^5 + 2880*x^7 + 7488*x^8 + 21120*x^9 + 23040*x^10 + 7680*x^11
+ 768*x^12 + E^(2*x)*(64*x^8 + 64*x^9) + E^x*(16*x^4 - 8*x^5 - 256*x^7 - 1216*x^8 - 2880*x^9 - 1280*x^10 - 12
8*x^11))/(32*x^7),x]

[Out]

1/(64*x^6) + 3/(4*x^4) + 1/(4*x^3) + 45/(4*x^2) + 9/x + 90*x + 117*x^2 + E^(2*x)*x^2 + 220*x^3 + 180*x^4 + 48*
x^5 + 4*x^6 - (E^x*(24 + x^(-2) + 8*x + 72*x^2 + 96*x^3 + 16*x^4))/4

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fricas [B]  time = 0.55, size = 99, normalized size = 3.96 \begin {gather*} \frac {256 \, x^{12} + 3072 \, x^{11} + 11520 \, x^{10} + 14080 \, x^{9} + 64 \, x^{8} e^{\left (2 \, x\right )} + 7488 \, x^{8} + 5760 \, x^{7} + 576 \, x^{5} + 720 \, x^{4} + 16 \, x^{3} + 48 \, x^{2} - 16 \, {\left (16 \, x^{10} + 96 \, x^{9} + 72 \, x^{8} + 8 \, x^{7} + 24 \, x^{6} + x^{4}\right )} e^{x} + 1}{64 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*((64*x^9+64*x^8)*exp(x)^2+(-128*x^11-1280*x^10-2880*x^9-1216*x^8-256*x^7-8*x^5+16*x^4)*exp(x)+7
68*x^12+7680*x^11+23040*x^10+21120*x^9+7488*x^8+2880*x^7-288*x^5-720*x^4-24*x^3-96*x^2-3)/x^7,x, algorithm="fr
icas")

[Out]

1/64*(256*x^12 + 3072*x^11 + 11520*x^10 + 14080*x^9 + 64*x^8*e^(2*x) + 7488*x^8 + 5760*x^7 + 576*x^5 + 720*x^4
+ 16*x^3 + 48*x^2 - 16*(16*x^10 + 96*x^9 + 72*x^8 + 8*x^7 + 24*x^6 + x^4)*e^x + 1)/x^6

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giac [B]  time = 0.19, size = 108, normalized size = 4.32 \begin {gather*} \frac {256 \, x^{12} + 3072 \, x^{11} - 256 \, x^{10} e^{x} + 11520 \, x^{10} - 1536 \, x^{9} e^{x} + 14080 \, x^{9} + 64 \, x^{8} e^{\left (2 \, x\right )} - 1152 \, x^{8} e^{x} + 7488 \, x^{8} - 128 \, x^{7} e^{x} + 5760 \, x^{7} - 384 \, x^{6} e^{x} + 576 \, x^{5} - 16 \, x^{4} e^{x} + 720 \, x^{4} + 16 \, x^{3} + 48 \, x^{2} + 1}{64 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*((64*x^9+64*x^8)*exp(x)^2+(-128*x^11-1280*x^10-2880*x^9-1216*x^8-256*x^7-8*x^5+16*x^4)*exp(x)+7
68*x^12+7680*x^11+23040*x^10+21120*x^9+7488*x^8+2880*x^7-288*x^5-720*x^4-24*x^3-96*x^2-3)/x^7,x, algorithm="gi
ac")

[Out]

1/64*(256*x^12 + 3072*x^11 - 256*x^10*e^x + 11520*x^10 - 1536*x^9*e^x + 14080*x^9 + 64*x^8*e^(2*x) - 1152*x^8*
e^x + 7488*x^8 - 128*x^7*e^x + 5760*x^7 - 384*x^6*e^x + 576*x^5 - 16*x^4*e^x + 720*x^4 + 16*x^3 + 48*x^2 + 1)/
x^6

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maple [B]  time = 0.04, size = 99, normalized size = 3.96

method	result	size
risch	\(4 x^{6}+48 x^{5}+180 x^{4}+220 x^{3}+117 x^{2}+90 x +\frac {288 x^{5}+360 x^{4}+8 x^{3}+24 x^{2}+\frac {1}{2}}{32 x^{6}}+{\mathrm e}^{2 x} x^{2}-\frac {\left (16 x^{6}+96 x^{5}+72 x^{4}+8 x^{3}+24 x^{2}+1\right ) {\mathrm e}^{x}}{4 x^{2}}\)	\(99\)
default	\(117 x^{2}+90 x +\frac {1}{64 x^{6}}+\frac {3}{4 x^{4}}+\frac {1}{4 x^{3}}+\frac {45}{4 x^{2}}+\frac {9}{x}+220 x^{3}+180 x^{4}+48 x^{5}+4 x^{6}-\frac {{\mathrm e}^{x}}{4 x^{2}}-2 \,{\mathrm e}^{x} x -6 \,{\mathrm e}^{x}-18 \,{\mathrm e}^{x} x^{2}-24 \,{\mathrm e}^{x} x^{3}-4 \,{\mathrm e}^{x} x^{4}+{\mathrm e}^{2 x} x^{2}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/32*((64*x^9+64*x^8)*exp(x)^2+(-128*x^11-1280*x^10-2880*x^9-1216*x^8-256*x^7-8*x^5+16*x^4)*exp(x)+768*x^1
2+7680*x^11+23040*x^10+21120*x^9+7488*x^8+2880*x^7-288*x^5-720*x^4-24*x^3-96*x^2-3)/x^7,x,method=_RETURNVERBOS
E)

[Out]

4*x^6+48*x^5+180*x^4+220*x^3+117*x^2+90*x+1/32*(288*x^5+360*x^4+8*x^3+24*x^2+1/2)/x^6+exp(2*x)*x^2-1/4*(16*x^6
+96*x^5+72*x^4+8*x^3+24*x^2+1)/x^2*exp(x)

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maxima [C]  time = 0.40, size = 157, normalized size = 6.28 \begin {gather*} 4 \, x^{6} + 48 \, x^{5} + 180 \, x^{4} + 220 \, x^{3} + 117 \, x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {1}{2} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} - 40 \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} - 90 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} - 38 \, {\left (x - 1\right )} e^{x} + 90 \, x + \frac {9}{x} + \frac {45}{4 \, x^{2}} + \frac {1}{4 \, x^{3}} + \frac {3}{4 \, x^{4}} + \frac {1}{64 \, x^{6}} - 8 \, e^{x} - \frac {1}{4} \, \Gamma \left (-1, -x\right ) - \frac {1}{2} \, \Gamma \left (-2, -x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*((64*x^9+64*x^8)*exp(x)^2+(-128*x^11-1280*x^10-2880*x^9-1216*x^8-256*x^7-8*x^5+16*x^4)*exp(x)+7
68*x^12+7680*x^11+23040*x^10+21120*x^9+7488*x^8+2880*x^7-288*x^5-720*x^4-24*x^3-96*x^2-3)/x^7,x, algorithm="ma
xima")

[Out]

4*x^6 + 48*x^5 + 180*x^4 + 220*x^3 + 117*x^2 + 1/2*(2*x^2 - 2*x + 1)*e^(2*x) + 1/2*(2*x - 1)*e^(2*x) - 4*(x^4
- 4*x^3 + 12*x^2 - 24*x + 24)*e^x - 40*(x^3 - 3*x^2 + 6*x - 6)*e^x - 90*(x^2 - 2*x + 2)*e^x - 38*(x - 1)*e^x +
90*x + 9/x + 45/4/x^2 + 1/4/x^3 + 3/4/x^4 + 1/64/x^6 - 8*e^x - 1/4*gamma(-1, -x) - 1/2*gamma(-2, -x)

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mupad [B]  time = 3.34, size = 92, normalized size = 3.68 \begin {gather*} \frac {\frac {3\,x^2}{4}-x^4\,\left (\frac {{\mathrm {e}}^x}{4}-\frac {45}{4}\right )+\frac {x^3}{4}+9\,x^5+\frac {1}{64}}{x^6}-6\,{\mathrm {e}}^x+x^2\,\left ({\mathrm {e}}^{2\,x}-18\,{\mathrm {e}}^x+117\right )-x\,\left (2\,{\mathrm {e}}^x-90\right )-x^4\,\left (4\,{\mathrm {e}}^x-180\right )-x^3\,\left (24\,{\mathrm {e}}^x-220\right )+48\,x^5+4\,x^6 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(2*x)*(64*x^8 + 64*x^9))/32 - (exp(x)*(8*x^5 - 16*x^4 + 256*x^7 + 1216*x^8 + 2880*x^9 + 1280*x^10 + 1
28*x^11))/32 - 3*x^2 - (3*x^3)/4 - (45*x^4)/2 - 9*x^5 + 90*x^7 + 234*x^8 + 660*x^9 + 720*x^10 + 240*x^11 + 24*
x^12 - 3/32)/x^7,x)

[Out]

((3*x^2)/4 - x^4*(exp(x)/4 - 45/4) + x^3/4 + 9*x^5 + 1/64)/x^6 - 6*exp(x) + x^2*(exp(2*x) - 18*exp(x) + 117) -
x*(2*exp(x) - 90) - x^4*(4*exp(x) - 180) - x^3*(24*exp(x) - 220) + 48*x^5 + 4*x^6

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sympy [B]  time = 0.21, size = 102, normalized size = 4.08 \begin {gather*} 4 x^{6} + 48 x^{5} + 180 x^{4} + 220 x^{3} + 117 x^{2} + 90 x + \frac {4 x^{4} e^{2 x} + \left (- 16 x^{6} - 96 x^{5} - 72 x^{4} - 8 x^{3} - 24 x^{2} - 1\right ) e^{x}}{4 x^{2}} + \frac {576 x^{5} + 720 x^{4} + 16 x^{3} + 48 x^{2} + 1}{64 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*((64*x**9+64*x**8)*exp(x)**2+(-128*x**11-1280*x**10-2880*x**9-1216*x**8-256*x**7-8*x**5+16*x**4
)*exp(x)+768*x**12+7680*x**11+23040*x**10+21120*x**9+7488*x**8+2880*x**7-288*x**5-720*x**4-24*x**3-96*x**2-3)/
x**7,x)

[Out]

4*x**6 + 48*x**5 + 180*x**4 + 220*x**3 + 117*x**2 + 90*x + (4*x**4*exp(2*x) + (-16*x**6 - 96*x**5 - 72*x**4 -
8*x**3 - 24*x**2 - 1)*exp(x))/(4*x**2) + (576*x**5 + 720*x**4 + 16*x**3 + 48*x**2 + 1)/(64*x**6)

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