3.27.95 \(\int \frac {-8 x^7+4 x^8+6 x^9-5 x^{10}+x^{11}+e^{\frac {256+32 x+x^2}{4 x^6-4 x^7+x^8}} (-8 x^7+12 x^8-6 x^9+x^{10})+e^{\frac {256+32 x+x^2}{4 x^6-4 x^7+x^8}} (3072+1344 x-1944 x^2-222 x^3-6 x^4) \log (1+x)}{-8 x^7+4 x^8+6 x^9-5 x^{10}+x^{11}} \, dx\)

Optimal. Leaf size=26 \[ -5+x+e^{\frac {(16+x)^2}{(2-x)^2 x^6}} \log (1+x) \]

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Rubi [B]  time = 5.49, antiderivative size = 128, normalized size of antiderivative = 4.92, number of steps used = 14, number of rules used = 6, integrand size = 150, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {6688, 6742, 44, 77, 88, 2288} \begin {gather*} x-\frac {3 e^{\frac {(x+16)^2}{(2-x)^2 x^6}} \left (x^4 (-\log (x+1))-37 x^3 \log (x+1)-324 x^2 \log (x+1)+224 x \log (x+1)+512 \log (x+1)\right )}{(2-x)^3 x^7 (x+1) \left (-\frac {3 (x+16)^2}{(2-x)^2 x^7}+\frac {(x+16)^2}{(2-x)^3 x^6}+\frac {x+16}{(2-x)^2 x^6}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8*x^7 + 4*x^8 + 6*x^9 - 5*x^10 + x^11 + E^((256 + 32*x + x^2)/(4*x^6 - 4*x^7 + x^8))*(-8*x^7 + 12*x^8 -
6*x^9 + x^10) + E^((256 + 32*x + x^2)/(4*x^6 - 4*x^7 + x^8))*(3072 + 1344*x - 1944*x^2 - 222*x^3 - 6*x^4)*Log[
1 + x])/(-8*x^7 + 4*x^8 + 6*x^9 - 5*x^10 + x^11),x]

[Out]

x - (3*E^((16 + x)^2/((2 - x)^2*x^6))*(512*Log[1 + x] + 224*x*Log[1 + x] - 324*x^2*Log[1 + x] - 37*x^3*Log[1 +
x] - x^4*Log[1 + x]))/((2 - x)^3*x^7*(1 + x)*((16 + x)/((2 - x)^2*x^6) - (3*(16 + x)^2)/((2 - x)^2*x^7) + (16
+ x)^2/((2 - x)^3*x^6)))

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

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 {8 x^7-e^{\frac {(16+x)^2}{(-2+x)^2 x^6}} (-2+x)^3 x^7-4 x^8-6 x^9+5 x^{10}-x^{11}+6 e^{\frac {(16+x)^2}{(-2+x)^2 x^6}} \left (-512-224 x+324 x^2+37 x^3+x^4\right ) \log (1+x)}{(2-x)^3 x^7 (1+x)} \, dx\\ &=\int \left (-\frac {8}{(-2+x)^3 (1+x)}+\frac {4 x}{(-2+x)^3 (1+x)}+\frac {6 x^2}{(-2+x)^3 (1+x)}-\frac {5 x^3}{(-2+x)^3 (1+x)}+\frac {x^4}{(-2+x)^3 (1+x)}+\frac {e^{\frac {(16+x)^2}{(-2+x)^2 x^6}} \left (-8 x^7+12 x^8-6 x^9+x^{10}+3072 \log (1+x)+1344 x \log (1+x)-1944 x^2 \log (1+x)-222 x^3 \log (1+x)-6 x^4 \log (1+x)\right )}{(-2+x)^3 x^7 (1+x)}\right ) \, dx\\ &=4 \int \frac {x}{(-2+x)^3 (1+x)} \, dx-5 \int \frac {x^3}{(-2+x)^3 (1+x)} \, dx+6 \int \frac {x^2}{(-2+x)^3 (1+x)} \, dx-8 \int \frac {1}{(-2+x)^3 (1+x)} \, dx+\int \frac {x^4}{(-2+x)^3 (1+x)} \, dx+\int \frac {e^{\frac {(16+x)^2}{(-2+x)^2 x^6}} \left (-8 x^7+12 x^8-6 x^9+x^{10}+3072 \log (1+x)+1344 x \log (1+x)-1944 x^2 \log (1+x)-222 x^3 \log (1+x)-6 x^4 \log (1+x)\right )}{(-2+x)^3 x^7 (1+x)} \, dx\\ &=-\frac {3 e^{\frac {(16+x)^2}{(2-x)^2 x^6}} \left (512 \log (1+x)+224 x \log (1+x)-324 x^2 \log (1+x)-37 x^3 \log (1+x)-x^4 \log (1+x)\right )}{(2-x)^3 x^7 (1+x) \left (\frac {16+x}{(2-x)^2 x^6}-\frac {3 (16+x)^2}{(2-x)^2 x^7}+\frac {(16+x)^2}{(2-x)^3 x^6}\right )}+4 \int \left (\frac {2}{3 (-2+x)^3}+\frac {1}{9 (-2+x)^2}-\frac {1}{27 (-2+x)}+\frac {1}{27 (1+x)}\right ) \, dx-5 \int \left (\frac {8}{3 (-2+x)^3}+\frac {28}{9 (-2+x)^2}+\frac {26}{27 (-2+x)}+\frac {1}{27 (1+x)}\right ) \, dx+6 \int \left (\frac {4}{3 (-2+x)^3}+\frac {8}{9 (-2+x)^2}+\frac {1}{27 (-2+x)}-\frac {1}{27 (1+x)}\right ) \, dx-8 \int \left (\frac {1}{3 (-2+x)^3}-\frac {1}{9 (-2+x)^2}+\frac {1}{27 (-2+x)}-\frac {1}{27 (1+x)}\right ) \, dx+\int \left (1+\frac {16}{3 (-2+x)^3}+\frac {80}{9 (-2+x)^2}+\frac {136}{27 (-2+x)}-\frac {1}{27 (1+x)}\right ) \, dx\\ &=x-\frac {3 e^{\frac {(16+x)^2}{(2-x)^2 x^6}} \left (512 \log (1+x)+224 x \log (1+x)-324 x^2 \log (1+x)-37 x^3 \log (1+x)-x^4 \log (1+x)\right )}{(2-x)^3 x^7 (1+x) \left (\frac {16+x}{(2-x)^2 x^6}-\frac {3 (16+x)^2}{(2-x)^2 x^7}+\frac {(16+x)^2}{(2-x)^3 x^6}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 23, normalized size = 0.88 \begin {gather*} x+e^{\frac {(16+x)^2}{(-2+x)^2 x^6}} \log (1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*x^7 + 4*x^8 + 6*x^9 - 5*x^10 + x^11 + E^((256 + 32*x + x^2)/(4*x^6 - 4*x^7 + x^8))*(-8*x^7 + 12*
x^8 - 6*x^9 + x^10) + E^((256 + 32*x + x^2)/(4*x^6 - 4*x^7 + x^8))*(3072 + 1344*x - 1944*x^2 - 222*x^3 - 6*x^4
)*Log[1 + x])/(-8*x^7 + 4*x^8 + 6*x^9 - 5*x^10 + x^11),x]

[Out]

x + E^((16 + x)^2/((-2 + x)^2*x^6))*Log[1 + x]

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fricas [A]  time = 0.58, size = 33, normalized size = 1.27 \begin {gather*} e^{\left (\frac {x^{2} + 32 \, x + 256}{x^{8} - 4 \, x^{7} + 4 \, x^{6}}\right )} \log \left (x + 1\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^4-222*x^3-1944*x^2+1344*x+3072)*exp((x^2+32*x+256)/(x^8-4*x^7+4*x^6))*log(x+1)+(x^10-6*x^9+12
*x^8-8*x^7)*exp((x^2+32*x+256)/(x^8-4*x^7+4*x^6))+x^11-5*x^10+6*x^9+4*x^8-8*x^7)/(x^11-5*x^10+6*x^9+4*x^8-8*x^
7),x, algorithm="fricas")

[Out]

e^((x^2 + 32*x + 256)/(x^8 - 4*x^7 + 4*x^6))*log(x + 1) + x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{11} - 5 \, x^{10} + 6 \, x^{9} + 4 \, x^{8} - 8 \, x^{7} - 6 \, {\left (x^{4} + 37 \, x^{3} + 324 \, x^{2} - 224 \, x - 512\right )} e^{\left (\frac {x^{2} + 32 \, x + 256}{x^{8} - 4 \, x^{7} + 4 \, x^{6}}\right )} \log \left (x + 1\right ) + {\left (x^{10} - 6 \, x^{9} + 12 \, x^{8} - 8 \, x^{7}\right )} e^{\left (\frac {x^{2} + 32 \, x + 256}{x^{8} - 4 \, x^{7} + 4 \, x^{6}}\right )}}{x^{11} - 5 \, x^{10} + 6 \, x^{9} + 4 \, x^{8} - 8 \, x^{7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^4-222*x^3-1944*x^2+1344*x+3072)*exp((x^2+32*x+256)/(x^8-4*x^7+4*x^6))*log(x+1)+(x^10-6*x^9+12
*x^8-8*x^7)*exp((x^2+32*x+256)/(x^8-4*x^7+4*x^6))+x^11-5*x^10+6*x^9+4*x^8-8*x^7)/(x^11-5*x^10+6*x^9+4*x^8-8*x^
7),x, algorithm="giac")

[Out]

integrate((x^11 - 5*x^10 + 6*x^9 + 4*x^8 - 8*x^7 - 6*(x^4 + 37*x^3 + 324*x^2 - 224*x - 512)*e^((x^2 + 32*x + 2
56)/(x^8 - 4*x^7 + 4*x^6))*log(x + 1) + (x^10 - 6*x^9 + 12*x^8 - 8*x^7)*e^((x^2 + 32*x + 256)/(x^8 - 4*x^7 + 4
*x^6)))/(x^11 - 5*x^10 + 6*x^9 + 4*x^8 - 8*x^7), x)

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maple [A]  time = 0.10, size = 23, normalized size = 0.88

method	result	size
risch	\({\mathrm e}^{\frac {\left (x +16\right )^{2}}{x^{6} \left (x -2\right )^{2}}} \ln \left (x +1\right )+x\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x^4-222*x^3-1944*x^2+1344*x+3072)*exp((x^2+32*x+256)/(x^8-4*x^7+4*x^6))*ln(x+1)+(x^10-6*x^9+12*x^8-8*
x^7)*exp((x^2+32*x+256)/(x^8-4*x^7+4*x^6))+x^11-5*x^10+6*x^9+4*x^8-8*x^7)/(x^11-5*x^10+6*x^9+4*x^8-8*x^7),x,me
thod=_RETURNVERBOSE)

[Out]

exp((x+16)^2/x^6/(x-2)^2)*ln(x+1)+x

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maxima [B]  time = 2.88, size = 141, normalized size = 5.42 \begin {gather*} e^{\left (\frac {81}{16 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {117}{8 \, {\left (x - 2\right )}} + \frac {117}{8 \, x} + \frac {387}{16 \, x^{2}} + \frac {153}{4 \, x^{3}} + \frac {225}{4 \, x^{4}} + \frac {72}{x^{5}} + \frac {64}{x^{6}}\right )} \log \left (x + 1\right ) + x - \frac {8 \, {\left (10 \, x - 17\right )}}{9 \, {\left (x^{2} - 4 \, x + 4\right )}} + \frac {20 \, {\left (7 \, x - 11\right )}}{9 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {4 \, {\left (4 \, x - 5\right )}}{3 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {4 \, {\left (2 \, x - 7\right )}}{9 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {4 \, {\left (x + 1\right )}}{9 \, {\left (x^{2} - 4 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^4-222*x^3-1944*x^2+1344*x+3072)*exp((x^2+32*x+256)/(x^8-4*x^7+4*x^6))*log(x+1)+(x^10-6*x^9+12
*x^8-8*x^7)*exp((x^2+32*x+256)/(x^8-4*x^7+4*x^6))+x^11-5*x^10+6*x^9+4*x^8-8*x^7)/(x^11-5*x^10+6*x^9+4*x^8-8*x^
7),x, algorithm="maxima")

[Out]

e^(81/16/(x^2 - 4*x + 4) - 117/8/(x - 2) + 117/8/x + 387/16/x^2 + 153/4/x^3 + 225/4/x^4 + 72/x^5 + 64/x^6)*log
(x + 1) + x - 8/9*(10*x - 17)/(x^2 - 4*x + 4) + 20/9*(7*x - 11)/(x^2 - 4*x + 4) - 4/3*(4*x - 5)/(x^2 - 4*x + 4
) - 4/9*(2*x - 7)/(x^2 - 4*x + 4) - 4/9*(x + 1)/(x^2 - 4*x + 4)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^{\frac {x^2+32\,x+256}{x^8-4\,x^7+4\,x^6}}\,\left (-x^{10}+6\,x^9-12\,x^8+8\,x^7\right )+8\,x^7-4\,x^8-6\,x^9+5\,x^{10}-x^{11}+\ln \left (x+1\right )\,{\mathrm {e}}^{\frac {x^2+32\,x+256}{x^8-4\,x^7+4\,x^6}}\,\left (6\,x^4+222\,x^3+1944\,x^2-1344\,x-3072\right )}{x^{11}-5\,x^{10}+6\,x^9+4\,x^8-8\,x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((32*x + x^2 + 256)/(4*x^6 - 4*x^7 + x^8))*(8*x^7 - 12*x^8 + 6*x^9 - x^10) + 8*x^7 - 4*x^8 - 6*x^9 +
5*x^10 - x^11 + log(x + 1)*exp((32*x + x^2 + 256)/(4*x^6 - 4*x^7 + x^8))*(1944*x^2 - 1344*x + 222*x^3 + 6*x^4
- 3072))/(4*x^8 - 8*x^7 + 6*x^9 - 5*x^10 + x^11),x)

[Out]

int(-(exp((32*x + x^2 + 256)/(4*x^6 - 4*x^7 + x^8))*(8*x^7 - 12*x^8 + 6*x^9 - x^10) + 8*x^7 - 4*x^8 - 6*x^9 +
5*x^10 - x^11 + log(x + 1)*exp((32*x + x^2 + 256)/(4*x^6 - 4*x^7 + x^8))*(1944*x^2 - 1344*x + 222*x^3 + 6*x^4
- 3072))/(4*x^8 - 8*x^7 + 6*x^9 - 5*x^10 + x^11), x)

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sympy [A]  time = 0.71, size = 29, normalized size = 1.12 \begin {gather*} x + e^{\frac {x^{2} + 32 x + 256}{x^{8} - 4 x^{7} + 4 x^{6}}} \log {\left (x + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x**4-222*x**3-1944*x**2+1344*x+3072)*exp((x**2+32*x+256)/(x**8-4*x**7+4*x**6))*ln(x+1)+(x**10-6
*x**9+12*x**8-8*x**7)*exp((x**2+32*x+256)/(x**8-4*x**7+4*x**6))+x**11-5*x**10+6*x**9+4*x**8-8*x**7)/(x**11-5*x
**10+6*x**9+4*x**8-8*x**7),x)

[Out]

x + exp((x**2 + 32*x + 256)/(x**8 - 4*x**7 + 4*x**6))*log(x + 1)

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