3.77.7 \(\int \frac {-600 x^7-900 x^8-300 x^9+150 x^{10}+e^{\log ^2(x)} (-750 x^4-2700 x^5-3600 x^6-1050 x^7+1350 x^8+750 x^9-300 x^{10}+(-300 x^4-1200 x^5-1800 x^6-600 x^7+900 x^8+600 x^9-300 x^{10}) \log (x))+e^{2 \log ^2(x)} (-150 x-900 x^2-2250 x^3-2550 x^4-450 x^5+1800 x^6+1200 x^7-450 x^8-450 x^9+150 x^{10}+(-300 x-1800 x^2-4500 x^3-5100 x^4-900 x^5+3600 x^6+2400 x^7-900 x^8-900 x^9+300 x^{10}) \log (x))}{-1-6 x-15 x^2-17 x^3-3 x^4+12 x^5+8 x^6-3 x^7-3 x^8+x^9} \, dx\)

Optimal. Leaf size=31 \[ 75 x^2 \left (e^{\log ^2(x)}-\frac {x}{x-\frac {(1+x)^2}{x^2}}\right )^2 \]

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Rubi [B]  time = 1.75, antiderivative size = 85, normalized size of antiderivative = 2.74, number of steps used = 7, number of rules used = 5, integrand size = 259, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6688, 12, 6742, 1588, 2288} \begin {gather*} 75 x^2 e^{2 \log ^2(x)}+\frac {75 x^8}{\left (-x^3+x^2+2 x+1\right )^2}+\frac {150 x^5 e^{\log ^2(x)} \left (x^3 (-\log (x))+x^2 \log (x)+2 x \log (x)+\log (x)\right )}{\left (-x^3+x^2+2 x+1\right )^2 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-600*x^7 - 900*x^8 - 300*x^9 + 150*x^10 + E^Log[x]^2*(-750*x^4 - 2700*x^5 - 3600*x^6 - 1050*x^7 + 1350*x^
8 + 750*x^9 - 300*x^10 + (-300*x^4 - 1200*x^5 - 1800*x^6 - 600*x^7 + 900*x^8 + 600*x^9 - 300*x^10)*Log[x]) + E
^(2*Log[x]^2)*(-150*x - 900*x^2 - 2250*x^3 - 2550*x^4 - 450*x^5 + 1800*x^6 + 1200*x^7 - 450*x^8 - 450*x^9 + 15
0*x^10 + (-300*x - 1800*x^2 - 4500*x^3 - 5100*x^4 - 900*x^5 + 3600*x^6 + 2400*x^7 - 900*x^8 - 900*x^9 + 300*x^
10)*Log[x]))/(-1 - 6*x - 15*x^2 - 17*x^3 - 3*x^4 + 12*x^5 + 8*x^6 - 3*x^7 - 3*x^8 + x^9),x]

[Out]

75*E^(2*Log[x]^2)*x^2 + (75*x^8)/(1 + 2*x + x^2 - x^3)^2 + (150*E^Log[x]^2*x^5*(Log[x] + 2*x*Log[x] + x^2*Log[
x] - x^3*Log[x]))/((1 + 2*x + x^2 - x^3)^2*Log[x])

Rule 12

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

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -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 {150 x \left (x^3-e^{\log ^2(x)} \left (-1-2 x-x^2+x^3\right )\right ) \left (e^{\log ^2(x)} \left (1+2 x+x^2-x^3\right )^2+x^3 \left (4+6 x+2 x^2-x^3\right )+2 e^{\log ^2(x)} \left (1+2 x+x^2-x^3\right )^2 \log (x)\right )}{\left (1+2 x+x^2-x^3\right )^3} \, dx\\ &=150 \int \frac {x \left (x^3-e^{\log ^2(x)} \left (-1-2 x-x^2+x^3\right )\right ) \left (e^{\log ^2(x)} \left (1+2 x+x^2-x^3\right )^2+x^3 \left (4+6 x+2 x^2-x^3\right )+2 e^{\log ^2(x)} \left (1+2 x+x^2-x^3\right )^2 \log (x)\right )}{\left (1+2 x+x^2-x^3\right )^3} \, dx\\ &=150 \int \left (\frac {x^7 \left (-4-6 x-2 x^2+x^3\right )}{\left (-1-2 x-x^2+x^3\right )^3}+e^{2 \log ^2(x)} x (1+2 \log (x))-\frac {e^{\log ^2(x)} x^4 \left (-5-8 x-3 x^2+2 x^3-2 \log (x)-4 x \log (x)-2 x^2 \log (x)+2 x^3 \log (x)\right )}{\left (-1-2 x-x^2+x^3\right )^2}\right ) \, dx\\ &=150 \int \frac {x^7 \left (-4-6 x-2 x^2+x^3\right )}{\left (-1-2 x-x^2+x^3\right )^3} \, dx+150 \int e^{2 \log ^2(x)} x (1+2 \log (x)) \, dx-150 \int \frac {e^{\log ^2(x)} x^4 \left (-5-8 x-3 x^2+2 x^3-2 \log (x)-4 x \log (x)-2 x^2 \log (x)+2 x^3 \log (x)\right )}{\left (-1-2 x-x^2+x^3\right )^2} \, dx\\ &=75 e^{2 \log ^2(x)} x^2+\frac {75 x^8}{\left (1+2 x+x^2-x^3\right )^2}+\frac {150 e^{\log ^2(x)} x^5 \left (\log (x)+2 x \log (x)+x^2 \log (x)-x^3 \log (x)\right )}{\left (1+2 x+x^2-x^3\right )^2 \log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.19, size = 100, normalized size = 3.23 \begin {gather*} 75 \left (2 x+x^2+e^{2 \log ^2(x)} x^2+\frac {28+69 x+60 x^2}{\left (1+2 x+x^2-x^3\right )^2}+\frac {2 e^{\log ^2(x)} x^5}{1+2 x+x^2-x^3}-\frac {35+29 x+18 x^2}{1+2 x+x^2-x^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-600*x^7 - 900*x^8 - 300*x^9 + 150*x^10 + E^Log[x]^2*(-750*x^4 - 2700*x^5 - 3600*x^6 - 1050*x^7 + 1
350*x^8 + 750*x^9 - 300*x^10 + (-300*x^4 - 1200*x^5 - 1800*x^6 - 600*x^7 + 900*x^8 + 600*x^9 - 300*x^10)*Log[x
]) + E^(2*Log[x]^2)*(-150*x - 900*x^2 - 2250*x^3 - 2550*x^4 - 450*x^5 + 1800*x^6 + 1200*x^7 - 450*x^8 - 450*x^
9 + 150*x^10 + (-300*x - 1800*x^2 - 4500*x^3 - 5100*x^4 - 900*x^5 + 3600*x^6 + 2400*x^7 - 900*x^8 - 900*x^9 +
300*x^10)*Log[x]))/(-1 - 6*x - 15*x^2 - 17*x^3 - 3*x^4 + 12*x^5 + 8*x^6 - 3*x^7 - 3*x^8 + x^9),x]

[Out]

75*(2*x + x^2 + E^(2*Log[x]^2)*x^2 + (28 + 69*x + 60*x^2)/(1 + 2*x + x^2 - x^3)^2 + (2*E^Log[x]^2*x^5)/(1 + 2*
x + x^2 - x^3) - (35 + 29*x + 18*x^2)/(1 + 2*x + x^2 - x^3))

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fricas [B]  time = 1.88, size = 131, normalized size = 4.23 \begin {gather*} \frac {75 \, {\left (x^{8} - 7 \, x^{6} + 14 \, x^{5} + 21 \, x^{4} - 14 \, x^{3} - 42 \, x^{2} + {\left (x^{8} - 2 \, x^{7} - 3 \, x^{6} + 2 \, x^{5} + 6 \, x^{4} + 4 \, x^{3} + x^{2}\right )} e^{\left (2 \, \log \relax (x)^{2}\right )} - 2 \, {\left (x^{8} - x^{7} - 2 \, x^{6} - x^{5}\right )} e^{\left (\log \relax (x)^{2}\right )} - 28 \, x - 7\right )}}{x^{6} - 2 \, x^{5} - 3 \, x^{4} + 2 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((300*x^10-900*x^9-900*x^8+2400*x^7+3600*x^6-900*x^5-5100*x^4-4500*x^3-1800*x^2-300*x)*log(x)+150*x
^10-450*x^9-450*x^8+1200*x^7+1800*x^6-450*x^5-2550*x^4-2250*x^3-900*x^2-150*x)*exp(log(x)^2)^2+((-300*x^10+600
*x^9+900*x^8-600*x^7-1800*x^6-1200*x^5-300*x^4)*log(x)-300*x^10+750*x^9+1350*x^8-1050*x^7-3600*x^6-2700*x^5-75
0*x^4)*exp(log(x)^2)+150*x^10-300*x^9-900*x^8-600*x^7)/(x^9-3*x^8-3*x^7+8*x^6+12*x^5-3*x^4-17*x^3-15*x^2-6*x-1
),x, algorithm="fricas")

[Out]

75*(x^8 - 7*x^6 + 14*x^5 + 21*x^4 - 14*x^3 - 42*x^2 + (x^8 - 2*x^7 - 3*x^6 + 2*x^5 + 6*x^4 + 4*x^3 + x^2)*e^(2
*log(x)^2) - 2*(x^8 - x^7 - 2*x^6 - x^5)*e^(log(x)^2) - 28*x - 7)/(x^6 - 2*x^5 - 3*x^4 + 2*x^3 + 6*x^2 + 4*x +
1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {150 \, {\left (x^{10} - 2 \, x^{9} - 6 \, x^{8} - 4 \, x^{7} + {\left (x^{10} - 3 \, x^{9} - 3 \, x^{8} + 8 \, x^{7} + 12 \, x^{6} - 3 \, x^{5} - 17 \, x^{4} - 15 \, x^{3} - 6 \, x^{2} + 2 \, {\left (x^{10} - 3 \, x^{9} - 3 \, x^{8} + 8 \, x^{7} + 12 \, x^{6} - 3 \, x^{5} - 17 \, x^{4} - 15 \, x^{3} - 6 \, x^{2} - x\right )} \log \relax (x) - x\right )} e^{\left (2 \, \log \relax (x)^{2}\right )} - {\left (2 \, x^{10} - 5 \, x^{9} - 9 \, x^{8} + 7 \, x^{7} + 24 \, x^{6} + 18 \, x^{5} + 5 \, x^{4} + 2 \, {\left (x^{10} - 2 \, x^{9} - 3 \, x^{8} + 2 \, x^{7} + 6 \, x^{6} + 4 \, x^{5} + x^{4}\right )} \log \relax (x)\right )} e^{\left (\log \relax (x)^{2}\right )}\right )}}{x^{9} - 3 \, x^{8} - 3 \, x^{7} + 8 \, x^{6} + 12 \, x^{5} - 3 \, x^{4} - 17 \, x^{3} - 15 \, x^{2} - 6 \, x - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((300*x^10-900*x^9-900*x^8+2400*x^7+3600*x^6-900*x^5-5100*x^4-4500*x^3-1800*x^2-300*x)*log(x)+150*x
^10-450*x^9-450*x^8+1200*x^7+1800*x^6-450*x^5-2550*x^4-2250*x^3-900*x^2-150*x)*exp(log(x)^2)^2+((-300*x^10+600
*x^9+900*x^8-600*x^7-1800*x^6-1200*x^5-300*x^4)*log(x)-300*x^10+750*x^9+1350*x^8-1050*x^7-3600*x^6-2700*x^5-75
0*x^4)*exp(log(x)^2)+150*x^10-300*x^9-900*x^8-600*x^7)/(x^9-3*x^8-3*x^7+8*x^6+12*x^5-3*x^4-17*x^3-15*x^2-6*x-1
),x, algorithm="giac")

[Out]

integrate(150*(x^10 - 2*x^9 - 6*x^8 - 4*x^7 + (x^10 - 3*x^9 - 3*x^8 + 8*x^7 + 12*x^6 - 3*x^5 - 17*x^4 - 15*x^3
- 6*x^2 + 2*(x^10 - 3*x^9 - 3*x^8 + 8*x^7 + 12*x^6 - 3*x^5 - 17*x^4 - 15*x^3 - 6*x^2 - x)*log(x) - x)*e^(2*lo
g(x)^2) - (2*x^10 - 5*x^9 - 9*x^8 + 7*x^7 + 24*x^6 + 18*x^5 + 5*x^4 + 2*(x^10 - 2*x^9 - 3*x^8 + 2*x^7 + 6*x^6
+ 4*x^5 + x^4)*log(x))*e^(log(x)^2))/(x^9 - 3*x^8 - 3*x^7 + 8*x^6 + 12*x^5 - 3*x^4 - 17*x^3 - 15*x^2 - 6*x - 1
), x)

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maple [B]  time = 0.09, size = 103, normalized size = 3.32

method	result	size
risch	\(75 x^{2}+150 x +\frac {1350 x^{5}+825 x^{4}-2250 x^{3}-3825 x^{2}-2250 x -525}{x^{6}-2 x^{5}-3 x^{4}+2 x^{3}+6 x^{2}+4 x +1}+75 \,{\mathrm e}^{2 \ln \relax (x )^{2}} x^{2}-\frac {150 x^{5} {\mathrm e}^{\ln \relax (x )^{2}}}{x^{3}-x^{2}-2 x -1}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((300*x^10-900*x^9-900*x^8+2400*x^7+3600*x^6-900*x^5-5100*x^4-4500*x^3-1800*x^2-300*x)*ln(x)+150*x^10-450
*x^9-450*x^8+1200*x^7+1800*x^6-450*x^5-2550*x^4-2250*x^3-900*x^2-150*x)*exp(ln(x)^2)^2+((-300*x^10+600*x^9+900
*x^8-600*x^7-1800*x^6-1200*x^5-300*x^4)*ln(x)-300*x^10+750*x^9+1350*x^8-1050*x^7-3600*x^6-2700*x^5-750*x^4)*ex
p(ln(x)^2)+150*x^10-300*x^9-900*x^8-600*x^7)/(x^9-3*x^8-3*x^7+8*x^6+12*x^5-3*x^4-17*x^3-15*x^2-6*x-1),x,method
=_RETURNVERBOSE)

[Out]

75*x^2+150*x+(1350*x^5+825*x^4-2250*x^3-3825*x^2-2250*x-525)/(x^6-2*x^5-3*x^4+2*x^3+6*x^2+4*x+1)+75*exp(2*ln(x
)^2)*x^2-150*x^5/(x^3-x^2-2*x-1)*exp(ln(x)^2)

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maxima [B]  time = 0.38, size = 131, normalized size = 4.23 \begin {gather*} \frac {75 \, {\left (x^{8} - 7 \, x^{6} + 14 \, x^{5} + 21 \, x^{4} - 14 \, x^{3} - 42 \, x^{2} + {\left (x^{8} - 2 \, x^{7} - 3 \, x^{6} + 2 \, x^{5} + 6 \, x^{4} + 4 \, x^{3} + x^{2}\right )} e^{\left (2 \, \log \relax (x)^{2}\right )} - 2 \, {\left (x^{8} - x^{7} - 2 \, x^{6} - x^{5}\right )} e^{\left (\log \relax (x)^{2}\right )} - 28 \, x - 7\right )}}{x^{6} - 2 \, x^{5} - 3 \, x^{4} + 2 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((300*x^10-900*x^9-900*x^8+2400*x^7+3600*x^6-900*x^5-5100*x^4-4500*x^3-1800*x^2-300*x)*log(x)+150*x
^10-450*x^9-450*x^8+1200*x^7+1800*x^6-450*x^5-2550*x^4-2250*x^3-900*x^2-150*x)*exp(log(x)^2)^2+((-300*x^10+600
*x^9+900*x^8-600*x^7-1800*x^6-1200*x^5-300*x^4)*log(x)-300*x^10+750*x^9+1350*x^8-1050*x^7-3600*x^6-2700*x^5-75
0*x^4)*exp(log(x)^2)+150*x^10-300*x^9-900*x^8-600*x^7)/(x^9-3*x^8-3*x^7+8*x^6+12*x^5-3*x^4-17*x^3-15*x^2-6*x-1
),x, algorithm="maxima")

[Out]

75*(x^8 - 7*x^6 + 14*x^5 + 21*x^4 - 14*x^3 - 42*x^2 + (x^8 - 2*x^7 - 3*x^6 + 2*x^5 + 6*x^4 + 4*x^3 + x^2)*e^(2
*log(x)^2) - 2*(x^8 - x^7 - 2*x^6 - x^5)*e^(log(x)^2) - 28*x - 7)/(x^6 - 2*x^5 - 3*x^4 + 2*x^3 + 6*x^2 + 4*x +
1)

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mupad [B]  time = 5.15, size = 103, normalized size = 3.32 \begin {gather*} 150\,x+75\,x^2\,{\mathrm {e}}^{2\,{\ln \relax (x)}^2}-\frac {-1350\,x^5-825\,x^4+2250\,x^3+3825\,x^2+2250\,x+525}{x^6-2\,x^5-3\,x^4+2\,x^3+6\,x^2+4\,x+1}+75\,x^2+\frac {150\,x^5\,{\mathrm {e}}^{{\ln \relax (x)}^2}}{-x^3+x^2+2\,x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(x)^2)*(log(x)*(300*x^4 + 1200*x^5 + 1800*x^6 + 600*x^7 - 900*x^8 - 600*x^9 + 300*x^10) + 750*x^4
+ 2700*x^5 + 3600*x^6 + 1050*x^7 - 1350*x^8 - 750*x^9 + 300*x^10) + 600*x^7 + 900*x^8 + 300*x^9 - 150*x^10 + e
xp(2*log(x)^2)*(150*x + log(x)*(300*x + 1800*x^2 + 4500*x^3 + 5100*x^4 + 900*x^5 - 3600*x^6 - 2400*x^7 + 900*x
^8 + 900*x^9 - 300*x^10) + 900*x^2 + 2250*x^3 + 2550*x^4 + 450*x^5 - 1800*x^6 - 1200*x^7 + 450*x^8 + 450*x^9 -
150*x^10))/(6*x + 15*x^2 + 17*x^3 + 3*x^4 - 12*x^5 - 8*x^6 + 3*x^7 + 3*x^8 - x^9 + 1),x)

[Out]

150*x + 75*x^2*exp(2*log(x)^2) - (2250*x + 3825*x^2 + 2250*x^3 - 825*x^4 - 1350*x^5 + 525)/(4*x + 6*x^2 + 2*x^
3 - 3*x^4 - 2*x^5 + x^6 + 1) + 75*x^2 + (150*x^5*exp(log(x)^2))/(2*x + x^2 - x^3 + 1)

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sympy [B]  time = 0.63, size = 114, normalized size = 3.68 \begin {gather*} 75 x^{2} + 150 x + \frac {- 150 x^{5} e^{\log {\relax (x )}^{2}} + \left (75 x^{5} - 75 x^{4} - 150 x^{3} - 75 x^{2}\right ) e^{2 \log {\relax (x )}^{2}}}{x^{3} - x^{2} - 2 x - 1} + \frac {1350 x^{5} + 825 x^{4} - 2250 x^{3} - 3825 x^{2} - 2250 x - 525}{x^{6} - 2 x^{5} - 3 x^{4} + 2 x^{3} + 6 x^{2} + 4 x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((300*x**10-900*x**9-900*x**8+2400*x**7+3600*x**6-900*x**5-5100*x**4-4500*x**3-1800*x**2-300*x)*ln(
x)+150*x**10-450*x**9-450*x**8+1200*x**7+1800*x**6-450*x**5-2550*x**4-2250*x**3-900*x**2-150*x)*exp(ln(x)**2)*
*2+((-300*x**10+600*x**9+900*x**8-600*x**7-1800*x**6-1200*x**5-300*x**4)*ln(x)-300*x**10+750*x**9+1350*x**8-10
50*x**7-3600*x**6-2700*x**5-750*x**4)*exp(ln(x)**2)+150*x**10-300*x**9-900*x**8-600*x**7)/(x**9-3*x**8-3*x**7+
8*x**6+12*x**5-3*x**4-17*x**3-15*x**2-6*x-1),x)

[Out]

75*x**2 + 150*x + (-150*x**5*exp(log(x)**2) + (75*x**5 - 75*x**4 - 150*x**3 - 75*x**2)*exp(2*log(x)**2))/(x**3
- x**2 - 2*x - 1) + (1350*x**5 + 825*x**4 - 2250*x**3 - 3825*x**2 - 2250*x - 525)/(x**6 - 2*x**5 - 3*x**4 + 2
*x**3 + 6*x**2 + 4*x + 1)

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