3.40.6 \(\int \frac {-18-18 e^9+36 x}{(8000+1200 x-1140 x^2-119 x^3+57 x^4+3 x^5-x^6+e^{27} (64+48 x+12 x^2+x^3)+e^{18} (960+528 x+36 x^2-21 x^3-3 x^4)+e^9 (4800+1680 x-348 x^2-141 x^3+6 x^4+3 x^5)) \log ^2(5)} \, dx\)

Optimal. Leaf size=21 \[ \frac {9}{(4+x)^2 \left (-5-e^9+x\right )^2 \log ^2(5)} \]

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Rubi [B]  time = 0.15, antiderivative size = 83, normalized size of antiderivative = 3.95, number of steps used = 3, number of rules used = 2, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {12, 2074} \begin {gather*} \frac {18}{\left (9+e^9\right )^3 (x+4) \log ^2(5)}+\frac {9}{\left (9+e^9\right )^2 (x+4)^2 \log ^2(5)}+\frac {18}{\left (9+e^9\right )^3 \left (-x+e^9+5\right ) \log ^2(5)}+\frac {9}{\left (9+e^9\right )^2 \left (-x+e^9+5\right )^2 \log ^2(5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-18 - 18*E^9 + 36*x)/((8000 + 1200*x - 1140*x^2 - 119*x^3 + 57*x^4 + 3*x^5 - x^6 + E^27*(64 + 48*x + 12*x
^2 + x^3) + E^18*(960 + 528*x + 36*x^2 - 21*x^3 - 3*x^4) + E^9*(4800 + 1680*x - 348*x^2 - 141*x^3 + 6*x^4 + 3*
x^5))*Log[5]^2),x]

[Out]

9/((9 + E^9)^2*(5 + E^9 - x)^2*Log[5]^2) + 18/((9 + E^9)^3*(5 + E^9 - x)*Log[5]^2) + 9/((9 + E^9)^2*(4 + x)^2*
Log[5]^2) + 18/((9 + E^9)^3*(4 + x)*Log[5]^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 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-18-18 e^9+36 x}{8000+1200 x-1140 x^2-119 x^3+57 x^4+3 x^5-x^6+e^{27} \left (64+48 x+12 x^2+x^3\right )+e^{18} \left (960+528 x+36 x^2-21 x^3-3 x^4\right )+e^9 \left (4800+1680 x-348 x^2-141 x^3+6 x^4+3 x^5\right )} \, dx}{\log ^2(5)}\\ &=\frac {\int \left (\frac {18}{\left (9+e^9\right )^2 \left (5+e^9-x\right )^3}+\frac {18}{\left (9+e^9\right )^3 \left (5+e^9-x\right )^2}-\frac {18}{\left (9+e^9\right )^2 (4+x)^3}-\frac {18}{\left (9+e^9\right )^3 (4+x)^2}\right ) \, dx}{\log ^2(5)}\\ &=\frac {9}{\left (9+e^9\right )^2 \left (5+e^9-x\right )^2 \log ^2(5)}+\frac {18}{\left (9+e^9\right )^3 \left (5+e^9-x\right ) \log ^2(5)}+\frac {9}{\left (9+e^9\right )^2 (4+x)^2 \log ^2(5)}+\frac {18}{\left (9+e^9\right )^3 (4+x) \log ^2(5)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 23, normalized size = 1.10 \begin {gather*} \frac {9}{\left (20+x-x^2+e^9 (4+x)\right )^2 \log ^2(5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-18 - 18*E^9 + 36*x)/((8000 + 1200*x - 1140*x^2 - 119*x^3 + 57*x^4 + 3*x^5 - x^6 + E^27*(64 + 48*x
+ 12*x^2 + x^3) + E^18*(960 + 528*x + 36*x^2 - 21*x^3 - 3*x^4) + E^9*(4800 + 1680*x - 348*x^2 - 141*x^3 + 6*x^
4 + 3*x^5))*Log[5]^2),x]

[Out]

9/((20 + x - x^2 + E^9*(4 + x))^2*Log[5]^2)

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fricas [B]  time = 0.73, size = 54, normalized size = 2.57 \begin {gather*} \frac {9}{{\left (x^{4} - 2 \, x^{3} - 39 \, x^{2} + {\left (x^{2} + 8 \, x + 16\right )} e^{18} - 2 \, {\left (x^{3} + 3 \, x^{2} - 24 \, x - 80\right )} e^{9} + 40 \, x + 400\right )} \log \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*exp(9)+36*x-18)/((x^3+12*x^2+48*x+64)*exp(9)^3+(-3*x^4-21*x^3+36*x^2+528*x+960)*exp(9)^2+(3*x^5
+6*x^4-141*x^3-348*x^2+1680*x+4800)*exp(9)-x^6+3*x^5+57*x^4-119*x^3-1140*x^2+1200*x+8000)/log(5)^2,x, algorith
m="fricas")

[Out]

9/((x^4 - 2*x^3 - 39*x^2 + (x^2 + 8*x + 16)*e^18 - 2*(x^3 + 3*x^2 - 24*x - 80)*e^9 + 40*x + 400)*log(5)^2)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*exp(9)+36*x-18)/((x^3+12*x^2+48*x+64)*exp(9)^3+(-3*x^4-21*x^3+36*x^2+528*x+960)*exp(9)^2+(3*x^5
+6*x^4-141*x^3-348*x^2+1680*x+4800)*exp(9)-x^6+3*x^5+57*x^4-119*x^3-1140*x^2+1200*x+8000)/log(5)^2,x, algorith
m="giac")

[Out]

Timed out

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maple [A]  time = 0.20, size = 21, normalized size = 1.00

method	result	size
norman	\(\frac {9}{\ln \relax (5)^{2} \left (4+x \right )^{2} \left (-x +{\mathrm e}^{9}+5\right )^{2}}\)	\(21\)
risch	\(\frac {9}{\ln \relax (5)^{2} \left (x^{2} {\mathrm e}^{18}-2 \,{\mathrm e}^{9} x^{3}+x^{4}+8 \,{\mathrm e}^{18} x -6 x^{2} {\mathrm e}^{9}-2 x^{3}+16 \,{\mathrm e}^{18}+48 x \,{\mathrm e}^{9}-39 x^{2}+160 \,{\mathrm e}^{9}+40 x +400\right )}\)	\(65\)
gosper	\(\frac {9}{\ln \relax (5)^{2} \left (x^{2} {\mathrm e}^{18}-2 \,{\mathrm e}^{9} x^{3}+x^{4}+8 \,{\mathrm e}^{18} x -6 x^{2} {\mathrm e}^{9}-2 x^{3}+16 \,{\mathrm e}^{18}+48 x \,{\mathrm e}^{9}-39 x^{2}+160 \,{\mathrm e}^{9}+40 x +400\right )}\)	\(71\)
default	\(\frac {-\frac {6 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}+\left (-3 \,{\mathrm e}^{9}-15\right ) \textit {\_Z}^{2}+\left (30 \,{\mathrm e}^{9}+3 \,{\mathrm e}^{18}+75\right ) \textit {\_Z} -75 \,{\mathrm e}^{9}-{\mathrm e}^{27}-15 \,{\mathrm e}^{18}-125\right )}{\sum }\frac {\left (7440174-531441 \textit {\_R} +6022998 \,{\mathrm e}^{9}-354294 \textit {\_R} \,{\mathrm e}^{9}+46170 \,{\mathrm e}^{36}-54 \textit {\_R} \,{\mathrm e}^{45}-{\mathrm e}^{54} \textit {\_R} -1215 \,{\mathrm e}^{36} \textit {\_R} +2086398 \,{\mathrm e}^{18}-98415 \,{\mathrm e}^{18} \textit {\_R} +400950 \,{\mathrm e}^{27}-14580 \textit {\_R} \,{\mathrm e}^{27}+3186 \,{\mathrm e}^{45}+2 \,{\mathrm e}^{63}+122 \,{\mathrm e}^{54}\right ) \ln \left (x -\textit {\_R} \right )}{25-2 \textit {\_R} \,{\mathrm e}^{9}+\textit {\_R}^{2}+10 \,{\mathrm e}^{9}+{\mathrm e}^{18}-10 \textit {\_R}}\right )}{\left (729+243 \,{\mathrm e}^{9}+27 \,{\mathrm e}^{18}+{\mathrm e}^{27}\right )^{3}}-\frac {18 \left (-354294 \,{\mathrm e}^{9}-1215 \,{\mathrm e}^{36}-54 \,{\mathrm e}^{45}-{\mathrm e}^{54}-14580 \,{\mathrm e}^{27}-98415 \,{\mathrm e}^{18}-531441\right )}{\left (729+243 \,{\mathrm e}^{9}+27 \,{\mathrm e}^{18}+{\mathrm e}^{27}\right )^{3} \left (4+x \right )}-\frac {9 \left (-3720087 \,{\mathrm e}^{9}-25515 \,{\mathrm e}^{36}-1701 \,{\mathrm e}^{45}-63 \,{\mathrm e}^{54}-229635 \,{\mathrm e}^{27}-1240029 \,{\mathrm e}^{18}-{\mathrm e}^{63}-4782969\right )}{\left (729+243 \,{\mathrm e}^{9}+27 \,{\mathrm e}^{18}+{\mathrm e}^{27}\right )^{3} \left (4+x \right )^{2}}}{\ln \relax (5)^{2}}\)	\(256\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-18*exp(9)+36*x-18)/((x^3+12*x^2+48*x+64)*exp(9)^3+(-3*x^4-21*x^3+36*x^2+528*x+960)*exp(9)^2+(3*x^5+6*x^4
-141*x^3-348*x^2+1680*x+4800)*exp(9)-x^6+3*x^5+57*x^4-119*x^3-1140*x^2+1200*x+8000)/ln(5)^2,x,method=_RETURNVE
RBOSE)

[Out]

9/ln(5)^2/(4+x)^2/(-x+exp(9)+5)^2

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maxima [B]  time = 0.35, size = 53, normalized size = 2.52 \begin {gather*} \frac {9}{{\left (x^{4} - 2 \, x^{3} {\left (e^{9} + 1\right )} + x^{2} {\left (e^{18} - 6 \, e^{9} - 39\right )} + 8 \, x {\left (e^{18} + 6 \, e^{9} + 5\right )} + 16 \, e^{18} + 160 \, e^{9} + 400\right )} \log \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*exp(9)+36*x-18)/((x^3+12*x^2+48*x+64)*exp(9)^3+(-3*x^4-21*x^3+36*x^2+528*x+960)*exp(9)^2+(3*x^5
+6*x^4-141*x^3-348*x^2+1680*x+4800)*exp(9)-x^6+3*x^5+57*x^4-119*x^3-1140*x^2+1200*x+8000)/log(5)^2,x, algorith
m="maxima")

[Out]

9/((x^4 - 2*x^3*(e^9 + 1) + x^2*(e^18 - 6*e^9 - 39) + 8*x*(e^18 + 6*e^9 + 5) + 16*e^18 + 160*e^9 + 400)*log(5)
^2)

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mupad [B]  time = 2.44, size = 78, normalized size = 3.71 \begin {gather*} \frac {9}{{\ln \relax (5)}^2\,{\left ({\mathrm {e}}^9+9\right )}^2\,{\left (x+4\right )}^2}+\frac {18}{{\ln \relax (5)}^2\,{\left ({\mathrm {e}}^9+9\right )}^3\,\left (x+4\right )}+\frac {9\,\left (3\,{\mathrm {e}}^9-2\,x+19\right )}{{\ln \relax (5)}^2\,{\left ({\mathrm {e}}^9+9\right )}^3\,\left (x^2+\left (-2\,{\mathrm {e}}^9-10\right )\,x+10\,{\mathrm {e}}^9+{\mathrm {e}}^{18}+25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(18*exp(9) - 36*x + 18)/(log(5)^2*(1200*x + exp(27)*(48*x + 12*x^2 + x^3 + 64) + exp(18)*(528*x + 36*x^2
- 21*x^3 - 3*x^4 + 960) + exp(9)*(1680*x - 348*x^2 - 141*x^3 + 6*x^4 + 3*x^5 + 4800) - 1140*x^2 - 119*x^3 + 57
*x^4 + 3*x^5 - x^6 + 8000)),x)

[Out]

9/(log(5)^2*(exp(9) + 9)^2*(x + 4)^2) + 18/(log(5)^2*(exp(9) + 9)^3*(x + 4)) + (9*(3*exp(9) - 2*x + 19))/(log(
5)^2*(exp(9) + 9)^3*(10*exp(9) + exp(18) + x^2 - x*(2*exp(9) + 10) + 25))

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sympy [B]  time = 1.28, size = 116, normalized size = 5.52 \begin {gather*} \frac {9}{x^{4} \log {\relax (5 )}^{2} + x^{3} \left (- 2 e^{9} \log {\relax (5 )}^{2} - 2 \log {\relax (5 )}^{2}\right ) + x^{2} \left (- 6 e^{9} \log {\relax (5 )}^{2} - 39 \log {\relax (5 )}^{2} + e^{18} \log {\relax (5 )}^{2}\right ) + x \left (40 \log {\relax (5 )}^{2} + 48 e^{9} \log {\relax (5 )}^{2} + 8 e^{18} \log {\relax (5 )}^{2}\right ) + 400 \log {\relax (5 )}^{2} + 160 e^{9} \log {\relax (5 )}^{2} + 16 e^{18} \log {\relax (5 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*exp(9)+36*x-18)/((x**3+12*x**2+48*x+64)*exp(9)**3+(-3*x**4-21*x**3+36*x**2+528*x+960)*exp(9)**2
+(3*x**5+6*x**4-141*x**3-348*x**2+1680*x+4800)*exp(9)-x**6+3*x**5+57*x**4-119*x**3-1140*x**2+1200*x+8000)/ln(5
)**2,x)

[Out]

9/(x**4*log(5)**2 + x**3*(-2*exp(9)*log(5)**2 - 2*log(5)**2) + x**2*(-6*exp(9)*log(5)**2 - 39*log(5)**2 + exp(
18)*log(5)**2) + x*(40*log(5)**2 + 48*exp(9)*log(5)**2 + 8*exp(18)*log(5)**2) + 400*log(5)**2 + 160*exp(9)*log
(5)**2 + 16*exp(18)*log(5)**2)

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