3.65.61 \(\int \frac {7040+4096 x+810 x^2+67 x^3+2 x^4+e^9 (7+2 x)+e^6 (210+81 x+6 x^2)+e^3 (2104+1020 x+141 x^2+6 x^3)}{1000+e^9+300 x+30 x^2+x^3+e^6 (30+3 x)+e^3 (300+60 x+3 x^2)} \, dx\)

Optimal. Leaf size=15 \[ x \left (7+x+\frac {4}{\left (10+e^3+x\right )^2}\right ) \]

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Rubi [B]  time = 0.11, antiderivative size = 32, normalized size of antiderivative = 2.13, number of steps used = 2, number of rules used = 1, integrand size = 104, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {2074} \begin {gather*} x^2+7 x+\frac {4}{x+e^3+10}-\frac {4 \left (10+e^3\right )}{\left (x+e^3+10\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(7040 + 4096*x + 810*x^2 + 67*x^3 + 2*x^4 + E^9*(7 + 2*x) + E^6*(210 + 81*x + 6*x^2) + E^3*(2104 + 1020*x
+ 141*x^2 + 6*x^3))/(1000 + E^9 + 300*x + 30*x^2 + x^3 + E^6*(30 + 3*x) + E^3*(300 + 60*x + 3*x^2)),x]

[Out]

7*x + x^2 - (4*(10 + E^3))/(10 + E^3 + x)^2 + 4/(10 + E^3 + 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} &=\int \left (7+2 x+\frac {8 \left (10+e^3\right )}{\left (10+e^3+x\right )^3}-\frac {4}{\left (10+e^3+x\right )^2}\right ) \, dx\\ &=7 x+x^2-\frac {4 \left (10+e^3\right )}{\left (10+e^3+x\right )^2}+\frac {4}{10+e^3+x}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.04, size = 48, normalized size = 3.20 \begin {gather*} -\frac {4 \left (10+e^3\right )}{\left (10+e^3+x\right )^2}+\frac {4}{10+e^3+x}+\left (-13-2 e^3\right ) \left (10+e^3+x\right )+\left (10+e^3+x\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(7040 + 4096*x + 810*x^2 + 67*x^3 + 2*x^4 + E^9*(7 + 2*x) + E^6*(210 + 81*x + 6*x^2) + E^3*(2104 + 1
020*x + 141*x^2 + 6*x^3))/(1000 + E^9 + 300*x + 30*x^2 + x^3 + E^6*(30 + 3*x) + E^3*(300 + 60*x + 3*x^2)),x]

[Out]

(-4*(10 + E^3))/(10 + E^3 + x)^2 + 4/(10 + E^3 + x) + (-13 - 2*E^3)*(10 + E^3 + x) + (10 + E^3 + x)^2

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fricas [B]  time = 0.68, size = 63, normalized size = 4.20 \begin {gather*} \frac {x^{4} + 27 \, x^{3} + 240 \, x^{2} + {\left (x^{2} + 7 \, x\right )} e^{6} + 2 \, {\left (x^{3} + 17 \, x^{2} + 70 \, x\right )} e^{3} + 704 \, x}{x^{2} + 2 \, {\left (x + 10\right )} e^{3} + 20 \, x + e^{6} + 100} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7+2*x)*exp(3)^3+(6*x^2+81*x+210)*exp(3)^2+(6*x^3+141*x^2+1020*x+2104)*exp(3)+2*x^4+67*x^3+810*x^2+
4096*x+7040)/(exp(3)^3+(3*x+30)*exp(3)^2+(3*x^2+60*x+300)*exp(3)+x^3+30*x^2+300*x+1000),x, algorithm="fricas")

[Out]

(x^4 + 27*x^3 + 240*x^2 + (x^2 + 7*x)*e^6 + 2*(x^3 + 17*x^2 + 70*x)*e^3 + 704*x)/(x^2 + 2*(x + 10)*e^3 + 20*x
+ e^6 + 100)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{4} + 67 \, x^{3} + 810 \, x^{2} + {\left (2 \, x + 7\right )} e^{9} + 3 \, {\left (2 \, x^{2} + 27 \, x + 70\right )} e^{6} + {\left (6 \, x^{3} + 141 \, x^{2} + 1020 \, x + 2104\right )} e^{3} + 4096 \, x + 7040}{x^{3} + 30 \, x^{2} + 3 \, {\left (x + 10\right )} e^{6} + 3 \, {\left (x^{2} + 20 \, x + 100\right )} e^{3} + 300 \, x + e^{9} + 1000}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7+2*x)*exp(3)^3+(6*x^2+81*x+210)*exp(3)^2+(6*x^3+141*x^2+1020*x+2104)*exp(3)+2*x^4+67*x^3+810*x^2+
4096*x+7040)/(exp(3)^3+(3*x+30)*exp(3)^2+(3*x^2+60*x+300)*exp(3)+x^3+30*x^2+300*x+1000),x, algorithm="giac")

[Out]

integrate((2*x^4 + 67*x^3 + 810*x^2 + (2*x + 7)*e^9 + 3*(2*x^2 + 27*x + 70)*e^6 + (6*x^3 + 141*x^2 + 1020*x +
2104)*e^3 + 4096*x + 7040)/(x^3 + 30*x^2 + 3*(x + 10)*e^6 + 3*(x^2 + 20*x + 100)*e^3 + 300*x + e^9 + 1000), x)

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maple [B]  time = 0.11, size = 32, normalized size = 2.13

method	result	size
risch	\(x^{2}+7 x +\frac {4 x}{{\mathrm e}^{6}+2 x \,{\mathrm e}^{3}+x^{2}+20 \,{\mathrm e}^{3}+20 x +100}\)	\(32\)
norman	\(\frac {x^{4}+\left (27+2 \,{\mathrm e}^{3}\right ) x^{3}-24000+\left (-2 \,{\mathrm e}^{9}-81 \,{\mathrm e}^{6}-1020 \,{\mathrm e}^{3}-4096\right ) x -{\mathrm e}^{12}-54 \,{\mathrm e}^{9}-1020 \,{\mathrm e}^{6}-8200 \,{\mathrm e}^{3}}{\left (10+{\mathrm e}^{3}+x \right )^{2}}\)	\(66\)
default	\(7 x +x^{2}+\frac {4 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}+\left (3 \,{\mathrm e}^{3}+30\right ) \textit {\_Z}^{2}+\left (60 \,{\mathrm e}^{3}+3 \,{\mathrm e}^{6}+300\right ) \textit {\_Z} +300 \,{\mathrm e}^{3}+{\mathrm e}^{9}+30 \,{\mathrm e}^{6}+1000\right )}{\sum }\frac {\left (10-\textit {\_R} +{\mathrm e}^{3}\right ) \ln \left (x -\textit {\_R} \right )}{100+{\mathrm e}^{6}+2 \textit {\_R} \,{\mathrm e}^{3}+\textit {\_R}^{2}+20 \,{\mathrm e}^{3}+20 \textit {\_R}}\right )}{3}\)	\(86\)
gosper	\(-\frac {{\mathrm e}^{12}+2 x \,{\mathrm e}^{9}-2 x^{3} {\mathrm e}^{3}-x^{4}+54 \,{\mathrm e}^{9}+81 x \,{\mathrm e}^{6}-27 x^{3}+1020 \,{\mathrm e}^{6}+1020 x \,{\mathrm e}^{3}+8200 \,{\mathrm e}^{3}+4096 x +24000}{{\mathrm e}^{6}+2 x \,{\mathrm e}^{3}+x^{2}+20 \,{\mathrm e}^{3}+20 x +100}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((7+2*x)*exp(3)^3+(6*x^2+81*x+210)*exp(3)^2+(6*x^3+141*x^2+1020*x+2104)*exp(3)+2*x^4+67*x^3+810*x^2+4096*x
+7040)/(exp(3)^3+(3*x+30)*exp(3)^2+(3*x^2+60*x+300)*exp(3)+x^3+30*x^2+300*x+1000),x,method=_RETURNVERBOSE)

[Out]

x^2+7*x+4*x/(exp(6)+2*x*exp(3)+x^2+20*exp(3)+20*x+100)

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maxima [B]  time = 0.38, size = 30, normalized size = 2.00 \begin {gather*} x^{2} + 7 \, x + \frac {4 \, x}{x^{2} + 2 \, x {\left (e^{3} + 10\right )} + e^{6} + 20 \, e^{3} + 100} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7+2*x)*exp(3)^3+(6*x^2+81*x+210)*exp(3)^2+(6*x^3+141*x^2+1020*x+2104)*exp(3)+2*x^4+67*x^3+810*x^2+
4096*x+7040)/(exp(3)^3+(3*x+30)*exp(3)^2+(3*x^2+60*x+300)*exp(3)+x^3+30*x^2+300*x+1000),x, algorithm="maxima")

[Out]

x^2 + 7*x + 4*x/(x^2 + 2*x*(e^3 + 10) + e^6 + 20*e^3 + 100)

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mupad [B]  time = 4.09, size = 31, normalized size = 2.07 \begin {gather*} 7\,x+\frac {4\,x}{x^2+\left (2\,{\mathrm {e}}^3+20\right )\,x+20\,{\mathrm {e}}^3+{\mathrm {e}}^6+100}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4096*x + exp(6)*(81*x + 6*x^2 + 210) + exp(3)*(1020*x + 141*x^2 + 6*x^3 + 2104) + 810*x^2 + 67*x^3 + 2*x^
4 + exp(9)*(2*x + 7) + 7040)/(300*x + exp(9) + exp(3)*(60*x + 3*x^2 + 300) + 30*x^2 + x^3 + exp(6)*(3*x + 30)
+ 1000),x)

[Out]

7*x + (4*x)/(20*exp(3) + exp(6) + x^2 + x*(2*exp(3) + 20) + 100) + x^2

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sympy [B]  time = 0.30, size = 31, normalized size = 2.07 \begin {gather*} x^{2} + 7 x + \frac {4 x}{x^{2} + x \left (20 + 2 e^{3}\right ) + 100 + 20 e^{3} + e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7+2*x)*exp(3)**3+(6*x**2+81*x+210)*exp(3)**2+(6*x**3+141*x**2+1020*x+2104)*exp(3)+2*x**4+67*x**3+8
10*x**2+4096*x+7040)/(exp(3)**3+(3*x+30)*exp(3)**2+(3*x**2+60*x+300)*exp(3)+x**3+30*x**2+300*x+1000),x)

[Out]

x**2 + 7*x + 4*x/(x**2 + x*(20 + 2*exp(3)) + 100 + 20*exp(3) + exp(6))

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