3.69.44 \(\int \frac {128-480 x+600 x^2-154 x^3-240 x^4+150 x^5+24 x^6-30 x^7+2 x^9+e^{10} (32 x-16 x^4)}{64-240 x+300 x^2-77 x^3-120 x^4+75 x^5+12 x^6-15 x^7+x^9} \, dx\)

Optimal. Leaf size=26 \[ 2 x+\frac {4 e^{10} x^2}{\left (4-x \left (5-x^2\right )\right )^2} \]

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Rubi [B]  time = 0.31, antiderivative size = 96, normalized size of antiderivative = 3.69, number of steps used = 11, number of rules used = 5, integrand size = 94, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2074, 638, 614, 618, 206} \begin {gather*} \frac {21 e^{10} (2 x+1)}{17 \left (-x^2-x+4\right )}+\frac {e^{10} (43 x+166)}{17 \left (-x^2-x+4\right )}+\frac {e^{10} (7 x+20)}{\left (-x^2-x+4\right )^2}+2 x-\frac {5 e^{10}}{1-x}+\frac {e^{10}}{(1-x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(128 - 480*x + 600*x^2 - 154*x^3 - 240*x^4 + 150*x^5 + 24*x^6 - 30*x^7 + 2*x^9 + E^10*(32*x - 16*x^4))/(64
- 240*x + 300*x^2 - 77*x^3 - 120*x^4 + 75*x^5 + 12*x^6 - 15*x^7 + x^9),x]

[Out]

E^10/(1 - x)^2 - (5*E^10)/(1 - x) + 2*x + (E^10*(20 + 7*x))/(4 - x - x^2)^2 + (21*E^10*(1 + 2*x))/(17*(4 - x -
x^2)) + (E^10*(166 + 43*x))/(17*(4 - x - x^2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

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 (2-\frac {2 e^{10}}{(-1+x)^3}-\frac {5 e^{10}}{(-1+x)^2}-\frac {2 e^{10} (76+33 x)}{\left (-4+x+x^2\right )^3}+\frac {e^{10} (30+17 x)}{\left (-4+x+x^2\right )^2}+\frac {5 e^{10}}{-4+x+x^2}\right ) \, dx\\ &=\frac {e^{10}}{(1-x)^2}-\frac {5 e^{10}}{1-x}+2 x+e^{10} \int \frac {30+17 x}{\left (-4+x+x^2\right )^2} \, dx-\left (2 e^{10}\right ) \int \frac {76+33 x}{\left (-4+x+x^2\right )^3} \, dx+\left (5 e^{10}\right ) \int \frac {1}{-4+x+x^2} \, dx\\ &=\frac {e^{10}}{(1-x)^2}-\frac {5 e^{10}}{1-x}+2 x+\frac {e^{10} (20+7 x)}{\left (4-x-x^2\right )^2}+\frac {e^{10} (166+43 x)}{17 \left (4-x-x^2\right )}-\frac {1}{17} \left (43 e^{10}\right ) \int \frac {1}{-4+x+x^2} \, dx-\left (10 e^{10}\right ) \operatorname {Subst}\left (\int \frac {1}{17-x^2} \, dx,x,1+2 x\right )+\left (21 e^{10}\right ) \int \frac {1}{\left (-4+x+x^2\right )^2} \, dx\\ &=\frac {e^{10}}{(1-x)^2}-\frac {5 e^{10}}{1-x}+2 x+\frac {e^{10} (20+7 x)}{\left (4-x-x^2\right )^2}+\frac {21 e^{10} (1+2 x)}{17 \left (4-x-x^2\right )}+\frac {e^{10} (166+43 x)}{17 \left (4-x-x^2\right )}-\frac {10 e^{10} \tanh ^{-1}\left (\frac {1+2 x}{\sqrt {17}}\right )}{\sqrt {17}}-\frac {1}{17} \left (42 e^{10}\right ) \int \frac {1}{-4+x+x^2} \, dx+\frac {1}{17} \left (86 e^{10}\right ) \operatorname {Subst}\left (\int \frac {1}{17-x^2} \, dx,x,1+2 x\right )\\ &=\frac {e^{10}}{(1-x)^2}-\frac {5 e^{10}}{1-x}+2 x+\frac {e^{10} (20+7 x)}{\left (4-x-x^2\right )^2}+\frac {21 e^{10} (1+2 x)}{17 \left (4-x-x^2\right )}+\frac {e^{10} (166+43 x)}{17 \left (4-x-x^2\right )}-\frac {84 e^{10} \tanh ^{-1}\left (\frac {1+2 x}{\sqrt {17}}\right )}{17 \sqrt {17}}+\frac {1}{17} \left (84 e^{10}\right ) \operatorname {Subst}\left (\int \frac {1}{17-x^2} \, dx,x,1+2 x\right )\\ &=\frac {e^{10}}{(1-x)^2}-\frac {5 e^{10}}{1-x}+2 x+\frac {e^{10} (20+7 x)}{\left (4-x-x^2\right )^2}+\frac {21 e^{10} (1+2 x)}{17 \left (4-x-x^2\right )}+\frac {e^{10} (166+43 x)}{17 \left (4-x-x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 22, normalized size = 0.85 \begin {gather*} 2 \left (x+\frac {2 e^{10} x^2}{\left (4-5 x+x^3\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(128 - 480*x + 600*x^2 - 154*x^3 - 240*x^4 + 150*x^5 + 24*x^6 - 30*x^7 + 2*x^9 + E^10*(32*x - 16*x^4
))/(64 - 240*x + 300*x^2 - 77*x^3 - 120*x^4 + 75*x^5 + 12*x^6 - 15*x^7 + x^9),x]

[Out]

2*(x + (2*E^10*x^2)/(4 - 5*x + x^3)^2)

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fricas [B]  time = 0.61, size = 61, normalized size = 2.35 \begin {gather*} \frac {2 \, {\left (x^{7} - 10 \, x^{5} + 8 \, x^{4} + 25 \, x^{3} + 2 \, x^{2} e^{10} - 40 \, x^{2} + 16 \, x\right )}}{x^{6} - 10 \, x^{4} + 8 \, x^{3} + 25 \, x^{2} - 40 \, x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^4+32*x)*exp(5)^2+2*x^9-30*x^7+24*x^6+150*x^5-240*x^4-154*x^3+600*x^2-480*x+128)/(x^9-15*x^7+
12*x^6+75*x^5-120*x^4-77*x^3+300*x^2-240*x+64),x, algorithm="fricas")

[Out]

2*(x^7 - 10*x^5 + 8*x^4 + 25*x^3 + 2*x^2*e^10 - 40*x^2 + 16*x)/(x^6 - 10*x^4 + 8*x^3 + 25*x^2 - 40*x + 16)

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giac [A]  time = 0.15, size = 21, normalized size = 0.81 \begin {gather*} 2 \, x + \frac {4 \, x^{2} e^{10}}{{\left (x^{3} - 5 \, x + 4\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^4+32*x)*exp(5)^2+2*x^9-30*x^7+24*x^6+150*x^5-240*x^4-154*x^3+600*x^2-480*x+128)/(x^9-15*x^7+
12*x^6+75*x^5-120*x^4-77*x^3+300*x^2-240*x+64),x, algorithm="giac")

[Out]

2*x + 4*x^2*e^10/(x^3 - 5*x + 4)^2

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maple [A]  time = 0.08, size = 37, normalized size = 1.42

method	result	size
risch	\(2 x +\frac {4 \,{\mathrm e}^{10} x^{2}}{x^{6}-10 x^{4}+8 x^{3}+25 x^{2}-40 x +16}\)	\(37\)
default	\(2 x +\frac {{\mathrm e}^{10} \left (-5 x^{3}-16 x^{2}+16 x +64\right )}{\left (x^{2}+x -4\right )^{2}}+\frac {{\mathrm e}^{10}}{\left (x -1\right )^{2}}+\frac {5 \,{\mathrm e}^{10}}{x -1}\)	\(48\)
norman	\(\frac {16 x^{4}+32 x +50 x^{3}+\left (4 \,{\mathrm e}^{10}-80\right ) x^{2}-20 x^{5}+2 x^{7}}{\left (x^{3}-5 x +4\right )^{2}}\)	\(48\)
gosper	\(\frac {2 x \left (x^{6}-10 x^{4}+2 x \,{\mathrm e}^{10}+8 x^{3}+25 x^{2}-40 x +16\right )}{x^{6}-10 x^{4}+8 x^{3}+25 x^{2}-40 x +16}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-16*x^4+32*x)*exp(5)^2+2*x^9-30*x^7+24*x^6+150*x^5-240*x^4-154*x^3+600*x^2-480*x+128)/(x^9-15*x^7+12*x^6
+75*x^5-120*x^4-77*x^3+300*x^2-240*x+64),x,method=_RETURNVERBOSE)

[Out]

2*x+4*exp(10)*x^2/(x^6-10*x^4+8*x^3+25*x^2-40*x+16)

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maxima [A]  time = 0.35, size = 36, normalized size = 1.38 \begin {gather*} \frac {4 \, x^{2} e^{10}}{x^{6} - 10 \, x^{4} + 8 \, x^{3} + 25 \, x^{2} - 40 \, x + 16} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^4+32*x)*exp(5)^2+2*x^9-30*x^7+24*x^6+150*x^5-240*x^4-154*x^3+600*x^2-480*x+128)/(x^9-15*x^7+
12*x^6+75*x^5-120*x^4-77*x^3+300*x^2-240*x+64),x, algorithm="maxima")

[Out]

4*x^2*e^10/(x^6 - 10*x^4 + 8*x^3 + 25*x^2 - 40*x + 16) + 2*x

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mupad [B]  time = 0.09, size = 21, normalized size = 0.81 \begin {gather*} 2\,x+\frac {4\,x^2\,{\mathrm {e}}^{10}}{{\left (x^3-5\,x+4\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(10)*(32*x - 16*x^4) - 480*x + 600*x^2 - 154*x^3 - 240*x^4 + 150*x^5 + 24*x^6 - 30*x^7 + 2*x^9 + 128)/
(300*x^2 - 240*x - 77*x^3 - 120*x^4 + 75*x^5 + 12*x^6 - 15*x^7 + x^9 + 64),x)

[Out]

2*x + (4*x^2*exp(10))/(x^3 - 5*x + 4)^2

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sympy [A]  time = 0.46, size = 34, normalized size = 1.31 \begin {gather*} \frac {4 x^{2} e^{10}}{x^{6} - 10 x^{4} + 8 x^{3} + 25 x^{2} - 40 x + 16} + 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x**4+32*x)*exp(5)**2+2*x**9-30*x**7+24*x**6+150*x**5-240*x**4-154*x**3+600*x**2-480*x+128)/(x*
*9-15*x**7+12*x**6+75*x**5-120*x**4-77*x**3+300*x**2-240*x+64),x)

[Out]

4*x**2*exp(10)/(x**6 - 10*x**4 + 8*x**3 + 25*x**2 - 40*x + 16) + 2*x

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