3.40.86 \(\int \frac {e^4 (480 x-144 x^2-120 x^3+12 x^4)+e^2 (1600 x^2-640 x^3-1200 x^4+160 x^5)}{25600 x^2+19200 x^4+4800 x^6+400 x^8+e^4 (576+432 x^2+108 x^4+9 x^6)+e^2 (7680 x+5760 x^3+1440 x^5+120 x^7)} \, dx\)

Optimal. Leaf size=28 \[ \frac {(5-x) x^2}{\left (\frac {3}{4}+\frac {5 x}{e^2}\right ) \left (4+x^2\right )^2} \]

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Rubi [C]  time = 0.26, antiderivative size = 243, normalized size of antiderivative = 8.68, number of steps used = 8, number of rules used = 4, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2074, 639, 199, 203} \begin {gather*} \frac {6 e^2 \left (100+3 e^2\right ) x}{\left (1600+9 e^4\right ) \left (x^2+4\right )}-\frac {10 e^2 \left (3 \left (100+3 e^2\right ) \left (320+3 e^4\right ) x+2 \left (25600+288 e^4-27 e^6\right )\right )}{\left (1600+9 e^4\right )^2 \left (x^2+4\right )}+\frac {16 e^2 \left (\left (100+3 e^2\right ) x+5 \left (16-3 e^2\right )\right )}{\left (1600+9 e^4\right ) \left (x^2+4\right )^2}+\frac {720 e^6 \left (100+3 e^2\right )}{\left (1600+9 e^4\right )^2 \left (20 x+3 e^2\right )}+\frac {3 e^2 \left (100+3 e^2\right ) \tan ^{-1}\left (\frac {x}{2}\right )}{1600+9 e^4}-\frac {15 e^2 \left (100+3 e^2\right ) \left (320+3 e^4\right ) \tan ^{-1}\left (\frac {x}{2}\right )}{\left (1600+9 e^4\right )^2}+\frac {18 e^6 \left (100+3 e^2\right ) \tan ^{-1}\left (\frac {x}{2}\right )}{\left (1600+9 e^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^4*(480*x - 144*x^2 - 120*x^3 + 12*x^4) + E^2*(1600*x^2 - 640*x^3 - 1200*x^4 + 160*x^5))/(25600*x^2 + 19
200*x^4 + 4800*x^6 + 400*x^8 + E^4*(576 + 432*x^2 + 108*x^4 + 9*x^6) + E^2*(7680*x + 5760*x^3 + 1440*x^5 + 120
*x^7)),x]

[Out]

(720*E^6*(100 + 3*E^2))/((1600 + 9*E^4)^2*(3*E^2 + 20*x)) + (16*E^2*(5*(16 - 3*E^2) + (100 + 3*E^2)*x))/((1600
+ 9*E^4)*(4 + x^2)^2) + (6*E^2*(100 + 3*E^2)*x)/((1600 + 9*E^4)*(4 + x^2)) - (10*E^2*(2*(25600 + 288*E^4 - 27
*E^6) + 3*(100 + 3*E^2)*(320 + 3*E^4)*x))/((1600 + 9*E^4)^2*(4 + x^2)) + (18*E^6*(100 + 3*E^2)*ArcTan[x/2])/(1
600 + 9*E^4)^2 - (15*E^2*(100 + 3*E^2)*(320 + 3*E^4)*ArcTan[x/2])/(1600 + 9*E^4)^2 + (3*E^2*(100 + 3*E^2)*ArcT
an[x/2])/(1600 + 9*E^4)

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 203

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

Rule 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& 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 (-\frac {14400 e^6 \left (100+3 e^2\right )}{\left (1600+9 e^4\right )^2 \left (3 e^2+20 x\right )^2}+\frac {64 \left (4 e^2 \left (100+3 e^2\right )-5 e^2 \left (16-3 e^2\right ) x\right )}{\left (1600+9 e^4\right ) \left (4+x^2\right )^3}+\frac {40 \left (-6 e^2 \left (100+3 e^2\right ) \left (320+3 e^4\right )+e^2 \left (25600+288 e^4-27 e^6\right ) x\right )}{\left (1600+9 e^4\right )^2 \left (4+x^2\right )^2}+\frac {36 e^6 \left (100+3 e^2\right )}{\left (1600+9 e^4\right )^2 \left (4+x^2\right )}\right ) \, dx\\ &=\frac {720 e^6 \left (100+3 e^2\right )}{\left (1600+9 e^4\right )^2 \left (3 e^2+20 x\right )}+\frac {40 \int \frac {-6 e^2 \left (100+3 e^2\right ) \left (320+3 e^4\right )+e^2 \left (25600+288 e^4-27 e^6\right ) x}{\left (4+x^2\right )^2} \, dx}{\left (1600+9 e^4\right )^2}+\frac {\left (36 e^6 \left (100+3 e^2\right )\right ) \int \frac {1}{4+x^2} \, dx}{\left (1600+9 e^4\right )^2}+\frac {64 \int \frac {4 e^2 \left (100+3 e^2\right )-5 e^2 \left (16-3 e^2\right ) x}{\left (4+x^2\right )^3} \, dx}{1600+9 e^4}\\ &=\frac {720 e^6 \left (100+3 e^2\right )}{\left (1600+9 e^4\right )^2 \left (3 e^2+20 x\right )}+\frac {16 e^2 \left (5 \left (16-3 e^2\right )+\left (100+3 e^2\right ) x\right )}{\left (1600+9 e^4\right ) \left (4+x^2\right )^2}-\frac {10 e^2 \left (2 \left (25600+288 e^4-27 e^6\right )+3 \left (100+3 e^2\right ) \left (320+3 e^4\right ) x\right )}{\left (1600+9 e^4\right )^2 \left (4+x^2\right )}+\frac {18 e^6 \left (100+3 e^2\right ) \tan ^{-1}\left (\frac {x}{2}\right )}{\left (1600+9 e^4\right )^2}-\frac {\left (30 e^2 \left (100+3 e^2\right ) \left (320+3 e^4\right )\right ) \int \frac {1}{4+x^2} \, dx}{\left (1600+9 e^4\right )^2}+\frac {\left (48 e^2 \left (100+3 e^2\right )\right ) \int \frac {1}{\left (4+x^2\right )^2} \, dx}{1600+9 e^4}\\ &=\frac {720 e^6 \left (100+3 e^2\right )}{\left (1600+9 e^4\right )^2 \left (3 e^2+20 x\right )}+\frac {16 e^2 \left (5 \left (16-3 e^2\right )+\left (100+3 e^2\right ) x\right )}{\left (1600+9 e^4\right ) \left (4+x^2\right )^2}+\frac {6 e^2 \left (100+3 e^2\right ) x}{\left (1600+9 e^4\right ) \left (4+x^2\right )}-\frac {10 e^2 \left (2 \left (25600+288 e^4-27 e^6\right )+3 \left (100+3 e^2\right ) \left (320+3 e^4\right ) x\right )}{\left (1600+9 e^4\right )^2 \left (4+x^2\right )}+\frac {18 e^6 \left (100+3 e^2\right ) \tan ^{-1}\left (\frac {x}{2}\right )}{\left (1600+9 e^4\right )^2}-\frac {15 e^2 \left (100+3 e^2\right ) \left (320+3 e^4\right ) \tan ^{-1}\left (\frac {x}{2}\right )}{\left (1600+9 e^4\right )^2}+\frac {\left (6 e^2 \left (100+3 e^2\right )\right ) \int \frac {1}{4+x^2} \, dx}{1600+9 e^4}\\ &=\frac {720 e^6 \left (100+3 e^2\right )}{\left (1600+9 e^4\right )^2 \left (3 e^2+20 x\right )}+\frac {16 e^2 \left (5 \left (16-3 e^2\right )+\left (100+3 e^2\right ) x\right )}{\left (1600+9 e^4\right ) \left (4+x^2\right )^2}+\frac {6 e^2 \left (100+3 e^2\right ) x}{\left (1600+9 e^4\right ) \left (4+x^2\right )}-\frac {10 e^2 \left (2 \left (25600+288 e^4-27 e^6\right )+3 \left (100+3 e^2\right ) \left (320+3 e^4\right ) x\right )}{\left (1600+9 e^4\right )^2 \left (4+x^2\right )}+\frac {18 e^6 \left (100+3 e^2\right ) \tan ^{-1}\left (\frac {x}{2}\right )}{\left (1600+9 e^4\right )^2}-\frac {15 e^2 \left (100+3 e^2\right ) \left (320+3 e^4\right ) \tan ^{-1}\left (\frac {x}{2}\right )}{\left (1600+9 e^4\right )^2}+\frac {3 e^2 \left (100+3 e^2\right ) \tan ^{-1}\left (\frac {x}{2}\right )}{1600+9 e^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 29, normalized size = 1.04 \begin {gather*} -\frac {4 e^2 (-5+x) x^2}{\left (3 e^2+20 x\right ) \left (4+x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^4*(480*x - 144*x^2 - 120*x^3 + 12*x^4) + E^2*(1600*x^2 - 640*x^3 - 1200*x^4 + 160*x^5))/(25600*x^
2 + 19200*x^4 + 4800*x^6 + 400*x^8 + E^4*(576 + 432*x^2 + 108*x^4 + 9*x^6) + E^2*(7680*x + 5760*x^3 + 1440*x^5
+ 120*x^7)),x]

[Out]

(-4*E^2*(-5 + x)*x^2)/((3*E^2 + 20*x)*(4 + x^2)^2)

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fricas [A]  time = 1.12, size = 43, normalized size = 1.54 \begin {gather*} -\frac {4 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{2}}{20 \, x^{5} + 160 \, x^{3} + 3 \, {\left (x^{4} + 8 \, x^{2} + 16\right )} e^{2} + 320 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^4-120*x^3-144*x^2+480*x)*exp(2)^2+(160*x^5-1200*x^4-640*x^3+1600*x^2)*exp(2))/((9*x^6+108*x^4
+432*x^2+576)*exp(2)^2+(120*x^7+1440*x^5+5760*x^3+7680*x)*exp(2)+400*x^8+4800*x^6+19200*x^4+25600*x^2),x, algo
rithm="fricas")

[Out]

-4*(x^3 - 5*x^2)*e^2/(20*x^5 + 160*x^3 + 3*(x^4 + 8*x^2 + 16)*e^2 + 320*x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^4-120*x^3-144*x^2+480*x)*exp(2)^2+(160*x^5-1200*x^4-640*x^3+1600*x^2)*exp(2))/((9*x^6+108*x^4
+432*x^2+576)*exp(2)^2+(120*x^7+1440*x^5+5760*x^3+7680*x)*exp(2)+400*x^8+4800*x^6+19200*x^4+25600*x^2),x, algo
rithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 4*((-39366000*exp(4)^5+39366000*exp(4)^4
*exp(2)^2+1679616000*exp(4)^4*exp(2)[phone]*exp(4)^4-1679616000*exp(4)^3*exp(2)^3+69984000000*exp(4)^3*ex
p(2)^2-447897600000*e

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maple [A]  time = 0.28, size = 34, normalized size = 1.21

method	result	size
norman	\(\frac {20 x^{2} {\mathrm e}^{2}-4 x^{3} {\mathrm e}^{2}}{\left (x^{2}+4\right )^{2} \left (3 \,{\mathrm e}^{2}+20 x \right )}\)	\(34\)
gosper	\(-\frac {4 x^{2} \left (x -5\right ) {\mathrm e}^{2}}{3 x^{4} {\mathrm e}^{2}+20 x^{5}+24 x^{2} {\mathrm e}^{2}+160 x^{3}+48 \,{\mathrm e}^{2}+320 x}\)	\(45\)
risch	\(\frac {-\frac {4 x^{3} {\mathrm e}^{2}}{3}+\frac {20 x^{2} {\mathrm e}^{2}}{3}}{x^{4} {\mathrm e}^{2}+\frac {20 x^{5}}{3}+8 x^{2} {\mathrm e}^{2}+\frac {160 x^{3}}{3}+16 \,{\mathrm e}^{2}+\frac {320 x}{3}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x^4-120*x^3-144*x^2+480*x)*exp(2)^2+(160*x^5-1200*x^4-640*x^3+1600*x^2)*exp(2))/((9*x^6+108*x^4+432*x
^2+576)*exp(2)^2+(120*x^7+1440*x^5+5760*x^3+7680*x)*exp(2)+400*x^8+4800*x^6+19200*x^4+25600*x^2),x,method=_RET
URNVERBOSE)

[Out]

(20*x^2*exp(2)-4*x^3*exp(2))/(x^2+4)^2/(3*exp(2)+20*x)

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maxima [B]  time = 0.37, size = 50, normalized size = 1.79 \begin {gather*} -\frac {4 \, {\left (x^{3} e^{2} - 5 \, x^{2} e^{2}\right )}}{20 \, x^{5} + 3 \, x^{4} e^{2} + 160 \, x^{3} + 24 \, x^{2} e^{2} + 320 \, x + 48 \, e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^4-120*x^3-144*x^2+480*x)*exp(2)^2+(160*x^5-1200*x^4-640*x^3+1600*x^2)*exp(2))/((9*x^6+108*x^4
+432*x^2+576)*exp(2)^2+(120*x^7+1440*x^5+5760*x^3+7680*x)*exp(2)+400*x^8+4800*x^6+19200*x^4+25600*x^2),x, algo
rithm="maxima")

[Out]

-4*(x^3*e^2 - 5*x^2*e^2)/(20*x^5 + 3*x^4*e^2 + 160*x^3 + 24*x^2*e^2 + 320*x + 48*e^2)

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mupad [B]  time = 2.74, size = 50, normalized size = 1.79 \begin {gather*} \frac {20\,x^2\,{\mathrm {e}}^2-4\,x^3\,{\mathrm {e}}^2}{20\,x^5+3\,{\mathrm {e}}^2\,x^4+160\,x^3+24\,{\mathrm {e}}^2\,x^2+320\,x+48\,{\mathrm {e}}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4)*(480*x - 144*x^2 - 120*x^3 + 12*x^4) + exp(2)*(1600*x^2 - 640*x^3 - 1200*x^4 + 160*x^5))/(exp(2)*(
7680*x + 5760*x^3 + 1440*x^5 + 120*x^7) + exp(4)*(432*x^2 + 108*x^4 + 9*x^6 + 576) + 25600*x^2 + 19200*x^4 + 4
800*x^6 + 400*x^8),x)

[Out]

(20*x^2*exp(2) - 4*x^3*exp(2))/(320*x + 48*exp(2) + 24*x^2*exp(2) + 3*x^4*exp(2) + 160*x^3 + 20*x^5)

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sympy [B]  time = 3.38, size = 51, normalized size = 1.82 \begin {gather*} \frac {- 4 x^{3} e^{2} + 20 x^{2} e^{2}}{20 x^{5} + 3 x^{4} e^{2} + 160 x^{3} + 24 x^{2} e^{2} + 320 x + 48 e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x**4-120*x**3-144*x**2+480*x)*exp(2)**2+(160*x**5-1200*x**4-640*x**3+1600*x**2)*exp(2))/((9*x**
6+108*x**4+432*x**2+576)*exp(2)**2+(120*x**7+1440*x**5+5760*x**3+7680*x)*exp(2)+400*x**8+4800*x**6+19200*x**4+
25600*x**2),x)

[Out]

(-4*x**3*exp(2) + 20*x**2*exp(2))/(20*x**5 + 3*x**4*exp(2) + 160*x**3 + 24*x**2*exp(2) + 320*x + 48*exp(2))

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