3.84.40 \(\int \frac {-1125000-757500 x-191250 x^2-21250 x^3+e^{e^{x^2}} (-281250-93750 x+e^{x^2} (-562500 x^2-375000 x^3-62500 x^4))}{9000000 x^3+140625 e^{3 e^{x^2}} x^3+4590000 x^4+780300 x^5+44217 x^6+e^{2 e^{x^2}} (1687500 x^3+286875 x^4)+e^{e^{x^2}} (6750000 x^3+2295000 x^4+195075 x^5)} \, dx\)

Optimal. Leaf size=31 \[ \frac {(3+x)^2}{x^2 \left (-\frac {126 x}{25}+3 \left (-4-e^{e^{x^2}}+x\right )\right )^2} \]

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Rubi [F]  time = 4.24, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-1125000-757500 x-191250 x^2-21250 x^3+e^{e^{x^2}} \left (-281250-93750 x+e^{x^2} \left (-562500 x^2-375000 x^3-62500 x^4\right )\right )}{9000000 x^3+140625 e^{3 e^{x^2}} x^3+4590000 x^4+780300 x^5+44217 x^6+e^{2 e^{x^2}} \left (1687500 x^3+286875 x^4\right )+e^{e^{x^2}} \left (6750000 x^3+2295000 x^4+195075 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1125000 - 757500*x - 191250*x^2 - 21250*x^3 + E^E^x^2*(-281250 - 93750*x + E^x^2*(-562500*x^2 - 375000*x
^3 - 62500*x^4)))/(9000000*x^3 + 140625*E^(3*E^x^2)*x^3 + 4590000*x^4 + 780300*x^5 + 44217*x^6 + E^(2*E^x^2)*(
1687500*x^3 + 286875*x^4) + E^E^x^2*(6750000*x^3 + 2295000*x^4 + 195075*x^5)),x]

[Out]

(-21250*Defer[Int][(100 + 25*E^E^x^2 + 17*x)^(-3), x])/9 - (125000*Defer[Int][E^(E^x^2 + x^2)/(100 + 25*E^E^x^
2 + 17*x)^3, x])/3 - 125000*Defer[Int][1/(x^3*(100 + 25*E^E^x^2 + 17*x)^3), x] - 31250*Defer[Int][E^E^x^2/(x^3
*(100 + 25*E^E^x^2 + 17*x)^3), x] - (252500*Defer[Int][1/(x^2*(100 + 25*E^E^x^2 + 17*x)^3), x])/3 - (31250*Def
er[Int][E^E^x^2/(x^2*(100 + 25*E^E^x^2 + 17*x)^3), x])/3 - 21250*Defer[Int][1/(x*(100 + 25*E^E^x^2 + 17*x)^3),
x] - 62500*Defer[Int][E^(E^x^2 + x^2)/(x*(100 + 25*E^E^x^2 + 17*x)^3), x] - (62500*Defer[Int][(E^(E^x^2 + x^2
)*x)/(100 + 25*E^E^x^2 + 17*x)^3, x])/9

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1250 (3+x) \left (-300-75 e^{e^{x^2}}-102 x-17 x^2-50 e^{e^{x^2}+x^2} x^2 (3+x)\right )}{9 x^3 \left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx\\ &=\frac {1250}{9} \int \frac {(3+x) \left (-300-75 e^{e^{x^2}}-102 x-17 x^2-50 e^{e^{x^2}+x^2} x^2 (3+x)\right )}{x^3 \left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx\\ &=\frac {1250}{9} \int \left (-\frac {300 (3+x)}{x^3 \left (100+25 e^{e^{x^2}}+17 x\right )^3}-\frac {75 e^{e^{x^2}} (3+x)}{x^3 \left (100+25 e^{e^{x^2}}+17 x\right )^3}-\frac {102 (3+x)}{x^2 \left (100+25 e^{e^{x^2}}+17 x\right )^3}-\frac {17 (3+x)}{x \left (100+25 e^{e^{x^2}}+17 x\right )^3}-\frac {50 e^{e^{x^2}+x^2} (3+x)^2}{x \left (100+25 e^{e^{x^2}}+17 x\right )^3}\right ) \, dx\\ &=-\left (\frac {21250}{9} \int \frac {3+x}{x \left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx\right )-\frac {62500}{9} \int \frac {e^{e^{x^2}+x^2} (3+x)^2}{x \left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx-\frac {31250}{3} \int \frac {e^{e^{x^2}} (3+x)}{x^3 \left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx-\frac {42500}{3} \int \frac {3+x}{x^2 \left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx-\frac {125000}{3} \int \frac {3+x}{x^3 \left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx\\ &=-\left (\frac {21250}{9} \int \left (\frac {1}{\left (100+25 e^{e^{x^2}}+17 x\right )^3}+\frac {3}{x \left (100+25 e^{e^{x^2}}+17 x\right )^3}\right ) \, dx\right )-\frac {62500}{9} \int \left (\frac {6 e^{e^{x^2}+x^2}}{\left (100+25 e^{e^{x^2}}+17 x\right )^3}+\frac {9 e^{e^{x^2}+x^2}}{x \left (100+25 e^{e^{x^2}}+17 x\right )^3}+\frac {e^{e^{x^2}+x^2} x}{\left (100+25 e^{e^{x^2}}+17 x\right )^3}\right ) \, dx-\frac {31250}{3} \int \left (\frac {3 e^{e^{x^2}}}{x^3 \left (100+25 e^{e^{x^2}}+17 x\right )^3}+\frac {e^{e^{x^2}}}{x^2 \left (100+25 e^{e^{x^2}}+17 x\right )^3}\right ) \, dx-\frac {42500}{3} \int \left (\frac {3}{x^2 \left (100+25 e^{e^{x^2}}+17 x\right )^3}+\frac {1}{x \left (100+25 e^{e^{x^2}}+17 x\right )^3}\right ) \, dx-\frac {125000}{3} \int \left (\frac {3}{x^3 \left (100+25 e^{e^{x^2}}+17 x\right )^3}+\frac {1}{x^2 \left (100+25 e^{e^{x^2}}+17 x\right )^3}\right ) \, dx\\ &=-\left (\frac {21250}{9} \int \frac {1}{\left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx\right )-\frac {62500}{9} \int \frac {e^{e^{x^2}+x^2} x}{\left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx-\frac {21250}{3} \int \frac {1}{x \left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx-\frac {31250}{3} \int \frac {e^{e^{x^2}}}{x^2 \left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx-\frac {42500}{3} \int \frac {1}{x \left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx-31250 \int \frac {e^{e^{x^2}}}{x^3 \left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx-\frac {125000}{3} \int \frac {e^{e^{x^2}+x^2}}{\left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx-\frac {125000}{3} \int \frac {1}{x^2 \left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx-42500 \int \frac {1}{x^2 \left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx-62500 \int \frac {e^{e^{x^2}+x^2}}{x \left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx-125000 \int \frac {1}{x^3 \left (100+25 e^{e^{x^2}}+17 x\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.91, size = 28, normalized size = 0.90 \begin {gather*} \frac {625 (3+x)^2}{9 x^2 \left (100+25 e^{e^{x^2}}+17 x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1125000 - 757500*x - 191250*x^2 - 21250*x^3 + E^E^x^2*(-281250 - 93750*x + E^x^2*(-562500*x^2 - 37
5000*x^3 - 62500*x^4)))/(9000000*x^3 + 140625*E^(3*E^x^2)*x^3 + 4590000*x^4 + 780300*x^5 + 44217*x^6 + E^(2*E^
x^2)*(1687500*x^3 + 286875*x^4) + E^E^x^2*(6750000*x^3 + 2295000*x^4 + 195075*x^5)),x]

[Out]

(625*(3 + x)^2)/(9*x^2*(100 + 25*E^E^x^2 + 17*x)^2)

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fricas [B]  time = 0.69, size = 58, normalized size = 1.87 \begin {gather*} \frac {625 \, {\left (x^{2} + 6 \, x + 9\right )}}{9 \, {\left (289 \, x^{4} + 3400 \, x^{3} + 625 \, x^{2} e^{\left (2 \, e^{\left (x^{2}\right )}\right )} + 10000 \, x^{2} + 50 \, {\left (17 \, x^{3} + 100 \, x^{2}\right )} e^{\left (e^{\left (x^{2}\right )}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-62500*x^4-375000*x^3-562500*x^2)*exp(x^2)-93750*x-281250)*exp(exp(x^2))-21250*x^3-191250*x^2-757
500*x-1125000)/(140625*x^3*exp(exp(x^2))^3+(286875*x^4+1687500*x^3)*exp(exp(x^2))^2+(195075*x^5+2295000*x^4+67
50000*x^3)*exp(exp(x^2))+44217*x^6+780300*x^5+4590000*x^4+9000000*x^3),x, algorithm="fricas")

[Out]

625/9*(x^2 + 6*x + 9)/(289*x^4 + 3400*x^3 + 625*x^2*e^(2*e^(x^2)) + 10000*x^2 + 50*(17*x^3 + 100*x^2)*e^(e^(x^
2)))

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giac [B]  time = 0.22, size = 60, normalized size = 1.94 \begin {gather*} \frac {625 \, {\left (x^{2} + 6 \, x + 9\right )}}{9 \, {\left (289 \, x^{4} + 850 \, x^{3} e^{\left (e^{\left (x^{2}\right )}\right )} + 3400 \, x^{3} + 625 \, x^{2} e^{\left (2 \, e^{\left (x^{2}\right )}\right )} + 5000 \, x^{2} e^{\left (e^{\left (x^{2}\right )}\right )} + 10000 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-62500*x^4-375000*x^3-562500*x^2)*exp(x^2)-93750*x-281250)*exp(exp(x^2))-21250*x^3-191250*x^2-757
500*x-1125000)/(140625*x^3*exp(exp(x^2))^3+(286875*x^4+1687500*x^3)*exp(exp(x^2))^2+(195075*x^5+2295000*x^4+67
50000*x^3)*exp(exp(x^2))+44217*x^6+780300*x^5+4590000*x^4+9000000*x^3),x, algorithm="giac")

[Out]

625/9*(x^2 + 6*x + 9)/(289*x^4 + 850*x^3*e^(e^(x^2)) + 3400*x^3 + 625*x^2*e^(2*e^(x^2)) + 5000*x^2*e^(e^(x^2))
+ 10000*x^2)

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maple [A]  time = 0.05, size = 28, normalized size = 0.90

method	result	size
risch	\(\frac {\frac {625}{9} x^{2}+\frac {1250}{3} x +625}{x^{2} \left (25 \,{\mathrm e}^{{\mathrm e}^{x^{2}}}+17 x +100\right )^{2}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-62500*x^4-375000*x^3-562500*x^2)*exp(x^2)-93750*x-281250)*exp(exp(x^2))-21250*x^3-191250*x^2-757500*x-
1125000)/(140625*x^3*exp(exp(x^2))^3+(286875*x^4+1687500*x^3)*exp(exp(x^2))^2+(195075*x^5+2295000*x^4+6750000*
x^3)*exp(exp(x^2))+44217*x^6+780300*x^5+4590000*x^4+9000000*x^3),x,method=_RETURNVERBOSE)

[Out]

625/9*(x^2+6*x+9)/x^2/(25*exp(exp(x^2))+17*x+100)^2

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maxima [B]  time = 0.41, size = 58, normalized size = 1.87 \begin {gather*} \frac {625 \, {\left (x^{2} + 6 \, x + 9\right )}}{9 \, {\left (289 \, x^{4} + 3400 \, x^{3} + 625 \, x^{2} e^{\left (2 \, e^{\left (x^{2}\right )}\right )} + 10000 \, x^{2} + 50 \, {\left (17 \, x^{3} + 100 \, x^{2}\right )} e^{\left (e^{\left (x^{2}\right )}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-62500*x^4-375000*x^3-562500*x^2)*exp(x^2)-93750*x-281250)*exp(exp(x^2))-21250*x^3-191250*x^2-757
500*x-1125000)/(140625*x^3*exp(exp(x^2))^3+(286875*x^4+1687500*x^3)*exp(exp(x^2))^2+(195075*x^5+2295000*x^4+67
50000*x^3)*exp(exp(x^2))+44217*x^6+780300*x^5+4590000*x^4+9000000*x^3),x, algorithm="maxima")

[Out]

625/9*(x^2 + 6*x + 9)/(289*x^4 + 3400*x^3 + 625*x^2*e^(2*e^(x^2)) + 10000*x^2 + 50*(17*x^3 + 100*x^2)*e^(e^(x^
2)))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {757500\,x+191250\,x^2+21250\,x^3+{\mathrm {e}}^{{\mathrm {e}}^{x^2}}\,\left (93750\,x+{\mathrm {e}}^{x^2}\,\left (62500\,x^4+375000\,x^3+562500\,x^2\right )+281250\right )+1125000}{{\mathrm {e}}^{{\mathrm {e}}^{x^2}}\,\left (195075\,x^5+2295000\,x^4+6750000\,x^3\right )+{\mathrm {e}}^{2\,{\mathrm {e}}^{x^2}}\,\left (286875\,x^4+1687500\,x^3\right )+140625\,x^3\,{\mathrm {e}}^{3\,{\mathrm {e}}^{x^2}}+9000000\,x^3+4590000\,x^4+780300\,x^5+44217\,x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(757500*x + 191250*x^2 + 21250*x^3 + exp(exp(x^2))*(93750*x + exp(x^2)*(562500*x^2 + 375000*x^3 + 62500*x
^4) + 281250) + 1125000)/(exp(exp(x^2))*(6750000*x^3 + 2295000*x^4 + 195075*x^5) + exp(2*exp(x^2))*(1687500*x^
3 + 286875*x^4) + 140625*x^3*exp(3*exp(x^2)) + 9000000*x^3 + 4590000*x^4 + 780300*x^5 + 44217*x^6),x)

[Out]

int(-(757500*x + 191250*x^2 + 21250*x^3 + exp(exp(x^2))*(93750*x + exp(x^2)*(562500*x^2 + 375000*x^3 + 62500*x
^4) + 281250) + 1125000)/(exp(exp(x^2))*(6750000*x^3 + 2295000*x^4 + 195075*x^5) + exp(2*exp(x^2))*(1687500*x^
3 + 286875*x^4) + 140625*x^3*exp(3*exp(x^2)) + 9000000*x^3 + 4590000*x^4 + 780300*x^5 + 44217*x^6), x)

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sympy [B]  time = 0.37, size = 54, normalized size = 1.74 \begin {gather*} \frac {625 x^{2} + 3750 x + 5625}{2601 x^{4} + 30600 x^{3} + 5625 x^{2} e^{2 e^{x^{2}}} + 90000 x^{2} + \left (7650 x^{3} + 45000 x^{2}\right ) e^{e^{x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-62500*x**4-375000*x**3-562500*x**2)*exp(x**2)-93750*x-281250)*exp(exp(x**2))-21250*x**3-191250*x
**2-757500*x-1125000)/(140625*x**3*exp(exp(x**2))**3+(286875*x**4+1687500*x**3)*exp(exp(x**2))**2+(195075*x**5
+2295000*x**4+6750000*x**3)*exp(exp(x**2))+44217*x**6+780300*x**5+4590000*x**4+9000000*x**3),x)

[Out]

(625*x**2 + 3750*x + 5625)/(2601*x**4 + 30600*x**3 + 5625*x**2*exp(2*exp(x**2)) + 90000*x**2 + (7650*x**3 + 45
000*x**2)*exp(exp(x**2)))

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