3.59.18 \(\int \frac {e^{2 x^2} (-576 x^3+504 x^5)+e^{3+x^2} (24 x^5+24 x^7)}{e^6 x^4+e^{3+x^2} (-24 x^2+42 x^4)+e^{2 x^2} (144-504 x^2+441 x^4)} \, dx\)

Optimal. Leaf size=27 \[ \frac {4 x^2}{7+\frac {e^{3-x^2}}{3}-\frac {4}{x^2}} \]

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Rubi [F]  time = 4.18, 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 {e^{2 x^2} \left (-576 x^3+504 x^5\right )+e^{3+x^2} \left (24 x^5+24 x^7\right )}{e^6 x^4+e^{3+x^2} \left (-24 x^2+42 x^4\right )+e^{2 x^2} \left (144-504 x^2+441 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*x^2)*(-576*x^3 + 504*x^5) + E^(3 + x^2)*(24*x^5 + 24*x^7))/(E^6*x^4 + E^(3 + x^2)*(-24*x^2 + 42*x^4)
+ E^(2*x^2)*(144 - 504*x^2 + 441*x^4)),x]

[Out]

(192*Defer[Subst][Defer[Int][E^(3 + x)/(-12*E^x + E^3*x + 21*E^x*x)^2, x], x, x^2])/49 + (48*Defer[Subst][Defe
r[Int][(E^(3 + x)*x)/(-12*E^x + E^3*x + 21*E^x*x)^2, x], x, x^2])/7 + 12*Defer[Subst][Defer[Int][(E^(3 + x)*x^
3)/(-12*E^x + E^3*x + 21*E^x*x)^2, x], x, x^2] + (768*Defer[Subst][Defer[Int][E^(3 + x)/((-4 + 7*x)*(-12*E^x +
E^3*x + 21*E^x*x)^2), x], x, x^2])/49 - (48*Defer[Subst][Defer[Int][E^x/(-12*E^x + E^3*x + 21*E^x*x), x], x,
x^2])/7 + 12*Defer[Subst][Defer[Int][(E^x*x)/(-12*E^x + E^3*x + 21*E^x*x), x], x, x^2] - (192*Defer[Subst][Def
er[Int][E^x/((-4 + 7*x)*(-12*E^x + E^3*x + 21*E^x*x)), x], x, x^2])/7

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {24 e^{x^2} x^3 \left (3 e^{x^2} \left (-8+7 x^2\right )+e^3 \left (x^2+x^4\right )\right )}{\left (e^3 x^2+3 e^{x^2} \left (-4+7 x^2\right )\right )^2} \, dx\\ &=24 \int \frac {e^{x^2} x^3 \left (3 e^{x^2} \left (-8+7 x^2\right )+e^3 \left (x^2+x^4\right )\right )}{\left (e^3 x^2+3 e^{x^2} \left (-4+7 x^2\right )\right )^2} \, dx\\ &=12 \operatorname {Subst}\left (\int \frac {e^x x \left (e^3 x (1+x)+3 e^x (-8+7 x)\right )}{\left (e^3 x+3 e^x (-4+7 x)\right )^2} \, dx,x,x^2\right )\\ &=12 \operatorname {Subst}\left (\int \left (\frac {e^x x (-8+7 x)}{(-4+7 x) \left (-12 e^x+e^3 x+21 e^x x\right )}+\frac {e^{3+x} x^2 \left (4-4 x+7 x^2\right )}{(-4+7 x) \left (-12 e^x+e^3 x+21 e^x x\right )^2}\right ) \, dx,x,x^2\right )\\ &=12 \operatorname {Subst}\left (\int \frac {e^x x (-8+7 x)}{(-4+7 x) \left (-12 e^x+e^3 x+21 e^x x\right )} \, dx,x,x^2\right )+12 \operatorname {Subst}\left (\int \frac {e^{3+x} x^2 \left (4-4 x+7 x^2\right )}{(-4+7 x) \left (-12 e^x+e^3 x+21 e^x x\right )^2} \, dx,x,x^2\right )\\ &=12 \operatorname {Subst}\left (\int \left (\frac {16 e^{3+x}}{49 \left (-12 e^x+e^3 x+21 e^x x\right )^2}+\frac {4 e^{3+x} x}{7 \left (-12 e^x+e^3 x+21 e^x x\right )^2}+\frac {e^{3+x} x^3}{\left (-12 e^x+e^3 x+21 e^x x\right )^2}+\frac {64 e^{3+x}}{49 (-4+7 x) \left (-12 e^x+e^3 x+21 e^x x\right )^2}\right ) \, dx,x,x^2\right )+12 \operatorname {Subst}\left (\int \left (-\frac {4 e^x}{7 \left (-12 e^x+e^3 x+21 e^x x\right )}+\frac {e^x x}{-12 e^x+e^3 x+21 e^x x}-\frac {16 e^x}{7 (-4+7 x) \left (-12 e^x+e^3 x+21 e^x x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {192}{49} \operatorname {Subst}\left (\int \frac {e^{3+x}}{\left (-12 e^x+e^3 x+21 e^x x\right )^2} \, dx,x,x^2\right )+\frac {48}{7} \operatorname {Subst}\left (\int \frac {e^{3+x} x}{\left (-12 e^x+e^3 x+21 e^x x\right )^2} \, dx,x,x^2\right )-\frac {48}{7} \operatorname {Subst}\left (\int \frac {e^x}{-12 e^x+e^3 x+21 e^x x} \, dx,x,x^2\right )+12 \operatorname {Subst}\left (\int \frac {e^{3+x} x^3}{\left (-12 e^x+e^3 x+21 e^x x\right )^2} \, dx,x,x^2\right )+12 \operatorname {Subst}\left (\int \frac {e^x x}{-12 e^x+e^3 x+21 e^x x} \, dx,x,x^2\right )+\frac {768}{49} \operatorname {Subst}\left (\int \frac {e^{3+x}}{(-4+7 x) \left (-12 e^x+e^3 x+21 e^x x\right )^2} \, dx,x,x^2\right )-\frac {192}{7} \operatorname {Subst}\left (\int \frac {e^x}{(-4+7 x) \left (-12 e^x+e^3 x+21 e^x x\right )} \, dx,x,x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.73, size = 56, normalized size = 2.07 \begin {gather*} \frac {4 \left (-4 e^3 x^2+3 e^{x^2} \left (16-28 x^2+49 x^4\right )\right )}{49 \left (e^3 x^2+3 e^{x^2} \left (-4+7 x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x^2)*(-576*x^3 + 504*x^5) + E^(3 + x^2)*(24*x^5 + 24*x^7))/(E^6*x^4 + E^(3 + x^2)*(-24*x^2 + 4
2*x^4) + E^(2*x^2)*(144 - 504*x^2 + 441*x^4)),x]

[Out]

(4*(-4*E^3*x^2 + 3*E^x^2*(16 - 28*x^2 + 49*x^4)))/(49*(E^3*x^2 + 3*E^x^2*(-4 + 7*x^2)))

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fricas [B]  time = 0.63, size = 54, normalized size = 2.00 \begin {gather*} -\frac {4 \, {\left (4 \, x^{2} e^{6} - 3 \, {\left (49 \, x^{4} - 28 \, x^{2} + 16\right )} e^{\left (x^{2} + 3\right )}\right )}}{49 \, {\left (x^{2} e^{6} + 3 \, {\left (7 \, x^{2} - 4\right )} e^{\left (x^{2} + 3\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((504*x^5-576*x^3)*exp(x^2)^2+(24*x^7+24*x^5)*exp(3)*exp(x^2))/((441*x^4-504*x^2+144)*exp(x^2)^2+(42
*x^4-24*x^2)*exp(3)*exp(x^2)+x^4*exp(3)^2),x, algorithm="fricas")

[Out]

-4/49*(4*x^2*e^6 - 3*(49*x^4 - 28*x^2 + 16)*e^(x^2 + 3))/(x^2*e^6 + 3*(7*x^2 - 4)*e^(x^2 + 3))

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giac [B]  time = 0.22, size = 58, normalized size = 2.15 \begin {gather*} \frac {4 \, {\left (147 \, x^{4} e^{\left (x^{2}\right )} - 4 \, x^{2} e^{3} - 84 \, x^{2} e^{\left (x^{2}\right )} + 48 \, e^{\left (x^{2}\right )}\right )}}{49 \, {\left (x^{2} e^{3} + 21 \, x^{2} e^{\left (x^{2}\right )} - 12 \, e^{\left (x^{2}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((504*x^5-576*x^3)*exp(x^2)^2+(24*x^7+24*x^5)*exp(3)*exp(x^2))/((441*x^4-504*x^2+144)*exp(x^2)^2+(42
*x^4-24*x^2)*exp(3)*exp(x^2)+x^4*exp(3)^2),x, algorithm="giac")

[Out]

4/49*(147*x^4*e^(x^2) - 4*x^2*e^3 - 84*x^2*e^(x^2) + 48*e^(x^2))/(x^2*e^3 + 21*x^2*e^(x^2) - 12*e^(x^2))

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maple [A]  time = 0.22, size = 34, normalized size = 1.26

method	result	size
norman	\(\frac {12 x^{4} {\mathrm e}^{x^{2}}}{21 x^{2} {\mathrm e}^{x^{2}}+x^{2} {\mathrm e}^{3}-12 \,{\mathrm e}^{x^{2}}}\)	\(34\)
risch	\(\frac {4 x^{2}}{7}+\frac {64}{343 \left (x^{2}-\frac {4}{7}\right )}-\frac {4 \,{\mathrm e}^{3} x^{6}}{\left (7 x^{2}-4\right ) \left (21 x^{2} {\mathrm e}^{x^{2}}+x^{2} {\mathrm e}^{3}-12 \,{\mathrm e}^{x^{2}}\right )}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((504*x^5-576*x^3)*exp(x^2)^2+(24*x^7+24*x^5)*exp(3)*exp(x^2))/((441*x^4-504*x^2+144)*exp(x^2)^2+(42*x^4-2
4*x^2)*exp(3)*exp(x^2)+x^4*exp(3)^2),x,method=_RETURNVERBOSE)

[Out]

12*x^4*exp(x^2)/(21*x^2*exp(x^2)+x^2*exp(3)-12*exp(x^2))

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maxima [B]  time = 0.41, size = 50, normalized size = 1.85 \begin {gather*} -\frac {4 \, {\left (4 \, x^{2} e^{3} - 3 \, {\left (49 \, x^{4} - 28 \, x^{2} + 16\right )} e^{\left (x^{2}\right )}\right )}}{49 \, {\left (x^{2} e^{3} + 3 \, {\left (7 \, x^{2} - 4\right )} e^{\left (x^{2}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((504*x^5-576*x^3)*exp(x^2)^2+(24*x^7+24*x^5)*exp(3)*exp(x^2))/((441*x^4-504*x^2+144)*exp(x^2)^2+(42
*x^4-24*x^2)*exp(3)*exp(x^2)+x^4*exp(3)^2),x, algorithm="maxima")

[Out]

-4/49*(4*x^2*e^3 - 3*(49*x^4 - 28*x^2 + 16)*e^(x^2))/(x^2*e^3 + 3*(7*x^2 - 4)*e^(x^2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {{\mathrm {e}}^{2\,x^2}\,\left (576\,x^3-504\,x^5\right )-{\mathrm {e}}^{x^2+3}\,\left (24\,x^7+24\,x^5\right )}{x^4\,{\mathrm {e}}^6+{\mathrm {e}}^{2\,x^2}\,\left (441\,x^4-504\,x^2+144\right )-{\mathrm {e}}^{x^2+3}\,\left (24\,x^2-42\,x^4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x^2)*(576*x^3 - 504*x^5) - exp(x^2)*exp(3)*(24*x^5 + 24*x^7))/(x^4*exp(6) + exp(2*x^2)*(441*x^4 -
504*x^2 + 144) - exp(x^2)*exp(3)*(24*x^2 - 42*x^4)),x)

[Out]

-int((exp(2*x^2)*(576*x^3 - 504*x^5) - exp(x^2 + 3)*(24*x^5 + 24*x^7))/(x^4*exp(6) + exp(2*x^2)*(441*x^4 - 504
*x^2 + 144) - exp(x^2 + 3)*(24*x^2 - 42*x^4)), x)

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sympy [B]  time = 0.27, size = 56, normalized size = 2.07 \begin {gather*} - \frac {4 x^{6} e^{3}}{7 x^{4} e^{3} - 4 x^{2} e^{3} + \left (147 x^{4} - 168 x^{2} + 48\right ) e^{x^{2}}} + \frac {4 x^{2}}{7} + \frac {64}{343 x^{2} - 196} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((504*x**5-576*x**3)*exp(x**2)**2+(24*x**7+24*x**5)*exp(3)*exp(x**2))/((441*x**4-504*x**2+144)*exp(x
**2)**2+(42*x**4-24*x**2)*exp(3)*exp(x**2)+x**4*exp(3)**2),x)

[Out]

-4*x**6*exp(3)/(7*x**4*exp(3) - 4*x**2*exp(3) + (147*x**4 - 168*x**2 + 48)*exp(x**2)) + 4*x**2/7 + 64/(343*x**
2 - 196)

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