3.2.67 \(\int \frac {-1024+13760 x^2-75025 x^4+193500 x^6-202500 x^8+e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} (1600 x+800 x^2-5375 x^4-22500 x^5+11250 x^6)}{1024-13760 x^2+75025 x^4-193500 x^6+202500 x^8} \, dx\)

Optimal. Leaf size=30 \[ e^{\frac {5 e^x}{5+10 \left (\frac {4}{5 x}-3 x\right )^2}}-x \]

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Rubi [F]  time = 16.80, 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 {-1024+13760 x^2-75025 x^4+193500 x^6-202500 x^8+e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} \left (1600 x+800 x^2-5375 x^4-22500 x^5+11250 x^6\right )}{1024-13760 x^2+75025 x^4-193500 x^6+202500 x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1024 + 13760*x^2 - 75025*x^4 + 193500*x^6 - 202500*x^8 + E^(x + (25*E^x*x^2)/(32 - 215*x^2 + 450*x^4))*(
1600*x + 800*x^2 - 5375*x^4 - 22500*x^5 + 11250*x^6))/(1024 - 13760*x^2 + 75025*x^4 - 193500*x^6 + 202500*x^8)
,x]

[Out]

-x - ((43*I + Sqrt[455])*Defer[Int][E^(x + (25*E^x*x^2)/(32 - 215*x^2 + 450*x^4))/(Sqrt[43 - I*Sqrt[455]] - 6*
Sqrt[5]*x), x])/(2*Sqrt[(91*(43 - I*Sqrt[455]))/5]) + ((43*I - Sqrt[455])*Defer[Int][E^(x + (25*E^x*x^2)/(32 -
215*x^2 + 450*x^4))/(Sqrt[43 + I*Sqrt[455]] - 6*Sqrt[5]*x), x])/(2*Sqrt[(91*(43 + I*Sqrt[455]))/5]) - ((43*I
+ Sqrt[455])*Defer[Int][E^(x + (25*E^x*x^2)/(32 - 215*x^2 + 450*x^4))/(Sqrt[43 - I*Sqrt[455]] + 6*Sqrt[5]*x),
x])/(2*Sqrt[(91*(43 - I*Sqrt[455]))/5]) + ((43*I - Sqrt[455])*Defer[Int][E^(x + (25*E^x*x^2)/(32 - 215*x^2 + 4
50*x^4))/(Sqrt[43 + I*Sqrt[455]] + 6*Sqrt[5]*x), x])/(2*Sqrt[(91*(43 + I*Sqrt[455]))/5]) - (20736000*Defer[Int
][(E^(x + (25*E^x*x^2)/(32 - 215*x^2 + 450*x^4))*x)/(215 + (5*I)*Sqrt[455] - 900*x^2)^2, x])/91 + (387000*(43
+ I*Sqrt[455])*Defer[Int][(E^(x + (25*E^x*x^2)/(32 - 215*x^2 + 450*x^4))*x)/(215 + (5*I)*Sqrt[455] - 900*x^2)^
2, x])/91 - (20736000*Defer[Int][(E^(x + (25*E^x*x^2)/(32 - 215*x^2 + 450*x^4))*x)/(-215 + (5*I)*Sqrt[455] + 9
00*x^2)^2, x])/91 + (387000*(43 - I*Sqrt[455])*Defer[Int][(E^(x + (25*E^x*x^2)/(32 - 215*x^2 + 450*x^4))*x)/(-
215 + (5*I)*Sqrt[455] + 900*x^2)^2, x])/91

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+\frac {25 e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x \left (64+32 x-215 x^3-900 x^4+450 x^5\right )}{\left (32-215 x^2+450 x^4\right )^2}\right ) \, dx\\ &=-x+25 \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x \left (64+32 x-215 x^3-900 x^4+450 x^5\right )}{\left (32-215 x^2+450 x^4\right )^2} \, dx\\ &=-x+25 \int \left (\frac {2 e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x \left (64-215 x^2\right )}{\left (32-215 x^2+450 x^4\right )^2}+\frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} (-2+x) x}{32-215 x^2+450 x^4}\right ) \, dx\\ &=-x+25 \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} (-2+x) x}{32-215 x^2+450 x^4} \, dx+50 \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x \left (64-215 x^2\right )}{\left (32-215 x^2+450 x^4\right )^2} \, dx\\ &=-x+25 \int \left (-\frac {2 e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{32-215 x^2+450 x^4}+\frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x^2}{32-215 x^2+450 x^4}\right ) \, dx+50 \int \left (\frac {64 e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{\left (32-215 x^2+450 x^4\right )^2}-\frac {215 e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x^3}{\left (32-215 x^2+450 x^4\right )^2}\right ) \, dx\\ &=-x+25 \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x^2}{32-215 x^2+450 x^4} \, dx-50 \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{32-215 x^2+450 x^4} \, dx+3200 \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{\left (32-215 x^2+450 x^4\right )^2} \, dx-10750 \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x^3}{\left (32-215 x^2+450 x^4\right )^2} \, dx\\ &=-x+25 \int \left (\frac {\left (1-\frac {43 i}{\sqrt {455}}\right ) e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}}}{-215-5 i \sqrt {455}+900 x^2}+\frac {\left (1+\frac {43 i}{\sqrt {455}}\right ) e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}}}{-215+5 i \sqrt {455}+900 x^2}\right ) \, dx-50 \int \left (\frac {36 i \sqrt {\frac {5}{91}} e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{215+5 i \sqrt {455}-900 x^2}+\frac {36 i \sqrt {\frac {5}{91}} e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{-215+5 i \sqrt {455}+900 x^2}\right ) \, dx+3200 \int \left (-\frac {6480 e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{91 \left (215+5 i \sqrt {455}-900 x^2\right )^2}+\frac {1296 i e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{91 \sqrt {455} \left (215+5 i \sqrt {455}-900 x^2\right )}-\frac {6480 e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{91 \left (-215+5 i \sqrt {455}+900 x^2\right )^2}+\frac {1296 i e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{91 \sqrt {455} \left (-215+5 i \sqrt {455}+900 x^2\right )}\right ) \, dx-10750 \int \left (-\frac {36 \left (215+5 i \sqrt {455}\right ) e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{455 \left (215+5 i \sqrt {455}-900 x^2\right )^2}+\frac {1548 i e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{455 \sqrt {455} \left (215+5 i \sqrt {455}-900 x^2\right )}+\frac {36 \left (-215+5 i \sqrt {455}\right ) e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{455 \left (-215+5 i \sqrt {455}+900 x^2\right )^2}+\frac {1548 i e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{455 \sqrt {455} \left (-215+5 i \sqrt {455}+900 x^2\right )}\right ) \, dx\\ &=-x-\frac {20736000}{91} \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{\left (215+5 i \sqrt {455}-900 x^2\right )^2} \, dx-\frac {20736000}{91} \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{\left (-215+5 i \sqrt {455}+900 x^2\right )^2} \, dx-\left (1800 i \sqrt {\frac {5}{91}}\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{215+5 i \sqrt {455}-900 x^2} \, dx-\left (1800 i \sqrt {\frac {5}{91}}\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{-215+5 i \sqrt {455}+900 x^2} \, dx-\frac {1}{91} \left (665640 i \sqrt {\frac {5}{91}}\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{215+5 i \sqrt {455}-900 x^2} \, dx-\frac {1}{91} \left (665640 i \sqrt {\frac {5}{91}}\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{-215+5 i \sqrt {455}+900 x^2} \, dx+\frac {1}{91} \left (829440 i \sqrt {\frac {5}{91}}\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{215+5 i \sqrt {455}-900 x^2} \, dx+\frac {1}{91} \left (829440 i \sqrt {\frac {5}{91}}\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{-215+5 i \sqrt {455}+900 x^2} \, dx+\frac {1}{91} \left (387000 \left (43-i \sqrt {455}\right )\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{\left (-215+5 i \sqrt {455}+900 x^2\right )^2} \, dx+\frac {1}{91} \left (387000 \left (43+i \sqrt {455}\right )\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{\left (215+5 i \sqrt {455}-900 x^2\right )^2} \, dx+\frac {1}{91} \left (5 \left (455-43 i \sqrt {455}\right )\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}}}{-215-5 i \sqrt {455}+900 x^2} \, dx+\frac {1}{91} \left (5 \left (455+43 i \sqrt {455}\right )\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}}}{-215+5 i \sqrt {455}+900 x^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.86, size = 28, normalized size = 0.93 \begin {gather*} e^{\frac {25 e^x x^2}{32-215 x^2+450 x^4}}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1024 + 13760*x^2 - 75025*x^4 + 193500*x^6 - 202500*x^8 + E^(x + (25*E^x*x^2)/(32 - 215*x^2 + 450*x
^4))*(1600*x + 800*x^2 - 5375*x^4 - 22500*x^5 + 11250*x^6))/(1024 - 13760*x^2 + 75025*x^4 - 193500*x^6 + 20250
0*x^8),x]

[Out]

E^((25*E^x*x^2)/(32 - 215*x^2 + 450*x^4)) - x

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fricas [A]  time = 0.68, size = 50, normalized size = 1.67 \begin {gather*} -{\left (x e^{x} - e^{\left (\frac {450 \, x^{5} - 215 \, x^{3} + 25 \, x^{2} e^{x} + 32 \, x}{450 \, x^{4} - 215 \, x^{2} + 32}\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((11250*x^6-22500*x^5-5375*x^4+800*x^2+1600*x)*exp(x)*exp(25*x^2*exp(x)/(450*x^4-215*x^2+32))-202500
*x^8+193500*x^6-75025*x^4+13760*x^2-1024)/(202500*x^8-193500*x^6+75025*x^4-13760*x^2+1024),x, algorithm="frica
s")

[Out]

-(x*e^x - e^((450*x^5 - 215*x^3 + 25*x^2*e^x + 32*x)/(450*x^4 - 215*x^2 + 32)))*e^(-x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {202500 \, x^{8} - 193500 \, x^{6} + 75025 \, x^{4} - 13760 \, x^{2} - 25 \, {\left (450 \, x^{6} - 900 \, x^{5} - 215 \, x^{4} + 32 \, x^{2} + 64 \, x\right )} e^{\left (\frac {25 \, x^{2} e^{x}}{450 \, x^{4} - 215 \, x^{2} + 32} + x\right )} + 1024}{202500 \, x^{8} - 193500 \, x^{6} + 75025 \, x^{4} - 13760 \, x^{2} + 1024}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((11250*x^6-22500*x^5-5375*x^4+800*x^2+1600*x)*exp(x)*exp(25*x^2*exp(x)/(450*x^4-215*x^2+32))-202500
*x^8+193500*x^6-75025*x^4+13760*x^2-1024)/(202500*x^8-193500*x^6+75025*x^4-13760*x^2+1024),x, algorithm="giac"
)

[Out]

integrate(-(202500*x^8 - 193500*x^6 + 75025*x^4 - 13760*x^2 - 25*(450*x^6 - 900*x^5 - 215*x^4 + 32*x^2 + 64*x)
*e^(25*x^2*e^x/(450*x^4 - 215*x^2 + 32) + x) + 1024)/(202500*x^8 - 193500*x^6 + 75025*x^4 - 13760*x^2 + 1024),
x)

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maple [A]  time = 0.77, size = 27, normalized size = 0.90

method	result	size
risch	\(-x +{\mathrm e}^{\frac {25 x^{2} {\mathrm e}^{x}}{450 x^{4}-215 x^{2}+32}}\)	\(27\)
norman	\(\frac {-32 x +215 x^{3}-450 x^{5}-215 x^{2} {\mathrm e}^{\frac {25 x^{2} {\mathrm e}^{x}}{450 x^{4}-215 x^{2}+32}}+450 x^{4} {\mathrm e}^{\frac {25 x^{2} {\mathrm e}^{x}}{450 x^{4}-215 x^{2}+32}}+32 \,{\mathrm e}^{\frac {25 x^{2} {\mathrm e}^{x}}{450 x^{4}-215 x^{2}+32}}}{450 x^{4}-215 x^{2}+32}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((11250*x^6-22500*x^5-5375*x^4+800*x^2+1600*x)*exp(x)*exp(25*x^2*exp(x)/(450*x^4-215*x^2+32))-202500*x^8+1
93500*x^6-75025*x^4+13760*x^2-1024)/(202500*x^8-193500*x^6+75025*x^4-13760*x^2+1024),x,method=_RETURNVERBOSE)

[Out]

-x+exp(25*x^2*exp(x)/(450*x^4-215*x^2+32))

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maxima [A]  time = 0.44, size = 26, normalized size = 0.87 \begin {gather*} -x + e^{\left (\frac {25 \, x^{2} e^{x}}{450 \, x^{4} - 215 \, x^{2} + 32}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((11250*x^6-22500*x^5-5375*x^4+800*x^2+1600*x)*exp(x)*exp(25*x^2*exp(x)/(450*x^4-215*x^2+32))-202500
*x^8+193500*x^6-75025*x^4+13760*x^2-1024)/(202500*x^8-193500*x^6+75025*x^4-13760*x^2+1024),x, algorithm="maxim
a")

[Out]

-x + e^(25*x^2*e^x/(450*x^4 - 215*x^2 + 32))

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mupad [B]  time = 0.50, size = 26, normalized size = 0.87 \begin {gather*} {\mathrm {e}}^{\frac {25\,x^2\,{\mathrm {e}}^x}{450\,x^4-215\,x^2+32}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((13760*x^2 - 75025*x^4 + 193500*x^6 - 202500*x^8 + exp((25*x^2*exp(x))/(450*x^4 - 215*x^2 + 32))*exp(x)*(1
600*x + 800*x^2 - 5375*x^4 - 22500*x^5 + 11250*x^6) - 1024)/(75025*x^4 - 13760*x^2 - 193500*x^6 + 202500*x^8 +
1024),x)

[Out]

exp((25*x^2*exp(x))/(450*x^4 - 215*x^2 + 32)) - x

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sympy [A]  time = 0.34, size = 22, normalized size = 0.73 \begin {gather*} - x + e^{\frac {25 x^{2} e^{x}}{450 x^{4} - 215 x^{2} + 32}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((11250*x**6-22500*x**5-5375*x**4+800*x**2+1600*x)*exp(x)*exp(25*x**2*exp(x)/(450*x**4-215*x**2+32))
-202500*x**8+193500*x**6-75025*x**4+13760*x**2-1024)/(202500*x**8-193500*x**6+75025*x**4-13760*x**2+1024),x)

[Out]

-x + exp(25*x**2*exp(x)/(450*x**4 - 215*x**2 + 32))

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