3.83.4 \(\int \frac {e^{\frac {50 x+25 e^{5+x} x}{25-15 x+16 x^2}} (1250-800 x^2+e^{5+x} (625+625 x-775 x^2+400 x^3))}{625-750 x+1025 x^2-480 x^3+256 x^4} \, dx\)

Optimal. Leaf size=27 \[ e^{\frac {2 x+e^{5+x} x}{\left (-1+\frac {4 x}{5}\right )^2+x}} \]

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Rubi [F]  time = 9.22, 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^{\frac {50 x+25 e^{5+x} x}{25-15 x+16 x^2}} \left (1250-800 x^2+e^{5+x} \left (625+625 x-775 x^2+400 x^3\right )\right )}{625-750 x+1025 x^2-480 x^3+256 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((50*x + 25*E^(5 + x)*x)/(25 - 15*x + 16*x^2))*(1250 - 800*x^2 + E^(5 + x)*(625 + 625*x - 775*x^2 + 400
*x^3)))/(625 - 750*x + 1025*x^2 - 480*x^3 + 256*x^4),x]

[Out]

(-20480*Defer[Int][E^((25*(2 + E^(5 + x))*x)/(25 - 15*x + 16*x^2))/(15 + (5*I)*Sqrt[55] - 32*x)^2, x])/11 + (9
60*(3 + I*Sqrt[55])*Defer[Int][E^((25*(2 + E^(5 + x))*x)/(25 - 15*x + 16*x^2))/(15 + (5*I)*Sqrt[55] - 32*x)^2,
x])/11 - (10240*Defer[Int][E^(5 + x + (25*(2 + E^(5 + x))*x)/(25 - 15*x + 16*x^2))/(15 + (5*I)*Sqrt[55] - 32*
x)^2, x])/11 + (480*(3 + I*Sqrt[55])*Defer[Int][E^(5 + x + (25*(2 + E^(5 + x))*x)/(25 - 15*x + 16*x^2))/(15 +
(5*I)*Sqrt[55] - 32*x)^2, x])/11 + (32*I)*Sqrt[5/11]*Defer[Int][E^(5 + x + (25*(2 + E^(5 + x))*x)/(25 - 15*x +
16*x^2))/(15 + (5*I)*Sqrt[55] - 32*x), x] + ((275 + (17*I)*Sqrt[55])*Defer[Int][E^(5 + x + (25*(2 + E^(5 + x)
)*x)/(25 - 15*x + 16*x^2))/(-15 - (5*I)*Sqrt[55] + 32*x), x])/11 - (20480*Defer[Int][E^((25*(2 + E^(5 + x))*x)
/(25 - 15*x + 16*x^2))/(-15 + (5*I)*Sqrt[55] + 32*x)^2, x])/11 + (960*(3 - I*Sqrt[55])*Defer[Int][E^((25*(2 +
E^(5 + x))*x)/(25 - 15*x + 16*x^2))/(-15 + (5*I)*Sqrt[55] + 32*x)^2, x])/11 - (10240*Defer[Int][E^(5 + x + (25
*(2 + E^(5 + x))*x)/(25 - 15*x + 16*x^2))/(-15 + (5*I)*Sqrt[55] + 32*x)^2, x])/11 + (480*(3 - I*Sqrt[55])*Defe
r[Int][E^(5 + x + (25*(2 + E^(5 + x))*x)/(25 - 15*x + 16*x^2))/(-15 + (5*I)*Sqrt[55] + 32*x)^2, x])/11 + (32*I
)*Sqrt[5/11]*Defer[Int][E^(5 + x + (25*(2 + E^(5 + x))*x)/(25 - 15*x + 16*x^2))/(-15 + (5*I)*Sqrt[55] + 32*x),
x] + ((275 - (17*I)*Sqrt[55])*Defer[Int][E^(5 + x + (25*(2 + E^(5 + x))*x)/(25 - 15*x + 16*x^2))/(-15 + (5*I)
*Sqrt[55] + 32*x), x])/11

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}} \left (1250-800 x^2+e^{5+x} \left (625+625 x-775 x^2+400 x^3\right )\right )}{625-750 x+1025 x^2-480 x^3+256 x^4} \, dx\\ &=\int \left (-\frac {50 e^{\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}} \left (-25+16 x^2\right )}{\left (25-15 x+16 x^2\right )^2}+\frac {25 e^{5+x+\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}} \left (25+25 x-31 x^2+16 x^3\right )}{\left (25-15 x+16 x^2\right )^2}\right ) \, dx\\ &=25 \int \frac {e^{5+x+\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}} \left (25+25 x-31 x^2+16 x^3\right )}{\left (25-15 x+16 x^2\right )^2} \, dx-50 \int \frac {e^{\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}} \left (-25+16 x^2\right )}{\left (25-15 x+16 x^2\right )^2} \, dx\\ &=25 \int \left (-\frac {5 e^{5+x+\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}} (-10+3 x)}{\left (25-15 x+16 x^2\right )^2}+\frac {e^{5+x+\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}} (-1+x)}{25-15 x+16 x^2}\right ) \, dx-50 \int \left (\frac {5 e^{\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}} (-10+3 x)}{\left (25-15 x+16 x^2\right )^2}+\frac {e^{\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}}}{25-15 x+16 x^2}\right ) \, dx\\ &=25 \int \frac {e^{5+x+\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}} (-1+x)}{25-15 x+16 x^2} \, dx-50 \int \frac {e^{\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}}}{25-15 x+16 x^2} \, dx-125 \int \frac {e^{5+x+\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}} (-10+3 x)}{\left (25-15 x+16 x^2\right )^2} \, dx-250 \int \frac {e^{\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}} (-10+3 x)}{\left (25-15 x+16 x^2\right )^2} \, dx\\ &=25 \int \left (\frac {\left (1+\frac {17 i}{5 \sqrt {55}}\right ) e^{5+x+\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}}}{-15-5 i \sqrt {55}+32 x}+\frac {\left (1-\frac {17 i}{5 \sqrt {55}}\right ) e^{5+x+\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}}}{-15+5 i \sqrt {55}+32 x}\right ) \, dx-50 \int \left (\frac {32 i e^{\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}}}{5 \sqrt {55} \left (15+5 i \sqrt {55}-32 x\right )}+\frac {32 i e^{\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}}}{5 \sqrt {55} \left (-15+5 i \sqrt {55}+32 x\right )}\right ) \, dx-125 \int \left (-\frac {10 e^{5+x+\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}}}{\left (25-15 x+16 x^2\right )^2}+\frac {3 e^{5+x+\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}} x}{\left (25-15 x+16 x^2\right )^2}\right ) \, dx-250 \int \left (-\frac {10 e^{\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}}}{\left (25-15 x+16 x^2\right )^2}+\frac {3 e^{\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}} x}{\left (25-15 x+16 x^2\right )^2}\right ) \, dx\\ &=-\left (375 \int \frac {e^{5+x+\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}} x}{\left (25-15 x+16 x^2\right )^2} \, dx\right )-750 \int \frac {e^{\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}} x}{\left (25-15 x+16 x^2\right )^2} \, dx+1250 \int \frac {e^{5+x+\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}}}{\left (25-15 x+16 x^2\right )^2} \, dx+2500 \int \frac {e^{\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}}}{\left (25-15 x+16 x^2\right )^2} \, dx-\left (64 i \sqrt {\frac {5}{11}}\right ) \int \frac {e^{\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}}}{15+5 i \sqrt {55}-32 x} \, dx-\left (64 i \sqrt {\frac {5}{11}}\right ) \int \frac {e^{\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}}}{-15+5 i \sqrt {55}+32 x} \, dx+\frac {1}{11} \left (275-17 i \sqrt {55}\right ) \int \frac {e^{5+x+\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}}}{-15+5 i \sqrt {55}+32 x} \, dx+\frac {1}{11} \left (275+17 i \sqrt {55}\right ) \int \frac {e^{5+x+\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}}}{-15-5 i \sqrt {55}+32 x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.87, size = 24, normalized size = 0.89 \begin {gather*} e^{\frac {25 \left (2+e^{5+x}\right ) x}{25-15 x+16 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((50*x + 25*E^(5 + x)*x)/(25 - 15*x + 16*x^2))*(1250 - 800*x^2 + E^(5 + x)*(625 + 625*x - 775*x^2
+ 400*x^3)))/(625 - 750*x + 1025*x^2 - 480*x^3 + 256*x^4),x]

[Out]

E^((25*(2 + E^(5 + x))*x)/(25 - 15*x + 16*x^2))

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fricas [A]  time = 0.71, size = 25, normalized size = 0.93 \begin {gather*} e^{\left (\frac {25 \, {\left (x e^{\left (x + 5\right )} + 2 \, x\right )}}{16 \, x^{2} - 15 \, x + 25}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((400*x^3-775*x^2+625*x+625)*exp(5+x)-800*x^2+1250)*exp((25*x*exp(5+x)+50*x)/(16*x^2-15*x+25))/(256*
x^4-480*x^3+1025*x^2-750*x+625),x, algorithm="fricas")

[Out]

e^(25*(x*e^(x + 5) + 2*x)/(16*x^2 - 15*x + 25))

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giac [A]  time = 0.31, size = 36, normalized size = 1.33 \begin {gather*} e^{\left (\frac {25 \, x e^{\left (x + 5\right )}}{16 \, x^{2} - 15 \, x + 25} + \frac {50 \, x}{16 \, x^{2} - 15 \, x + 25}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((400*x^3-775*x^2+625*x+625)*exp(5+x)-800*x^2+1250)*exp((25*x*exp(5+x)+50*x)/(16*x^2-15*x+25))/(256*
x^4-480*x^3+1025*x^2-750*x+625),x, algorithm="giac")

[Out]

e^(25*x*e^(x + 5)/(16*x^2 - 15*x + 25) + 50*x/(16*x^2 - 15*x + 25))

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maple [A]  time = 0.12, size = 23, normalized size = 0.85

method	result	size
risch	\({\mathrm e}^{\frac {25 x \left ({\mathrm e}^{5+x}+2\right )}{16 x^{2}-15 x +25}}\)	\(23\)
norman	\(\frac {-15 x \,{\mathrm e}^{\frac {25 x \,{\mathrm e}^{5+x}+50 x}{16 x^{2}-15 x +25}}+16 x^{2} {\mathrm e}^{\frac {25 x \,{\mathrm e}^{5+x}+50 x}{16 x^{2}-15 x +25}}+25 \,{\mathrm e}^{\frac {25 x \,{\mathrm e}^{5+x}+50 x}{16 x^{2}-15 x +25}}}{16 x^{2}-15 x +25}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((400*x^3-775*x^2+625*x+625)*exp(5+x)-800*x^2+1250)*exp((25*x*exp(5+x)+50*x)/(16*x^2-15*x+25))/(256*x^4-48
0*x^3+1025*x^2-750*x+625),x,method=_RETURNVERBOSE)

[Out]

exp(25*x*(exp(5+x)+2)/(16*x^2-15*x+25))

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maxima [A]  time = 0.48, size = 36, normalized size = 1.33 \begin {gather*} e^{\left (\frac {25 \, x e^{\left (x + 5\right )}}{16 \, x^{2} - 15 \, x + 25} + \frac {50 \, x}{16 \, x^{2} - 15 \, x + 25}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((400*x^3-775*x^2+625*x+625)*exp(5+x)-800*x^2+1250)*exp((25*x*exp(5+x)+50*x)/(16*x^2-15*x+25))/(256*
x^4-480*x^3+1025*x^2-750*x+625),x, algorithm="maxima")

[Out]

e^(25*x*e^(x + 5)/(16*x^2 - 15*x + 25) + 50*x/(16*x^2 - 15*x + 25))

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mupad [B]  time = 5.66, size = 25, normalized size = 0.93 \begin {gather*} {\mathrm {e}}^{\frac {50\,x+25\,x\,{\mathrm {e}}^5\,{\mathrm {e}}^x}{16\,x^2-15\,x+25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((50*x + 25*x*exp(x + 5))/(16*x^2 - 15*x + 25))*(exp(x + 5)*(625*x - 775*x^2 + 400*x^3 + 625) - 800*x^
2 + 1250))/(1025*x^2 - 750*x - 480*x^3 + 256*x^4 + 625),x)

[Out]

exp((50*x + 25*x*exp(5)*exp(x))/(16*x^2 - 15*x + 25))

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sympy [A]  time = 0.36, size = 22, normalized size = 0.81 \begin {gather*} e^{\frac {25 x e^{x + 5} + 50 x}{16 x^{2} - 15 x + 25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((400*x**3-775*x**2+625*x+625)*exp(5+x)-800*x**2+1250)*exp((25*x*exp(5+x)+50*x)/(16*x**2-15*x+25))/(
256*x**4-480*x**3+1025*x**2-750*x+625),x)

[Out]

exp((25*x*exp(x + 5) + 50*x)/(16*x**2 - 15*x + 25))

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