3.75.73 \(\int \frac {e^{\frac {2 (2+4 x+4 x^2)}{x+x^2}} (-2-4 x)+15 x^2+49 x^3+62 x^4+39 x^5+13 x^6+2 x^7+e^{\frac {2+4 x+4 x^2}{x+x^2}} (-10-26 x-11 x^2+4 x^3+7 x^4+2 x^5)}{8 x^2+16 x^3+8 x^4} \, dx\)

Optimal. Leaf size=30 \[ \frac {1}{16} \left (5+e^{4+\frac {2}{x (1+x)}}-x+x (4+x)\right )^2 \]

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Rubi [F]  time = 2.81, 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 {2 \left (2+4 x+4 x^2\right )}{x+x^2}} (-2-4 x)+15 x^2+49 x^3+62 x^4+39 x^5+13 x^6+2 x^7+e^{\frac {2+4 x+4 x^2}{x+x^2}} \left (-10-26 x-11 x^2+4 x^3+7 x^4+2 x^5\right )}{8 x^2+16 x^3+8 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((2*(2 + 4*x + 4*x^2))/(x + x^2))*(-2 - 4*x) + 15*x^2 + 49*x^3 + 62*x^4 + 39*x^5 + 13*x^6 + 2*x^7 + E^(
(2 + 4*x + 4*x^2)/(x + x^2))*(-10 - 26*x - 11*x^2 + 4*x^3 + 7*x^4 + 2*x^5))/(8*x^2 + 16*x^3 + 8*x^4),x]

[Out]

(15*x)/8 + (19*x^2)/16 + (3*x^3)/8 + x^4/16 + (3*Defer[Int][E^((2*(1 + 2*x + 2*x^2))/(x*(1 + x))), x])/8 - (5*
Defer[Int][E^((2*(1 + 2*x + 2*x^2))/(x*(1 + x)))/x^2, x])/4 - Defer[Int][E^((4*(1 + 2*x + 2*x^2))/(x*(1 + x)))
/x^2, x]/4 - (3*Defer[Int][E^((2*(1 + 2*x + 2*x^2))/(x*(1 + x)))/x, x])/4 + Defer[Int][E^((2*(1 + 2*x + 2*x^2)
)/(x*(1 + x)))*x, x]/4 + (3*Defer[Int][E^((2*(1 + 2*x + 2*x^2))/(x*(1 + x)))/(1 + x)^2, x])/4 + Defer[Int][E^(
(4*(1 + 2*x + 2*x^2))/(x*(1 + x)))/(1 + x)^2, x]/4 + Defer[Int][E^((2*(1 + 2*x + 2*x^2))/(x*(1 + x)))/(1 + x),
x]/4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {2 \left (2+4 x+4 x^2\right )}{x+x^2}} (-2-4 x)+15 x^2+49 x^3+62 x^4+39 x^5+13 x^6+2 x^7+e^{\frac {2+4 x+4 x^2}{x+x^2}} \left (-10-26 x-11 x^2+4 x^3+7 x^4+2 x^5\right )}{x^2 \left (8+16 x+8 x^2\right )} \, dx\\ &=\int \frac {e^{\frac {2 \left (2+4 x+4 x^2\right )}{x+x^2}} (-2-4 x)+15 x^2+49 x^3+62 x^4+39 x^5+13 x^6+2 x^7+e^{\frac {2+4 x+4 x^2}{x+x^2}} \left (-10-26 x-11 x^2+4 x^3+7 x^4+2 x^5\right )}{8 x^2 (1+x)^2} \, dx\\ &=\frac {1}{8} \int \frac {e^{\frac {2 \left (2+4 x+4 x^2\right )}{x+x^2}} (-2-4 x)+15 x^2+49 x^3+62 x^4+39 x^5+13 x^6+2 x^7+e^{\frac {2+4 x+4 x^2}{x+x^2}} \left (-10-26 x-11 x^2+4 x^3+7 x^4+2 x^5\right )}{x^2 (1+x)^2} \, dx\\ &=\frac {1}{8} \int \left (\frac {15}{(1+x)^2}+\frac {2 e^{\frac {4 \left (1+2 x+2 x^2\right )}{x (1+x)}} (-1-2 x)}{x^2 (1+x)^2}+\frac {49 x}{(1+x)^2}+\frac {62 x^2}{(1+x)^2}+\frac {39 x^3}{(1+x)^2}+\frac {13 x^4}{(1+x)^2}+\frac {2 x^5}{(1+x)^2}+\frac {e^{\frac {2 \left (1+2 x+2 x^2\right )}{x (1+x)}} \left (-10-26 x-11 x^2+4 x^3+7 x^4+2 x^5\right )}{x^2 (1+x)^2}\right ) \, dx\\ &=-\frac {15}{8 (1+x)}+\frac {1}{8} \int \frac {e^{\frac {2 \left (1+2 x+2 x^2\right )}{x (1+x)}} \left (-10-26 x-11 x^2+4 x^3+7 x^4+2 x^5\right )}{x^2 (1+x)^2} \, dx+\frac {1}{4} \int \frac {e^{\frac {4 \left (1+2 x+2 x^2\right )}{x (1+x)}} (-1-2 x)}{x^2 (1+x)^2} \, dx+\frac {1}{4} \int \frac {x^5}{(1+x)^2} \, dx+\frac {13}{8} \int \frac {x^4}{(1+x)^2} \, dx+\frac {39}{8} \int \frac {x^3}{(1+x)^2} \, dx+\frac {49}{8} \int \frac {x}{(1+x)^2} \, dx+\frac {31}{4} \int \frac {x^2}{(1+x)^2} \, dx\\ &=-\frac {15}{8 (1+x)}+\frac {1}{8} \int \left (3 e^{\frac {2 \left (1+2 x+2 x^2\right )}{x (1+x)}}-\frac {10 e^{\frac {2 \left (1+2 x+2 x^2\right )}{x (1+x)}}}{x^2}-\frac {6 e^{\frac {2 \left (1+2 x+2 x^2\right )}{x (1+x)}}}{x}+2 e^{\frac {2 \left (1+2 x+2 x^2\right )}{x (1+x)}} x+\frac {6 e^{\frac {2 \left (1+2 x+2 x^2\right )}{x (1+x)}}}{(1+x)^2}+\frac {2 e^{\frac {2 \left (1+2 x+2 x^2\right )}{x (1+x)}}}{1+x}\right ) \, dx+\frac {1}{4} \int \left (-\frac {e^{\frac {4 \left (1+2 x+2 x^2\right )}{x (1+x)}}}{x^2}+\frac {e^{\frac {4 \left (1+2 x+2 x^2\right )}{x (1+x)}}}{(1+x)^2}\right ) \, dx+\frac {1}{4} \int \left (-4+3 x-2 x^2+x^3-\frac {1}{(1+x)^2}+\frac {5}{1+x}\right ) \, dx+\frac {13}{8} \int \left (3-2 x+x^2+\frac {1}{(1+x)^2}-\frac {4}{1+x}\right ) \, dx+\frac {39}{8} \int \left (-2+x-\frac {1}{(1+x)^2}+\frac {3}{1+x}\right ) \, dx+\frac {49}{8} \int \left (-\frac {1}{(1+x)^2}+\frac {1}{1+x}\right ) \, dx+\frac {31}{4} \int \left (1+\frac {1}{(1+x)^2}-\frac {2}{1+x}\right ) \, dx\\ &=\frac {15 x}{8}+\frac {19 x^2}{16}+\frac {3 x^3}{8}+\frac {x^4}{16}-\frac {1}{4} \int \frac {e^{\frac {4 \left (1+2 x+2 x^2\right )}{x (1+x)}}}{x^2} \, dx+\frac {1}{4} \int e^{\frac {2 \left (1+2 x+2 x^2\right )}{x (1+x)}} x \, dx+\frac {1}{4} \int \frac {e^{\frac {4 \left (1+2 x+2 x^2\right )}{x (1+x)}}}{(1+x)^2} \, dx+\frac {1}{4} \int \frac {e^{\frac {2 \left (1+2 x+2 x^2\right )}{x (1+x)}}}{1+x} \, dx+\frac {3}{8} \int e^{\frac {2 \left (1+2 x+2 x^2\right )}{x (1+x)}} \, dx-\frac {3}{4} \int \frac {e^{\frac {2 \left (1+2 x+2 x^2\right )}{x (1+x)}}}{x} \, dx+\frac {3}{4} \int \frac {e^{\frac {2 \left (1+2 x+2 x^2\right )}{x (1+x)}}}{(1+x)^2} \, dx-\frac {5}{4} \int \frac {e^{\frac {2 \left (1+2 x+2 x^2\right )}{x (1+x)}}}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.11, size = 71, normalized size = 2.37 \begin {gather*} \frac {1}{16} e^{-\frac {4}{1+x}} \left (e^{8+\frac {4}{x}}+2 e^{2 \left (2+\frac {1}{x}+\frac {1}{1+x}\right )} \left (5+3 x+x^2\right )+e^{\frac {4}{1+x}} x \left (30+19 x+6 x^2+x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*(2 + 4*x + 4*x^2))/(x + x^2))*(-2 - 4*x) + 15*x^2 + 49*x^3 + 62*x^4 + 39*x^5 + 13*x^6 + 2*x^7
+ E^((2 + 4*x + 4*x^2)/(x + x^2))*(-10 - 26*x - 11*x^2 + 4*x^3 + 7*x^4 + 2*x^5))/(8*x^2 + 16*x^3 + 8*x^4),x]

[Out]

(E^(8 + 4/x) + 2*E^(2*(2 + x^(-1) + (1 + x)^(-1)))*(5 + 3*x + x^2) + E^(4/(1 + x))*x*(30 + 19*x + 6*x^2 + x^3)
)/(16*E^(4/(1 + x)))

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fricas [B]  time = 0.64, size = 71, normalized size = 2.37 \begin {gather*} \frac {1}{16} \, x^{4} + \frac {3}{8} \, x^{3} + \frac {19}{16} \, x^{2} + \frac {1}{8} \, {\left (x^{2} + 3 \, x + 5\right )} e^{\left (\frac {2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )}}{x^{2} + x}\right )} + \frac {15}{8} \, x + \frac {1}{16} \, e^{\left (\frac {4 \, {\left (2 \, x^{2} + 2 \, x + 1\right )}}{x^{2} + x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x-2)*exp((4*x^2+4*x+2)/(x^2+x))^2+(2*x^5+7*x^4+4*x^3-11*x^2-26*x-10)*exp((4*x^2+4*x+2)/(x^2+x))
+2*x^7+13*x^6+39*x^5+62*x^4+49*x^3+15*x^2)/(8*x^4+16*x^3+8*x^2),x, algorithm="fricas")

[Out]

1/16*x^4 + 3/8*x^3 + 19/16*x^2 + 1/8*(x^2 + 3*x + 5)*e^(2*(2*x^2 + 2*x + 1)/(x^2 + x)) + 15/8*x + 1/16*e^(4*(2
*x^2 + 2*x + 1)/(x^2 + x))

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giac [B]  time = 0.31, size = 124, normalized size = 4.13 \begin {gather*} \frac {1}{16} \, x^{4} + \frac {3}{8} \, x^{3} + \frac {1}{8} \, x^{2} e^{\left (\frac {2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )}}{x^{2} + x}\right )} + \frac {19}{16} \, x^{2} + \frac {3}{8} \, x e^{\left (\frac {2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )}}{x^{2} + x}\right )} + \frac {15}{8} \, x + \frac {1}{16} \, e^{\left (\frac {8 \, x^{2}}{x^{2} + x} + \frac {8 \, x}{x^{2} + x} + \frac {4}{x^{2} + x}\right )} + \frac {5}{8} \, e^{\left (\frac {2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )}}{x^{2} + x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x-2)*exp((4*x^2+4*x+2)/(x^2+x))^2+(2*x^5+7*x^4+4*x^3-11*x^2-26*x-10)*exp((4*x^2+4*x+2)/(x^2+x))
+2*x^7+13*x^6+39*x^5+62*x^4+49*x^3+15*x^2)/(8*x^4+16*x^3+8*x^2),x, algorithm="giac")

[Out]

1/16*x^4 + 3/8*x^3 + 1/8*x^2*e^(2*(2*x^2 + 2*x + 1)/(x^2 + x)) + 19/16*x^2 + 3/8*x*e^(2*(2*x^2 + 2*x + 1)/(x^2
+ x)) + 15/8*x + 1/16*e^(8*x^2/(x^2 + x) + 8*x/(x^2 + x) + 4/(x^2 + x)) + 5/8*e^(2*(2*x^2 + 2*x + 1)/(x^2 + x
))

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maple [A]  time = 0.26, size = 75, normalized size = 2.50

method	result	size
risch	\(\frac {15 x}{8}+\frac {19 x^{2}}{16}+\frac {3 x^{3}}{8}+\frac {x^{4}}{16}+\frac {{\mathrm e}^{\frac {8 x^{2}+8 x +4}{\left (x +1\right ) x}}}{16}+\left (\frac {5}{8}+\frac {3}{8} x +\frac {1}{8} x^{2}\right ) {\mathrm e}^{\frac {4 x^{2}+4 x +2}{\left (x +1\right ) x}}\)	\(75\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {4 x^{2}+4 x +2}{x^{2}+x}}-\frac {15 x}{8}+\frac {49 x^{3}}{16}+\frac {25 x^{4}}{16}+\frac {7 x^{5}}{16}+\frac {x^{6}}{16}+\frac {5 x \,{\mathrm e}^{\frac {4 x^{2}+4 x +2}{x^{2}+x}}}{8}+\frac {x \,{\mathrm e}^{\frac {8 x^{2}+8 x +4}{x^{2}+x}}}{16}+\frac {x^{2} {\mathrm e}^{\frac {8 x^{2}+8 x +4}{x^{2}+x}}}{16}+\frac {x^{3} {\mathrm e}^{\frac {4 x^{2}+4 x +2}{x^{2}+x}}}{2}+\frac {x^{4} {\mathrm e}^{\frac {4 x^{2}+4 x +2}{x^{2}+x}}}{8}}{\left (x +1\right ) x}\)	\(177\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x-2)*exp((4*x^2+4*x+2)/(x^2+x))^2+(2*x^5+7*x^4+4*x^3-11*x^2-26*x-10)*exp((4*x^2+4*x+2)/(x^2+x))+2*x^7
+13*x^6+39*x^5+62*x^4+49*x^3+15*x^2)/(8*x^4+16*x^3+8*x^2),x,method=_RETURNVERBOSE)

[Out]

15/8*x+19/16*x^2+3/8*x^3+1/16*x^4+1/16*exp(4*(2*x^2+2*x+1)/(x+1)/x)+(5/8+3/8*x+1/8*x^2)*exp(2*(2*x^2+2*x+1)/(x
+1)/x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{16} \, x^{4} + \frac {3}{8} \, x^{3} + \frac {19}{16} \, x^{2} + \frac {15}{8} \, x + \frac {1}{16} \, e^{\left (-\frac {4}{x + 1} + \frac {4}{x} + 8\right )} + \frac {1}{8} \, \int \frac {{\left (2 \, x^{5} e^{4} + 7 \, x^{4} e^{4} + 4 \, x^{3} e^{4} - 11 \, x^{2} e^{4} - 26 \, x e^{4} - 10 \, e^{4}\right )} e^{\left (-\frac {2}{x + 1} + \frac {2}{x}\right )}}{x^{4} + 2 \, x^{3} + x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x-2)*exp((4*x^2+4*x+2)/(x^2+x))^2+(2*x^5+7*x^4+4*x^3-11*x^2-26*x-10)*exp((4*x^2+4*x+2)/(x^2+x))
+2*x^7+13*x^6+39*x^5+62*x^4+49*x^3+15*x^2)/(8*x^4+16*x^3+8*x^2),x, algorithm="maxima")

[Out]

1/16*x^4 + 3/8*x^3 + 19/16*x^2 + 15/8*x + 1/16*e^(-4/(x + 1) + 4/x + 8) + 1/8*integrate((2*x^5*e^4 + 7*x^4*e^4
+ 4*x^3*e^4 - 11*x^2*e^4 - 26*x*e^4 - 10*e^4)*e^(-2/(x + 1) + 2/x)/(x^4 + 2*x^3 + x^2), x)

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mupad [B]  time = 5.15, size = 161, normalized size = 5.37 \begin {gather*} \frac {5\,{\mathrm {e}}^{\frac {2}{x^2+x}+\frac {4\,x^2}{x^2+x}+\frac {4\,x}{x^2+x}}}{8}+\frac {{\mathrm {e}}^{\frac {4}{x^2+x}+\frac {8\,x^2}{x^2+x}+\frac {8\,x}{x^2+x}}}{16}+x^2\,\left (\frac {{\mathrm {e}}^{\frac {2}{x^2+x}+\frac {4\,x^2}{x^2+x}+\frac {4\,x}{x^2+x}}}{8}+\frac {19}{16}\right )+\frac {3\,x^3}{8}+\frac {x^4}{16}+x\,\left (\frac {3\,{\mathrm {e}}^{\frac {2}{x^2+x}+\frac {4\,x^2}{x^2+x}+\frac {4\,x}{x^2+x}}}{8}+\frac {15}{8}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((15*x^2 - exp((2*(4*x + 4*x^2 + 2))/(x + x^2))*(4*x + 2) - exp((4*x + 4*x^2 + 2)/(x + x^2))*(26*x + 11*x^2
- 4*x^3 - 7*x^4 - 2*x^5 + 10) + 49*x^3 + 62*x^4 + 39*x^5 + 13*x^6 + 2*x^7)/(8*x^2 + 16*x^3 + 8*x^4),x)

[Out]

(5*exp(2/(x + x^2) + (4*x^2)/(x + x^2) + (4*x)/(x + x^2)))/8 + exp(4/(x + x^2) + (8*x^2)/(x + x^2) + (8*x)/(x
+ x^2))/16 + x^2*(exp(2/(x + x^2) + (4*x^2)/(x + x^2) + (4*x)/(x + x^2))/8 + 19/16) + (3*x^3)/8 + x^4/16 + x*(
(3*exp(2/(x + x^2) + (4*x^2)/(x + x^2) + (4*x)/(x + x^2)))/8 + 15/8)

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sympy [A]  time = 0.34, size = 70, normalized size = 2.33 \begin {gather*} \frac {x^{4}}{16} + \frac {3 x^{3}}{8} + \frac {19 x^{2}}{16} + \frac {15 x}{8} + \frac {\left (16 x^{2} + 48 x + 80\right ) e^{\frac {4 x^{2} + 4 x + 2}{x^{2} + x}}}{128} + \frac {e^{\frac {2 \left (4 x^{2} + 4 x + 2\right )}{x^{2} + x}}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x-2)*exp((4*x**2+4*x+2)/(x**2+x))**2+(2*x**5+7*x**4+4*x**3-11*x**2-26*x-10)*exp((4*x**2+4*x+2)/
(x**2+x))+2*x**7+13*x**6+39*x**5+62*x**4+49*x**3+15*x**2)/(8*x**4+16*x**3+8*x**2),x)

[Out]

x**4/16 + 3*x**3/8 + 19*x**2/16 + 15*x/8 + (16*x**2 + 48*x + 80)*exp((4*x**2 + 4*x + 2)/(x**2 + x))/128 + exp(
2*(4*x**2 + 4*x + 2)/(x**2 + x))/16

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