3.3.75 \(\int \frac {e^x (e^{1+\frac {1}{4} (-2+x)} (36-27 x+3 x^2)+e (144-144 x+16 x^2))}{192 x^2+96 e^{\frac {1}{4} (-2+x)} x^2+12 e^{\frac {1}{2} (-2+x)} x^2} \, dx\)

Optimal. Leaf size=28 \[ \frac {e^{1+x} (-9+x)}{3 \left (4+e^{\frac {1}{4} (-2+x)}\right ) x} \]

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Rubi [F]  time = 3.88, 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^x \left (e^{1+\frac {1}{4} (-2+x)} \left (36-27 x+3 x^2\right )+e \left (144-144 x+16 x^2\right )\right )}{192 x^2+96 e^{\frac {1}{4} (-2+x)} x^2+12 e^{\frac {1}{2} (-2+x)} x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(E^(1 + (-2 + x)/4)*(36 - 27*x + 3*x^2) + E*(144 - 144*x + 16*x^2)))/(192*x^2 + 96*E^((-2 + x)/4)*x^2
+ 12*E^((-2 + x)/2)*x^2),x]

[Out]

(16*E^(5/2 + x/4))/3 - (4*E^(2 + x/2))/3 + E^(3/2 + (3*x)/4)/3 + (256*E^(7/2))/(3*(4*Sqrt[E] + E^(x/4))) + 3*D
efer[Int][E^(3/2 + x)/((4*Sqrt[E] + E^(x/4))*x^2), x] - 3*Defer[Int][E^(2 + x)/((4*Sqrt[E] + E^(x/4))^2*x), x]
- (9*Defer[Int][E^(3/2 + x)/((4*Sqrt[E] + E^(x/4))*x), x])/4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{1+x} \left (e^{1+\frac {1}{4} (-2+x)} \left (36-27 x+3 x^2\right )+e \left (144-144 x+16 x^2\right )\right )}{12 \left (4 \sqrt {e}+e^{x/4}\right )^2 x^2} \, dx\\ &=\frac {1}{12} \int \frac {e^{1+x} \left (e^{1+\frac {1}{4} (-2+x)} \left (36-27 x+3 x^2\right )+e \left (144-144 x+16 x^2\right )\right )}{\left (4 \sqrt {e}+e^{x/4}\right )^2 x^2} \, dx\\ &=\frac {1}{12} \int \left (\frac {4 e^{2+x} (-9+x)}{\left (4 \sqrt {e}+e^{x/4}\right )^2 x}+\frac {3 e^{\frac {3}{2}+x} \left (12-9 x+x^2\right )}{\left (4 \sqrt {e}+e^{x/4}\right ) x^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {e^{\frac {3}{2}+x} \left (12-9 x+x^2\right )}{\left (4 \sqrt {e}+e^{x/4}\right ) x^2} \, dx+\frac {1}{3} \int \frac {e^{2+x} (-9+x)}{\left (4 \sqrt {e}+e^{x/4}\right )^2 x} \, dx\\ &=\frac {1}{4} \int \left (\frac {e^{\frac {3}{2}+x}}{4 \sqrt {e}+e^{x/4}}+\frac {12 e^{\frac {3}{2}+x}}{\left (4 \sqrt {e}+e^{x/4}\right ) x^2}-\frac {9 e^{\frac {3}{2}+x}}{\left (4 \sqrt {e}+e^{x/4}\right ) x}\right ) \, dx+\frac {1}{3} \int \left (\frac {e^{2+x}}{\left (4 \sqrt {e}+e^{x/4}\right )^2}-\frac {9 e^{2+x}}{\left (4 \sqrt {e}+e^{x/4}\right )^2 x}\right ) \, dx\\ &=\frac {1}{4} \int \frac {e^{\frac {3}{2}+x}}{4 \sqrt {e}+e^{x/4}} \, dx+\frac {1}{3} \int \frac {e^{2+x}}{\left (4 \sqrt {e}+e^{x/4}\right )^2} \, dx-\frac {9}{4} \int \frac {e^{\frac {3}{2}+x}}{\left (4 \sqrt {e}+e^{x/4}\right ) x} \, dx+3 \int \frac {e^{\frac {3}{2}+x}}{\left (4 \sqrt {e}+e^{x/4}\right ) x^2} \, dx-3 \int \frac {e^{2+x}}{\left (4 \sqrt {e}+e^{x/4}\right )^2 x} \, dx\\ &=-\left (\frac {9}{4} \int \frac {e^{\frac {3}{2}+x}}{\left (4 \sqrt {e}+e^{x/4}\right ) x} \, dx\right )+3 \int \frac {e^{\frac {3}{2}+x}}{\left (4 \sqrt {e}+e^{x/4}\right ) x^2} \, dx-3 \int \frac {e^{2+x}}{\left (4 \sqrt {e}+e^{x/4}\right )^2 x} \, dx+e^{3/2} \operatorname {Subst}\left (\int \frac {x^3}{4 \sqrt {e}+x} \, dx,x,e^{x/4}\right )+\frac {1}{3} \left (4 e^2\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (4 \sqrt {e}+x\right )^2} \, dx,x,e^{x/4}\right )\\ &=-\left (\frac {9}{4} \int \frac {e^{\frac {3}{2}+x}}{\left (4 \sqrt {e}+e^{x/4}\right ) x} \, dx\right )+3 \int \frac {e^{\frac {3}{2}+x}}{\left (4 \sqrt {e}+e^{x/4}\right ) x^2} \, dx-3 \int \frac {e^{2+x}}{\left (4 \sqrt {e}+e^{x/4}\right )^2 x} \, dx+e^{3/2} \operatorname {Subst}\left (\int \left (16 e-4 \sqrt {e} x+x^2-\frac {64 e^{3/2}}{4 \sqrt {e}+x}\right ) \, dx,x,e^{x/4}\right )+\frac {1}{3} \left (4 e^2\right ) \operatorname {Subst}\left (\int \left (-8 \sqrt {e}+x-\frac {64 e^{3/2}}{\left (4 \sqrt {e}+x\right )^2}+\frac {48 e}{4 \sqrt {e}+x}\right ) \, dx,x,e^{x/4}\right )\\ &=\frac {16}{3} e^{\frac {5}{2}+\frac {x}{4}}-\frac {4}{3} e^{2+\frac {x}{2}}+\frac {1}{3} e^{\frac {3}{2}+\frac {3 x}{4}}+\frac {256 e^{7/2}}{3 \left (4 \sqrt {e}+e^{x/4}\right )}-\frac {9}{4} \int \frac {e^{\frac {3}{2}+x}}{\left (4 \sqrt {e}+e^{x/4}\right ) x} \, dx+3 \int \frac {e^{\frac {3}{2}+x}}{\left (4 \sqrt {e}+e^{x/4}\right ) x^2} \, dx-3 \int \frac {e^{2+x}}{\left (4 \sqrt {e}+e^{x/4}\right )^2 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.44, size = 55, normalized size = 1.96 \begin {gather*} \frac {e^{3/2} \left (e^x (-9+x)+256 e^2 x+64 e^{\frac {6+x}{4}} x\right )}{3 \left (4 \sqrt {e}+e^{x/4}\right ) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(E^(1 + (-2 + x)/4)*(36 - 27*x + 3*x^2) + E*(144 - 144*x + 16*x^2)))/(192*x^2 + 96*E^((-2 + x)/
4)*x^2 + 12*E^((-2 + x)/2)*x^2),x]

[Out]

(E^(3/2)*(E^x*(-9 + x) + 256*E^2*x + 64*E^((6 + x)/4)*x))/(3*(4*Sqrt[E] + E^(x/4))*x)

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fricas [A]  time = 0.63, size = 41, normalized size = 1.46 \begin {gather*} \frac {256 \, x e^{4} + {\left (x - 9\right )} e^{\left (x + 2\right )} + 64 \, x e^{\left (\frac {1}{4} \, x + \frac {7}{2}\right )}}{3 \, {\left (4 \, x e + x e^{\left (\frac {1}{4} \, x + \frac {1}{2}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2-27*x+36)*exp(1)*exp(1/4*x-1/2)+(16*x^2-144*x+144)*exp(1))*exp(x)/(12*x^2*exp(1/4*x-1/2)^2+96
*x^2*exp(1/4*x-1/2)+192*x^2),x, algorithm="fricas")

[Out]

1/3*(256*x*e^4 + (x - 9)*e^(x + 2) + 64*x*e^(1/4*x + 7/2))/(4*x*e + x*e^(1/4*x + 1/2))

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giac [A]  time = 0.34, size = 43, normalized size = 1.54 \begin {gather*} \frac {256 \, x e^{\frac {7}{2}} + x e^{\left (x + \frac {3}{2}\right )} + 64 \, x e^{\left (\frac {1}{4} \, x + 3\right )} - 9 \, e^{\left (x + \frac {3}{2}\right )}}{3 \, {\left (4 \, x e^{\frac {1}{2}} + x e^{\left (\frac {1}{4} \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2-27*x+36)*exp(1)*exp(1/4*x-1/2)+(16*x^2-144*x+144)*exp(1))*exp(x)/(12*x^2*exp(1/4*x-1/2)^2+96
*x^2*exp(1/4*x-1/2)+192*x^2),x, algorithm="giac")

[Out]

1/3*(256*x*e^(7/2) + x*e^(x + 3/2) + 64*x*e^(1/4*x + 3) - 9*e^(x + 3/2))/(4*x*e^(1/2) + x*e^(1/4*x))

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maple [A]  time = 0.20, size = 42, normalized size = 1.50

method	result	size
norman	\(\frac {-3 \,{\mathrm e} \,{\mathrm e}^{\frac {1}{2}} {\mathrm e}^{x}+\frac {{\mathrm e} \,{\mathrm e}^{\frac {1}{2}} x \,{\mathrm e}^{x}}{3}}{x \left (4 \,{\mathrm e}^{\frac {1}{2}}+{\mathrm e}^{\frac {x}{4}}\right )}\)	\(42\)
risch	\(\frac {192 \,{\mathrm e}^{3}}{x}+\frac {\left (x -9\right ) {\mathrm e}^{\frac {3}{2}+\frac {3 x}{4}}}{3 x}-\frac {4 \left (x -9\right ) {\mathrm e}^{2+\frac {x}{2}}}{3 x}+\frac {16 \left (x -9\right ) {\mathrm e}^{\frac {5}{2}+\frac {x}{4}}}{3 x}+\frac {256 \,{\mathrm e}^{3} \left (x -9\right )}{3 x \left (4+{\mathrm e}^{\frac {x}{4}-\frac {1}{2}}\right )}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^2-27*x+36)*exp(1)*exp(1/4*x-1/2)+(16*x^2-144*x+144)*exp(1))*exp(x)/(12*x^2*exp(1/4*x-1/2)^2+96*x^2*e
xp(1/4*x-1/2)+192*x^2),x,method=_RETURNVERBOSE)

[Out]

(-3*exp(1)*exp(1/2)*exp(1/4*x)^4+1/3*exp(1)*exp(1/2)*x*exp(1/4*x)^4)/x/(4*exp(1/2)+exp(1/4*x))

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maxima [B]  time = 0.55, size = 45, normalized size = 1.61 \begin {gather*} \frac {256 \, x e^{4} + {\left (x e^{2} - 9 \, e^{2}\right )} e^{x} + 64 \, x e^{\left (\frac {1}{4} \, x + \frac {7}{2}\right )}}{3 \, {\left (4 \, x e + x e^{\left (\frac {1}{4} \, x + \frac {1}{2}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2-27*x+36)*exp(1)*exp(1/4*x-1/2)+(16*x^2-144*x+144)*exp(1))*exp(x)/(12*x^2*exp(1/4*x-1/2)^2+96
*x^2*exp(1/4*x-1/2)+192*x^2),x, algorithm="maxima")

[Out]

1/3*(256*x*e^4 + (x*e^2 - 9*e^2)*e^x + 64*x*e^(1/4*x + 7/2))/(4*x*e + x*e^(1/4*x + 1/2))

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mupad [B]  time = 0.63, size = 93, normalized size = 3.32 \begin {gather*} \frac {192\,{\mathrm {e}}^3}{x}-\frac {256\,\left (9\,x\,{\mathrm {e}}^{7/2}-x^2\,{\mathrm {e}}^{7/2}\right )}{3\,x^2\,\left ({\mathrm {e}}^{x/4}+4\,\sqrt {\mathrm {e}}\right )}-\frac {{\mathrm {e}}^{\frac {3\,x}{4}}\,\left (3\,{\mathrm {e}}^{3/2}-\frac {x\,{\mathrm {e}}^{3/2}}{3}\right )}{x}+\frac {{\mathrm {e}}^{x/2}\,\left (12\,{\mathrm {e}}^2-\frac {4\,x\,{\mathrm {e}}^2}{3}\right )}{x}-\frac {{\mathrm {e}}^{x/4}\,\left (48\,{\mathrm {e}}^{5/2}-\frac {16\,x\,{\mathrm {e}}^{5/2}}{3}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(exp(1)*(16*x^2 - 144*x + 144) + exp(1)*exp(x/4 - 1/2)*(3*x^2 - 27*x + 36)))/(12*x^2*exp(x/2 - 1)
+ 96*x^2*exp(x/4 - 1/2) + 192*x^2),x)

[Out]

(192*exp(3))/x - (256*(9*x*exp(7/2) - x^2*exp(7/2)))/(3*x^2*(exp(x/4) + 4*exp(1/2))) - (exp((3*x)/4)*(3*exp(3/
2) - (x*exp(3/2))/3))/x + (exp(x/2)*(12*exp(2) - (4*x*exp(2))/3))/x - (exp(x/4)*(48*exp(5/2) - (16*x*exp(5/2))
/3))/x

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**2-27*x+36)*exp(1)*exp(1/4*x-1/2)+(16*x**2-144*x+144)*exp(1))*exp(x)/(12*x**2*exp(1/4*x-1/2)**
2+96*x**2*exp(1/4*x-1/2)+192*x**2),x)

[Out]

Exception raised: PolynomialError

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