3.32.51 \(\int \frac {24 x^2+6 x^3+e^{\frac {e^{\frac {1}{2} (-5-3 x+\log (4+x))}+x^2}{x}} (16 x+12 x^2+2 x^3+e^{\frac {1}{2} (-5-3 x+\log (4+x))} (-8-13 x-3 x^2))}{8+2 x} \, dx\)

Optimal. Leaf size=33 \[ x^2 \left (e^{\frac {e^{-x+\frac {1}{2} (-5-x+\log (4+x))}}{x}+x}+x\right ) \]

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Rubi [F]  time = 4.88, 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 {24 x^2+6 x^3+e^{\frac {e^{\frac {1}{2} (-5-3 x+\log (4+x))}+x^2}{x}} \left (16 x+12 x^2+2 x^3+e^{\frac {1}{2} (-5-3 x+\log (4+x))} \left (-8-13 x-3 x^2\right )\right )}{8+2 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(24*x^2 + 6*x^3 + E^((E^((-5 - 3*x + Log[4 + x])/2) + x^2)/x)*(16*x + 12*x^2 + 2*x^3 + E^((-5 - 3*x + Log[
4 + x])/2)*(-8 - 13*x - 3*x^2)))/(8 + 2*x),x]

[Out]

x^3 - 4*Defer[Subst][Defer[Int][E^(-1/2 - x^2/2 + (E^(7/2 - (3*x^2)/2)*x)/(-4 + x^2)), x], x, Sqrt[4 + x]] + 1
6*Defer[Subst][Defer[Int][E^(-4 + x^2 + (E^(7/2 - (3*x^2)/2)*x)/(-4 + x^2))*x, x], x, Sqrt[4 + x]] + 11*Defer[
Subst][Defer[Int][E^(-1/2 - x^2/2 + (E^(7/2 - (3*x^2)/2)*x)/(-4 + x^2))*x^2, x], x, Sqrt[4 + x]] - 12*Defer[Su
bst][Defer[Int][E^(-4 + x^2 + (E^(7/2 - (3*x^2)/2)*x)/(-4 + x^2))*x^3, x], x, Sqrt[4 + x]] - 3*Defer[Subst][De
fer[Int][E^(-1/2 - x^2/2 + (E^(7/2 - (3*x^2)/2)*x)/(-4 + x^2))*x^4, x], x, Sqrt[4 + x]] + 2*Defer[Subst][Defer
[Int][E^(-4 + x^2 + (E^(7/2 - (3*x^2)/2)*x)/(-4 + x^2))*x^5, x], x, Sqrt[4 + x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (x \left (2 e^{x+\frac {e^{-\frac {5}{2}-\frac {3 x}{2}} \sqrt {4+x}}{x}}+3 x+e^{x+\frac {e^{-\frac {5}{2}-\frac {3 x}{2}} \sqrt {4+x}}{x}} x\right )-\frac {e^{-\frac {5}{2}-\frac {x}{2}+\frac {e^{-\frac {5}{2}-\frac {3 x}{2}} \sqrt {4+x}}{x}} \left (8+13 x+3 x^2\right )}{2 \sqrt {4+x}}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {e^{-\frac {5}{2}-\frac {x}{2}+\frac {e^{-\frac {5}{2}-\frac {3 x}{2}} \sqrt {4+x}}{x}} \left (8+13 x+3 x^2\right )}{\sqrt {4+x}} \, dx\right )+\int x \left (2 e^{x+\frac {e^{-\frac {5}{2}-\frac {3 x}{2}} \sqrt {4+x}}{x}}+3 x+e^{x+\frac {e^{-\frac {5}{2}-\frac {3 x}{2}} \sqrt {4+x}}{x}} x\right ) \, dx\\ &=\int \left (3 x^2+e^{x+\frac {e^{-\frac {5}{2}-\frac {3 x}{2}} \sqrt {4+x}}{x}} x (2+x)\right ) \, dx-\operatorname {Subst}\left (\int e^{-\frac {1}{2}-\frac {x^2}{2}+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} \left (4-11 x^2+3 x^4\right ) \, dx,x,\sqrt {4+x}\right )\\ &=x^3+\int e^{x+\frac {e^{-\frac {5}{2}-\frac {3 x}{2}} \sqrt {4+x}}{x}} x (2+x) \, dx-\operatorname {Subst}\left (\int \left (4 e^{-\frac {1}{2}-\frac {x^2}{2}+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}}-11 e^{-\frac {1}{2}-\frac {x^2}{2}+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} x^2+3 e^{-\frac {1}{2}-\frac {x^2}{2}+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} x^4\right ) \, dx,x,\sqrt {4+x}\right )\\ &=x^3+2 \operatorname {Subst}\left (\int e^{-4+x^2+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} x \left (8-6 x^2+x^4\right ) \, dx,x,\sqrt {4+x}\right )-3 \operatorname {Subst}\left (\int e^{-\frac {1}{2}-\frac {x^2}{2}+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} x^4 \, dx,x,\sqrt {4+x}\right )-4 \operatorname {Subst}\left (\int e^{-\frac {1}{2}-\frac {x^2}{2}+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} \, dx,x,\sqrt {4+x}\right )+11 \operatorname {Subst}\left (\int e^{-\frac {1}{2}-\frac {x^2}{2}+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} x^2 \, dx,x,\sqrt {4+x}\right )\\ &=x^3+2 \operatorname {Subst}\left (\int \left (8 e^{-4+x^2+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} x-6 e^{-4+x^2+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} x^3+e^{-4+x^2+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} x^5\right ) \, dx,x,\sqrt {4+x}\right )-3 \operatorname {Subst}\left (\int e^{-\frac {1}{2}-\frac {x^2}{2}+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} x^4 \, dx,x,\sqrt {4+x}\right )-4 \operatorname {Subst}\left (\int e^{-\frac {1}{2}-\frac {x^2}{2}+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} \, dx,x,\sqrt {4+x}\right )+11 \operatorname {Subst}\left (\int e^{-\frac {1}{2}-\frac {x^2}{2}+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} x^2 \, dx,x,\sqrt {4+x}\right )\\ &=x^3+2 \operatorname {Subst}\left (\int e^{-4+x^2+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} x^5 \, dx,x,\sqrt {4+x}\right )-3 \operatorname {Subst}\left (\int e^{-\frac {1}{2}-\frac {x^2}{2}+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} x^4 \, dx,x,\sqrt {4+x}\right )-4 \operatorname {Subst}\left (\int e^{-\frac {1}{2}-\frac {x^2}{2}+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} \, dx,x,\sqrt {4+x}\right )+11 \operatorname {Subst}\left (\int e^{-\frac {1}{2}-\frac {x^2}{2}+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} x^2 \, dx,x,\sqrt {4+x}\right )-12 \operatorname {Subst}\left (\int e^{-4+x^2+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} x^3 \, dx,x,\sqrt {4+x}\right )+16 \operatorname {Subst}\left (\int e^{-4+x^2+\frac {e^{\frac {7}{2}-\frac {3 x^2}{2}} x}{-4+x^2}} x \, dx,x,\sqrt {4+x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.58, size = 32, normalized size = 0.97 \begin {gather*} x^2 \left (e^{x+\frac {e^{-\frac {5}{2}-\frac {3 x}{2}} \sqrt {4+x}}{x}}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24*x^2 + 6*x^3 + E^((E^((-5 - 3*x + Log[4 + x])/2) + x^2)/x)*(16*x + 12*x^2 + 2*x^3 + E^((-5 - 3*x
+ Log[4 + x])/2)*(-8 - 13*x - 3*x^2)))/(8 + 2*x),x]

[Out]

x^2*(E^(x + (E^(-5/2 - (3*x)/2)*Sqrt[4 + x])/x) + x)

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fricas [A]  time = 0.50, size = 29, normalized size = 0.88 \begin {gather*} x^{3} + x^{2} e^{\left (\frac {x^{2} + e^{\left (-\frac {3}{2} \, x + \frac {1}{2} \, \log \left (x + 4\right ) - \frac {5}{2}\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^2-13*x-8)*exp(1/2*log(4+x)-3/2*x-5/2)+2*x^3+12*x^2+16*x)*exp((exp(1/2*log(4+x)-3/2*x-5/2)+x^
2)/x)+6*x^3+24*x^2)/(2*x+8),x, algorithm="fricas")

[Out]

x^3 + x^2*e^((x^2 + e^(-3/2*x + 1/2*log(x + 4) - 5/2))/x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {6 \, x^{3} + 24 \, x^{2} + {\left (2 \, x^{3} + 12 \, x^{2} - {\left (3 \, x^{2} + 13 \, x + 8\right )} e^{\left (-\frac {3}{2} \, x + \frac {1}{2} \, \log \left (x + 4\right ) - \frac {5}{2}\right )} + 16 \, x\right )} e^{\left (\frac {x^{2} + e^{\left (-\frac {3}{2} \, x + \frac {1}{2} \, \log \left (x + 4\right ) - \frac {5}{2}\right )}}{x}\right )}}{2 \, {\left (x + 4\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^2-13*x-8)*exp(1/2*log(4+x)-3/2*x-5/2)+2*x^3+12*x^2+16*x)*exp((exp(1/2*log(4+x)-3/2*x-5/2)+x^
2)/x)+6*x^3+24*x^2)/(2*x+8),x, algorithm="giac")

[Out]

integrate(1/2*(6*x^3 + 24*x^2 + (2*x^3 + 12*x^2 - (3*x^2 + 13*x + 8)*e^(-3/2*x + 1/2*log(x + 4) - 5/2) + 16*x)
*e^((x^2 + e^(-3/2*x + 1/2*log(x + 4) - 5/2))/x))/(x + 4), x)

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maple [A]  time = 0.87, size = 30, normalized size = 0.91

method	result	size
risch	\(x^{3}+{\mathrm e}^{\frac {\sqrt {4+x}\, {\mathrm e}^{-\frac {5}{2}-\frac {3 x}{2}}+x^{2}}{x}} x^{2}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-3*x^2-13*x-8)*exp(1/2*ln(4+x)-3/2*x-5/2)+2*x^3+12*x^2+16*x)*exp((exp(1/2*ln(4+x)-3/2*x-5/2)+x^2)/x)+6*
x^3+24*x^2)/(2*x+8),x,method=_RETURNVERBOSE)

[Out]

x^3+exp(((4+x)^(1/2)*exp(-5/2-3/2*x)+x^2)/x)*x^2

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maxima [A]  time = 1.62, size = 27, normalized size = 0.82 \begin {gather*} x^{3} + x^{2} e^{\left (x + \frac {\sqrt {x + 4} e^{\left (-\frac {3}{2} \, x - \frac {5}{2}\right )}}{x} + \frac {1}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^2-13*x-8)*exp(1/2*log(4+x)-3/2*x-5/2)+2*x^3+12*x^2+16*x)*exp((exp(1/2*log(4+x)-3/2*x-5/2)+x^
2)/x)+6*x^3+24*x^2)/(2*x+8),x, algorithm="maxima")

[Out]

x^3 + x^2*e^(x + sqrt(x + 4)*e^(-3/2*x - 5/2)/x + 1/2)

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mupad [B]  time = 2.16, size = 26, normalized size = 0.79 \begin {gather*} x^3+x^2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{-\frac {5}{2}}\,\sqrt {x+4}}{x\,{\left ({\mathrm {e}}^x\right )}^{3/2}}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((exp(log(x + 4)/2 - (3*x)/2 - 5/2) + x^2)/x)*(16*x - exp(log(x + 4)/2 - (3*x)/2 - 5/2)*(13*x + 3*x^2
+ 8) + 12*x^2 + 2*x^3) + 24*x^2 + 6*x^3)/(2*x + 8),x)

[Out]

x^3 + x^2*exp((exp(-5/2)*(x + 4)^(1/2))/(x*exp(x)^(3/2)))*exp(x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x**2-13*x-8)*exp(1/2*ln(4+x)-3/2*x-5/2)+2*x**3+12*x**2+16*x)*exp((exp(1/2*ln(4+x)-3/2*x-5/2)+x
**2)/x)+6*x**3+24*x**2)/(2*x+8),x)

[Out]

Timed out

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