3.32.96 \(\int \frac {8 x^4+e^{\frac {-1-51 x+4 x^3}{4 x}} (4-17 x+32 x^3-8 x^4)}{4 x^3} \, dx\)

Optimal. Leaf size=31 \[ 3+\frac {e^{-13+\frac {-1+x}{4 x}+x^2} (4-x)}{x}+x^2 \]

________________________________________________________________________________________

Rubi [F]  time = 0.43, 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 {8 x^4+e^{\frac {-1-51 x+4 x^3}{4 x}} \left (4-17 x+32 x^3-8 x^4\right )}{4 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8*x^4 + E^((-1 - 51*x + 4*x^3)/(4*x))*(4 - 17*x + 32*x^3 - 8*x^4))/(4*x^3),x]

[Out]

x^2 + 8*Defer[Int][E^(-51/4 - 1/(4*x) + x^2), x] + Defer[Int][E^(-51/4 - 1/(4*x) + x^2)/x^3, x] - (17*Defer[In
t][E^(-51/4 - 1/(4*x) + x^2)/x^2, x])/4 - 2*Defer[Int][E^(-51/4 - 1/(4*x) + x^2)*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {8 x^4+e^{\frac {-1-51 x+4 x^3}{4 x}} \left (4-17 x+32 x^3-8 x^4\right )}{x^3} \, dx\\ &=\frac {1}{4} \int \left (8 x-\frac {e^{-\frac {51}{4}-\frac {1}{4 x}+x^2} \left (-4+17 x-32 x^3+8 x^4\right )}{x^3}\right ) \, dx\\ &=x^2-\frac {1}{4} \int \frac {e^{-\frac {51}{4}-\frac {1}{4 x}+x^2} \left (-4+17 x-32 x^3+8 x^4\right )}{x^3} \, dx\\ &=x^2-\frac {1}{4} \int \left (-32 e^{-\frac {51}{4}-\frac {1}{4 x}+x^2}-\frac {4 e^{-\frac {51}{4}-\frac {1}{4 x}+x^2}}{x^3}+\frac {17 e^{-\frac {51}{4}-\frac {1}{4 x}+x^2}}{x^2}+8 e^{-\frac {51}{4}-\frac {1}{4 x}+x^2} x\right ) \, dx\\ &=x^2-2 \int e^{-\frac {51}{4}-\frac {1}{4 x}+x^2} x \, dx-\frac {17}{4} \int \frac {e^{-\frac {51}{4}-\frac {1}{4 x}+x^2}}{x^2} \, dx+8 \int e^{-\frac {51}{4}-\frac {1}{4 x}+x^2} \, dx+\int \frac {e^{-\frac {51}{4}-\frac {1}{4 x}+x^2}}{x^3} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.15, size = 39, normalized size = 1.26 \begin {gather*} -\frac {1}{4} e^{-\frac {1}{4 x}+x^2} \left (\frac {4}{e^{51/4}}-\frac {16}{e^{51/4} x}\right )+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*x^4 + E^((-1 - 51*x + 4*x^3)/(4*x))*(4 - 17*x + 32*x^3 - 8*x^4))/(4*x^3),x]

[Out]

-1/4*(E^(-1/4*1/x + x^2)*(4/E^(51/4) - 16/(E^(51/4)*x))) + x^2

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 29, normalized size = 0.94 \begin {gather*} \frac {x^{3} - {\left (x - 4\right )} e^{\left (\frac {4 \, x^{3} - 51 \, x - 1}{4 \, x}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-8*x^4+32*x^3-17*x+4)*exp(1/4*(4*x^3-51*x-1)/x)+8*x^4)/x^3,x, algorithm="fricas")

[Out]

(x^3 - (x - 4)*e^(1/4*(4*x^3 - 51*x - 1)/x))/x

________________________________________________________________________________________

giac [A]  time = 0.27, size = 45, normalized size = 1.45 \begin {gather*} \frac {x^{3} - x e^{\left (\frac {4 \, x^{3} - 51 \, x - 1}{4 \, x}\right )} + 4 \, e^{\left (\frac {4 \, x^{3} - 51 \, x - 1}{4 \, x}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-8*x^4+32*x^3-17*x+4)*exp(1/4*(4*x^3-51*x-1)/x)+8*x^4)/x^3,x, algorithm="giac")

[Out]

(x^3 - x*e^(1/4*(4*x^3 - 51*x - 1)/x) + 4*e^(1/4*(4*x^3 - 51*x - 1)/x))/x

________________________________________________________________________________________

maple [A]  time = 0.06, size = 29, normalized size = 0.94

method	result	size
risch	\(x^{2}-\frac {\left (x -4\right ) {\mathrm e}^{\frac {4 x^{3}-51 x -1}{4 x}}}{x}\)	\(29\)
norman	\(\frac {x^{4}+4 \,{\mathrm e}^{\frac {4 x^{3}-51 x -1}{4 x}} x -{\mathrm e}^{\frac {4 x^{3}-51 x -1}{4 x}} x^{2}}{x^{2}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*((-8*x^4+32*x^3-17*x+4)*exp(1/4*(4*x^3-51*x-1)/x)+8*x^4)/x^3,x,method=_RETURNVERBOSE)

[Out]

x^2-(x-4)/x*exp(1/4*(4*x^3-51*x-1)/x)

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 29, normalized size = 0.94 \begin {gather*} x^{2} - \frac {{\left (x e^{\frac {1}{4}} - 4 \, e^{\frac {1}{4}}\right )} e^{\left (x^{2} - \frac {1}{4 \, x} - 13\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-8*x^4+32*x^3-17*x+4)*exp(1/4*(4*x^3-51*x-1)/x)+8*x^4)/x^3,x, algorithm="maxima")

[Out]

x^2 - (x*e^(1/4) - 4*e^(1/4))*e^(x^2 - 1/4/x - 13)/x

________________________________________________________________________________________

mupad [B]  time = 1.91, size = 33, normalized size = 1.06 \begin {gather*} \frac {4\,{\mathrm {e}}^{x^2-\frac {1}{4\,x}-\frac {51}{4}}}{x}-{\mathrm {e}}^{x^2-\frac {1}{4\,x}-\frac {51}{4}}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((exp(-((51*x)/4 - x^3 + 1/4)/x)*(17*x - 32*x^3 + 8*x^4 - 4))/4 - 2*x^4)/x^3,x)

[Out]

(4*exp(x^2 - 1/(4*x) - 51/4))/x - exp(x^2 - 1/(4*x) - 51/4) + x^2

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 22, normalized size = 0.71 \begin {gather*} x^{2} + \frac {\left (4 - x\right ) e^{\frac {x^{3} - \frac {51 x}{4} - \frac {1}{4}}{x}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-8*x**4+32*x**3-17*x+4)*exp(1/4*(4*x**3-51*x-1)/x)+8*x**4)/x**3,x)

[Out]

x**2 + (4 - x)*exp((x**3 - 51*x/4 - 1/4)/x)/x

________________________________________________________________________________________