3.52.90 \(\int \frac {16 x^2-32 x^3+96 x^7+e^{e^{5 x/4}} (-4+5 e^{5 x/4} x)}{16 x^2} \, dx\)

Optimal. Leaf size=28 \[ 4+\frac {e^{e^{5 x/4}}}{4 x}+x+x \left (-x+x^5\right ) \]

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Rubi [F]  time = 0.14, 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 {16 x^2-32 x^3+96 x^7+e^{e^{5 x/4}} \left (-4+5 e^{5 x/4} x\right )}{16 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(16*x^2 - 32*x^3 + 96*x^7 + E^E^((5*x)/4)*(-4 + 5*E^((5*x)/4)*x))/(16*x^2),x]

[Out]

x - x^2 + x^6 - Defer[Int][E^E^((5*x)/4)/x^2, x]/4 + (5*Defer[Int][E^(E^((5*x)/4) + (5*x)/4)/x, x])/16

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{16} \int \frac {16 x^2-32 x^3+96 x^7+e^{e^{5 x/4}} \left (-4+5 e^{5 x/4} x\right )}{x^2} \, dx\\ &=\frac {1}{16} \int \left (\frac {5 e^{e^{5 x/4}+\frac {5 x}{4}}}{x}+\frac {4 \left (-e^{e^{5 x/4}}+4 x^2-8 x^3+24 x^7\right )}{x^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {-e^{e^{5 x/4}}+4 x^2-8 x^3+24 x^7}{x^2} \, dx+\frac {5}{16} \int \frac {e^{e^{5 x/4}+\frac {5 x}{4}}}{x} \, dx\\ &=\frac {1}{4} \int \left (-\frac {e^{e^{5 x/4}}}{x^2}+4 \left (1-2 x+6 x^5\right )\right ) \, dx+\frac {5}{16} \int \frac {e^{e^{5 x/4}+\frac {5 x}{4}}}{x} \, dx\\ &=-\left (\frac {1}{4} \int \frac {e^{e^{5 x/4}}}{x^2} \, dx\right )+\frac {5}{16} \int \frac {e^{e^{5 x/4}+\frac {5 x}{4}}}{x} \, dx+\int \left (1-2 x+6 x^5\right ) \, dx\\ &=x-x^2+x^6-\frac {1}{4} \int \frac {e^{e^{5 x/4}}}{x^2} \, dx+\frac {5}{16} \int \frac {e^{e^{5 x/4}+\frac {5 x}{4}}}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 26, normalized size = 0.93 \begin {gather*} \frac {e^{e^{5 x/4}}}{4 x}+x-x^2+x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16*x^2 - 32*x^3 + 96*x^7 + E^E^((5*x)/4)*(-4 + 5*E^((5*x)/4)*x))/(16*x^2),x]

[Out]

E^E^((5*x)/4)/(4*x) + x - x^2 + x^6

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fricas [A]  time = 0.50, size = 26, normalized size = 0.93 \begin {gather*} \frac {4 \, x^{7} - 4 \, x^{3} + 4 \, x^{2} + e^{\left (e^{\left (\frac {5}{4} \, x\right )}\right )}}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*((5*x*exp(5/4*x)-4)*exp(exp(5/4*x))+96*x^7-32*x^3+16*x^2)/x^2,x, algorithm="fricas")

[Out]

1/4*(4*x^7 - 4*x^3 + 4*x^2 + e^(e^(5/4*x)))/x

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giac [B]  time = 4.86, size = 46, normalized size = 1.64 \begin {gather*} \frac {{\left (4 \, x^{7} e^{\left (\frac {5}{4} \, x\right )} - 4 \, x^{3} e^{\left (\frac {5}{4} \, x\right )} + 4 \, x^{2} e^{\left (\frac {5}{4} \, x\right )} + e^{\left (\frac {5}{4} \, x + e^{\left (\frac {5}{4} \, x\right )}\right )}\right )} e^{\left (-\frac {5}{4} \, x\right )}}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*((5*x*exp(5/4*x)-4)*exp(exp(5/4*x))+96*x^7-32*x^3+16*x^2)/x^2,x, algorithm="giac")

[Out]

1/4*(4*x^7*e^(5/4*x) - 4*x^3*e^(5/4*x) + 4*x^2*e^(5/4*x) + e^(5/4*x + e^(5/4*x)))*e^(-5/4*x)/x

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maple [A]  time = 0.04, size = 21, normalized size = 0.75

method	result	size
risch	\(x^{6}-x^{2}+x +\frac {{\mathrm e}^{{\mathrm e}^{\frac {5 x}{4}}}}{4 x}\)	\(21\)
norman	\(\frac {x^{2}+x^{7}-x^{3}+\frac {{\mathrm e}^{{\mathrm e}^{\frac {5 x}{4}}}}{4}}{x}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/16*((5*x*exp(5/4*x)-4)*exp(exp(5/4*x))+96*x^7-32*x^3+16*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

x^6-x^2+x+1/4*exp(exp(5/4*x))/x

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maxima [A]  time = 0.39, size = 20, normalized size = 0.71 \begin {gather*} x^{6} - x^{2} + x + \frac {e^{\left (e^{\left (\frac {5}{4} \, x\right )}\right )}}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*((5*x*exp(5/4*x)-4)*exp(exp(5/4*x))+96*x^7-32*x^3+16*x^2)/x^2,x, algorithm="maxima")

[Out]

x^6 - x^2 + x + 1/4*e^(e^(5/4*x))/x

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mupad [B]  time = 0.10, size = 20, normalized size = 0.71 \begin {gather*} x-x^2+x^6+\frac {{\mathrm {e}}^{{\left ({\mathrm {e}}^x\right )}^{5/4}}}{4\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(exp((5*x)/4))*(5*x*exp((5*x)/4) - 4))/16 + x^2 - 2*x^3 + 6*x^7)/x^2,x)

[Out]

x - x^2 + x^6 + exp(exp(x)^(5/4))/(4*x)

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sympy [A]  time = 0.12, size = 19, normalized size = 0.68 \begin {gather*} x^{6} - x^{2} + x + \frac {e^{e^{\frac {5 x}{4}}}}{4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*((5*x*exp(5/4*x)-4)*exp(exp(5/4*x))+96*x**7-32*x**3+16*x**2)/x**2,x)

[Out]

x**6 - x**2 + x + exp(exp(5*x/4))/(4*x)

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