3.48.61 \(\int \frac {26+128 x-520 x^3-576 x^4-96 x^5-4 x^6+e^4 (-8 x+32 x^3+4 x^4)}{x^3} \, dx\)

Optimal. Leaf size=40 \[ 3+\frac {\frac {3}{x}-x}{x}-\left (e^4-x^2-\frac {(2+4 x)^2}{x}\right )^2 \]

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Rubi [A]  time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 1, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {14} \begin {gather*} -x^4-32 x^3-2 \left (144-e^4\right ) x^2-\frac {13}{x^2}-8 \left (65-4 e^4\right ) x-\frac {8 \left (16-e^4\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(26 + 128*x - 520*x^3 - 576*x^4 - 96*x^5 - 4*x^6 + E^4*(-8*x + 32*x^3 + 4*x^4))/x^3,x]

[Out]

-13/x^2 - (8*(16 - E^4))/x - 8*(65 - 4*E^4)*x - 2*(144 - E^4)*x^2 - 32*x^3 - x^4

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (8 \left (-65+4 e^4\right )+\frac {26}{x^3}-\frac {8 \left (-16+e^4\right )}{x^2}+4 \left (-144+e^4\right ) x-96 x^2-4 x^3\right ) \, dx\\ &=-\frac {13}{x^2}-\frac {8 \left (16-e^4\right )}{x}-8 \left (65-4 e^4\right ) x-2 \left (144-e^4\right ) x^2-32 x^3-x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 51, normalized size = 1.28 \begin {gather*} -\frac {13}{x^2}-\frac {128}{x}+\frac {8 e^4}{x}-520 x+32 e^4 x-288 x^2+2 e^4 x^2-32 x^3-x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(26 + 128*x - 520*x^3 - 576*x^4 - 96*x^5 - 4*x^6 + E^4*(-8*x + 32*x^3 + 4*x^4))/x^3,x]

[Out]

-13/x^2 - 128/x + (8*E^4)/x - 520*x + 32*E^4*x - 288*x^2 + 2*E^4*x^2 - 32*x^3 - x^4

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fricas [A]  time = 0.74, size = 44, normalized size = 1.10 \begin {gather*} -\frac {x^{6} + 32 \, x^{5} + 288 \, x^{4} + 520 \, x^{3} - 2 \, {\left (x^{4} + 16 \, x^{3} + 4 \, x\right )} e^{4} + 128 \, x + 13}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+32*x^3-8*x)*exp(4)-4*x^6-96*x^5-576*x^4-520*x^3+128*x+26)/x^3,x, algorithm="fricas")

[Out]

-(x^6 + 32*x^5 + 288*x^4 + 520*x^3 - 2*(x^4 + 16*x^3 + 4*x)*e^4 + 128*x + 13)/x^2

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giac [A]  time = 0.14, size = 45, normalized size = 1.12 \begin {gather*} -x^{4} - 32 \, x^{3} + 2 \, x^{2} e^{4} - 288 \, x^{2} + 32 \, x e^{4} - 520 \, x + \frac {8 \, x e^{4} - 128 \, x - 13}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+32*x^3-8*x)*exp(4)-4*x^6-96*x^5-576*x^4-520*x^3+128*x+26)/x^3,x, algorithm="giac")

[Out]

-x^4 - 32*x^3 + 2*x^2*e^4 - 288*x^2 + 32*x*e^4 - 520*x + (8*x*e^4 - 128*x - 13)/x^2

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maple [A]  time = 0.04, size = 45, normalized size = 1.12

method	result	size
norman	\(\frac {-13+\left (2 \,{\mathrm e}^{4}-288\right ) x^{4}+\left (8 \,{\mathrm e}^{4}-128\right ) x +\left (32 \,{\mathrm e}^{4}-520\right ) x^{3}-32 x^{5}-x^{6}}{x^{2}}\)	\(45\)
risch	\(-x^{4}+2 x^{2} {\mathrm e}^{4}-32 x^{3}+32 x \,{\mathrm e}^{4}-288 x^{2}-520 x +\frac {\left (8 \,{\mathrm e}^{4}-128\right ) x -13}{x^{2}}\)	\(46\)
default	\(-x^{4}+2 x^{2} {\mathrm e}^{4}-32 x^{3}+32 x \,{\mathrm e}^{4}-288 x^{2}-520 x -\frac {2 \left (-4 \,{\mathrm e}^{4}+64\right )}{x}-\frac {13}{x^{2}}\)	\(48\)
gosper	\(\frac {-x^{6}+2 x^{4} {\mathrm e}^{4}-32 x^{5}+32 x^{3} {\mathrm e}^{4}-288 x^{4}-520 x^{3}+8 x \,{\mathrm e}^{4}-128 x -13}{x^{2}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^4+32*x^3-8*x)*exp(4)-4*x^6-96*x^5-576*x^4-520*x^3+128*x+26)/x^3,x,method=_RETURNVERBOSE)

[Out]

(-13+(2*exp(4)-288)*x^4+(8*exp(4)-128)*x+(32*exp(4)-520)*x^3-32*x^5-x^6)/x^2

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maxima [A]  time = 0.36, size = 42, normalized size = 1.05 \begin {gather*} -x^{4} - 32 \, x^{3} + 2 \, x^{2} {\left (e^{4} - 144\right )} + 8 \, x {\left (4 \, e^{4} - 65\right )} + \frac {8 \, x {\left (e^{4} - 16\right )} - 13}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+32*x^3-8*x)*exp(4)-4*x^6-96*x^5-576*x^4-520*x^3+128*x+26)/x^3,x, algorithm="maxima")

[Out]

-x^4 - 32*x^3 + 2*x^2*(e^4 - 144) + 8*x*(4*e^4 - 65) + (8*x*(e^4 - 16) - 13)/x^2

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mupad [B]  time = 0.06, size = 43, normalized size = 1.08 \begin {gather*} x^2\,\left (2\,{\mathrm {e}}^4-288\right )+\frac {x\,\left (8\,{\mathrm {e}}^4-128\right )-13}{x^2}-32\,x^3-x^4+x\,\left (32\,{\mathrm {e}}^4-520\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(520*x^3 - exp(4)*(32*x^3 - 8*x + 4*x^4) - 128*x + 576*x^4 + 96*x^5 + 4*x^6 - 26)/x^3,x)

[Out]

x^2*(2*exp(4) - 288) + (x*(8*exp(4) - 128) - 13)/x^2 - 32*x^3 - x^4 + x*(32*exp(4) - 520)

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sympy [A]  time = 0.15, size = 41, normalized size = 1.02 \begin {gather*} - x^{4} - 32 x^{3} - x^{2} \left (288 - 2 e^{4}\right ) - x \left (520 - 32 e^{4}\right ) - \frac {x \left (128 - 8 e^{4}\right ) + 13}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**4+32*x**3-8*x)*exp(4)-4*x**6-96*x**5-576*x**4-520*x**3+128*x+26)/x**3,x)

[Out]

-x**4 - 32*x**3 - x**2*(288 - 2*exp(4)) - x*(520 - 32*exp(4)) - (x*(128 - 8*exp(4)) + 13)/x**2

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