3.96.88 \(\int \frac {64+8 x^2+10 x^5-2 e x^5+2 x^6}{x^5} \, dx\)

Optimal. Leaf size=19 \[ -\frac {16}{x^4}-\frac {4}{x^2}+(5-e+x)^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {6, 14} \begin {gather*} -\frac {16}{x^4}+x^2-\frac {4}{x^2}+2 (5-e) x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(64 + 8*x^2 + 10*x^5 - 2*E*x^5 + 2*x^6)/x^5,x]

[Out]

-16/x^4 - 4/x^2 + 2*(5 - E)*x + x^2

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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 \frac {64+8 x^2+(10-2 e) x^5+2 x^6}{x^5} \, dx\\ &=\int \left (-2 (-5+e)+\frac {64}{x^5}+\frac {8}{x^3}+2 x\right ) \, dx\\ &=-\frac {16}{x^4}-\frac {4}{x^2}+2 (5-e) x+x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.42 \begin {gather*} 2 \left (-\frac {8}{x^4}-\frac {2}{x^2}+5 x-e x+\frac {x^2}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(64 + 8*x^2 + 10*x^5 - 2*E*x^5 + 2*x^6)/x^5,x]

[Out]

2*(-8/x^4 - 2/x^2 + 5*x - E*x + x^2/2)

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fricas [A]  time = 0.58, size = 26, normalized size = 1.37 \begin {gather*} \frac {x^{6} - 2 \, x^{5} e + 10 \, x^{5} - 4 \, x^{2} - 16}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5*exp(1)+2*x^6+10*x^5+8*x^2+64)/x^5,x, algorithm="fricas")

[Out]

(x^6 - 2*x^5*e + 10*x^5 - 4*x^2 - 16)/x^4

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giac [A]  time = 0.12, size = 22, normalized size = 1.16 \begin {gather*} x^{2} - 2 \, x e + 10 \, x - \frac {4 \, {\left (x^{2} + 4\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5*exp(1)+2*x^6+10*x^5+8*x^2+64)/x^5,x, algorithm="giac")

[Out]

x^2 - 2*x*e + 10*x - 4*(x^2 + 4)/x^4

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maple [A]  time = 0.04, size = 23, normalized size = 1.21

method	result	size
default	\(x^{2}-2 x \,{\mathrm e}+10 x -\frac {16}{x^{4}}-\frac {4}{x^{2}}\)	\(23\)
risch	\(-2 x \,{\mathrm e}+x^{2}+10 x +\frac {-4 x^{2}-16}{x^{4}}\)	\(24\)
norman	\(\frac {-16+x^{6}+\left (-2 \,{\mathrm e}+10\right ) x^{5}-4 x^{2}}{x^{4}}\)	\(25\)
gosper	\(-\frac {2 x^{5} {\mathrm e}-x^{6}-10 x^{5}+4 x^{2}+16}{x^{4}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^5*exp(1)+2*x^6+10*x^5+8*x^2+64)/x^5,x,method=_RETURNVERBOSE)

[Out]

x^2-2*x*exp(1)+10*x-16/x^4-4/x^2

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maxima [A]  time = 0.35, size = 21, normalized size = 1.11 \begin {gather*} x^{2} - 2 \, x {\left (e - 5\right )} - \frac {4 \, {\left (x^{2} + 4\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5*exp(1)+2*x^6+10*x^5+8*x^2+64)/x^5,x, algorithm="maxima")

[Out]

x^2 - 2*x*(e - 5) - 4*(x^2 + 4)/x^4

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mupad [B]  time = 8.25, size = 25, normalized size = 1.32 \begin {gather*} x^2-\frac {4\,x^2+16}{x^4}-x\,\left (2\,\mathrm {e}-10\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^2 - 2*x^5*exp(1) + 10*x^5 + 2*x^6 + 64)/x^5,x)

[Out]

x^2 - (4*x^2 + 16)/x^4 - x*(2*exp(1) - 10)

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sympy [A]  time = 0.10, size = 22, normalized size = 1.16 \begin {gather*} x^{2} + x \left (10 - 2 e\right ) + \frac {- 4 x^{2} - 16}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**5*exp(1)+2*x**6+10*x**5+8*x**2+64)/x**5,x)

[Out]

x**2 + x*(10 - 2*E) + (-4*x**2 - 16)/x**4

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