3.76.33 \(\int \frac {e^2 (-72+288 x^2-2016 x^6+5040 x^8-6048 x^{10}+4032 x^{12}-1440 x^{14}+216 x^{16})}{5 x^5} \, dx\)

Optimal. Leaf size=19 \[ \frac {18}{5} e^2 x^4 \left (-\frac {1}{x}+x\right )^8 \]

________________________________________________________________________________________

Rubi [B]  time = 0.03, antiderivative size = 79, normalized size of antiderivative = 4.16, number of steps used = 3, number of rules used = 2, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {12, 14} \begin {gather*} \frac {18 e^2 x^{12}}{5}-\frac {144 e^2 x^{10}}{5}+\frac {504 e^2 x^8}{5}-\frac {1008 e^2 x^6}{5}+252 e^2 x^4+\frac {18 e^2}{5 x^4}-\frac {1008 e^2 x^2}{5}-\frac {144 e^2}{5 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^2*(-72 + 288*x^2 - 2016*x^6 + 5040*x^8 - 6048*x^10 + 4032*x^12 - 1440*x^14 + 216*x^16))/(5*x^5),x]

[Out]

(18*E^2)/(5*x^4) - (144*E^2)/(5*x^2) - (1008*E^2*x^2)/5 + 252*E^2*x^4 - (1008*E^2*x^6)/5 + (504*E^2*x^8)/5 - (
144*E^2*x^10)/5 + (18*E^2*x^12)/5

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {1}{5} e^2 \int \frac {-72+288 x^2-2016 x^6+5040 x^8-6048 x^{10}+4032 x^{12}-1440 x^{14}+216 x^{16}}{x^5} \, dx\\ &=\frac {1}{5} e^2 \int \left (-\frac {72}{x^5}+\frac {288}{x^3}-2016 x+5040 x^3-6048 x^5+4032 x^7-1440 x^9+216 x^{11}\right ) \, dx\\ &=\frac {18 e^2}{5 x^4}-\frac {144 e^2}{5 x^2}-\frac {1008 e^2 x^2}{5}+252 e^2 x^4-\frac {1008 e^2 x^6}{5}+\frac {504 e^2 x^8}{5}-\frac {144 e^2 x^{10}}{5}+\frac {18 e^2 x^{12}}{5}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.01, size = 54, normalized size = 2.84 \begin {gather*} \frac {72}{5} e^2 \left (\frac {1}{4 x^4}-\frac {2}{x^2}-14 x^2+\frac {35 x^4}{2}-14 x^6+7 x^8-2 x^{10}+\frac {x^{12}}{4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^2*(-72 + 288*x^2 - 2016*x^6 + 5040*x^8 - 6048*x^10 + 4032*x^12 - 1440*x^14 + 216*x^16))/(5*x^5),x
]

[Out]

(72*E^2*(1/(4*x^4) - 2/x^2 - 14*x^2 + (35*x^4)/2 - 14*x^6 + 7*x^8 - 2*x^10 + x^12/4))/5

________________________________________________________________________________________

fricas [B]  time = 1.34, size = 42, normalized size = 2.21 \begin {gather*} \frac {18 \, {\left (x^{16} - 8 \, x^{14} + 28 \, x^{12} - 56 \, x^{10} + 70 \, x^{8} - 56 \, x^{6} - 8 \, x^{2} + 1\right )} e^{2}}{5 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(216*x^16-1440*x^14+4032*x^12-6048*x^10+5040*x^8-2016*x^6+288*x^2-72)*exp(2)/x^5,x, algorithm="f
ricas")

[Out]

18/5*(x^16 - 8*x^14 + 28*x^12 - 56*x^10 + 70*x^8 - 56*x^6 - 8*x^2 + 1)*e^2/x^4

________________________________________________________________________________________

giac [B]  time = 0.14, size = 45, normalized size = 2.37 \begin {gather*} \frac {18}{5} \, {\left (x^{12} - 8 \, x^{10} + 28 \, x^{8} - 56 \, x^{6} + 70 \, x^{4} - 56 \, x^{2} - \frac {8 \, x^{2} - 1}{x^{4}}\right )} e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(216*x^16-1440*x^14+4032*x^12-6048*x^10+5040*x^8-2016*x^6+288*x^2-72)*exp(2)/x^5,x, algorithm="g
iac")

[Out]

18/5*(x^12 - 8*x^10 + 28*x^8 - 56*x^6 + 70*x^4 - 56*x^2 - (8*x^2 - 1)/x^4)*e^2

________________________________________________________________________________________

maple [B]  time = 0.04, size = 43, normalized size = 2.26

method	result	size
gosper	\(\frac {18 \,{\mathrm e}^{2} \left (x^{16}-8 x^{14}+28 x^{12}-56 x^{10}+70 x^{8}-56 x^{6}-8 x^{2}+1\right )}{5 x^{4}}\)	\(43\)
default	\(\frac {72 \,{\mathrm e}^{2} \left (\frac {x^{12}}{4}-2 x^{10}+7 x^{8}-14 x^{6}+\frac {35 x^{4}}{2}-14 x^{2}+\frac {1}{4 x^{4}}-\frac {2}{x^{2}}\right )}{5}\)	\(46\)
risch	\(\frac {18 \,{\mathrm e}^{2} x^{12}}{5}-\frac {144 \,{\mathrm e}^{2} x^{10}}{5}+\frac {504 x^{8} {\mathrm e}^{2}}{5}-\frac {1008 x^{6} {\mathrm e}^{2}}{5}+252 x^{4} {\mathrm e}^{2}-\frac {1008 x^{2} {\mathrm e}^{2}}{5}+\frac {{\mathrm e}^{2} \left (-144 x^{2}+18\right )}{5 x^{4}}\)	\(58\)
norman	\(\frac {-\frac {144 x^{2} {\mathrm e}^{2}}{5}-\frac {1008 x^{6} {\mathrm e}^{2}}{5}+252 x^{8} {\mathrm e}^{2}-\frac {1008 \,{\mathrm e}^{2} x^{10}}{5}+\frac {504 \,{\mathrm e}^{2} x^{12}}{5}-\frac {144 \,{\mathrm e}^{2} x^{14}}{5}+\frac {18 \,{\mathrm e}^{2} x^{16}}{5}+\frac {18 \,{\mathrm e}^{2}}{5}}{x^{4}}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(216*x^16-1440*x^14+4032*x^12-6048*x^10+5040*x^8-2016*x^6+288*x^2-72)*exp(2)/x^5,x,method=_RETURNVERBO
SE)

[Out]

18/5*exp(2)*(x^16-8*x^14+28*x^12-56*x^10+70*x^8-56*x^6-8*x^2+1)/x^4

________________________________________________________________________________________

maxima [B]  time = 0.36, size = 45, normalized size = 2.37 \begin {gather*} \frac {18}{5} \, {\left (x^{12} - 8 \, x^{10} + 28 \, x^{8} - 56 \, x^{6} + 70 \, x^{4} - 56 \, x^{2} - \frac {8 \, x^{2} - 1}{x^{4}}\right )} e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(216*x^16-1440*x^14+4032*x^12-6048*x^10+5040*x^8-2016*x^6+288*x^2-72)*exp(2)/x^5,x, algorithm="m
axima")

[Out]

18/5*(x^12 - 8*x^10 + 28*x^8 - 56*x^6 + 70*x^4 - 56*x^2 - (8*x^2 - 1)/x^4)*e^2

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 60, normalized size = 3.16 \begin {gather*} 252\,x^4\,{\mathrm {e}}^2-\frac {1008\,x^2\,{\mathrm {e}}^2}{5}-\frac {1008\,x^6\,{\mathrm {e}}^2}{5}+\frac {504\,x^8\,{\mathrm {e}}^2}{5}-\frac {144\,x^{10}\,{\mathrm {e}}^2}{5}+\frac {18\,x^{12}\,{\mathrm {e}}^2}{5}+\frac {18\,{\mathrm {e}}^2-144\,x^2\,{\mathrm {e}}^2}{5\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2)*(288*x^2 - 2016*x^6 + 5040*x^8 - 6048*x^10 + 4032*x^12 - 1440*x^14 + 216*x^16 - 72))/(5*x^5),x)

[Out]

252*x^4*exp(2) - (1008*x^2*exp(2))/5 - (1008*x^6*exp(2))/5 + (504*x^8*exp(2))/5 - (144*x^10*exp(2))/5 + (18*x^
12*exp(2))/5 + (18*exp(2) - 144*x^2*exp(2))/(5*x^4)

________________________________________________________________________________________

sympy [B]  time = 0.18, size = 76, normalized size = 4.00 \begin {gather*} \frac {18 x^{12} e^{2}}{5} - \frac {144 x^{10} e^{2}}{5} + \frac {504 x^{8} e^{2}}{5} - \frac {1008 x^{6} e^{2}}{5} + 252 x^{4} e^{2} - \frac {1008 x^{2} e^{2}}{5} + \frac {- 144 x^{2} e^{2} + 18 e^{2}}{5 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(216*x**16-1440*x**14+4032*x**12-6048*x**10+5040*x**8-2016*x**6+288*x**2-72)*exp(2)/x**5,x)

[Out]

18*x**12*exp(2)/5 - 144*x**10*exp(2)/5 + 504*x**8*exp(2)/5 - 1008*x**6*exp(2)/5 + 252*x**4*exp(2) - 1008*x**2*
exp(2)/5 + (-144*x**2*exp(2) + 18*exp(2))/(5*x**4)

________________________________________________________________________________________