3.2.25 \(\int \frac {-32400000000+10800000000 x^2+60 e^{16} x^4}{e^{16} x^7} \, dx\)

Optimal. Leaf size=22 \[ \frac {3 \left (5-\frac {900000000}{e^{16} x^4}\right ) \left (-\frac {2}{x}+x\right )}{x} \]

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {12, 14} \begin {gather*} \frac {5400000000}{e^{16} x^6}-\frac {2700000000}{e^{16} x^4}-\frac {30}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-32400000000 + 10800000000*x^2 + 60*E^16*x^4)/(E^16*x^7),x]

[Out]

5400000000/(E^16*x^6) - 2700000000/(E^16*x^4) - 30/x^2

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 {\int \frac {-32400000000+10800000000 x^2+60 e^{16} x^4}{x^7} \, dx}{e^{16}}\\ &=\frac {\int \left (-\frac {32400000000}{x^7}+\frac {10800000000}{x^5}+\frac {60 e^{16}}{x^3}\right ) \, dx}{e^{16}}\\ &=\frac {5400000000}{e^{16} x^6}-\frac {2700000000}{e^{16} x^4}-\frac {30}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 26, normalized size = 1.18 \begin {gather*} \frac {60 \left (\frac {90000000}{x^6}-\frac {45000000}{x^4}-\frac {e^{16}}{2 x^2}\right )}{e^{16}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-32400000000 + 10800000000*x^2 + 60*E^16*x^4)/(E^16*x^7),x]

[Out]

(60*(90000000/x^6 - 45000000/x^4 - E^16/(2*x^2)))/E^16

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fricas [A]  time = 0.61, size = 20, normalized size = 0.91 \begin {gather*} -\frac {30 \, {\left (x^{4} e^{16} + 90000000 \, x^{2} - 180000000\right )} e^{\left (-16\right )}}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((60*x^4*exp(4)^4+10800000000*x^2-32400000000)/x^7/exp(4)^4,x, algorithm="fricas")

[Out]

-30*(x^4*e^16 + 90000000*x^2 - 180000000)*e^(-16)/x^6

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giac [A]  time = 0.34, size = 20, normalized size = 0.91 \begin {gather*} -\frac {30 \, {\left (x^{4} e^{16} + 90000000 \, x^{2} - 180000000\right )} e^{\left (-16\right )}}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((60*x^4*exp(4)^4+10800000000*x^2-32400000000)/x^7/exp(4)^4,x, algorithm="giac")

[Out]

-30*(x^4*e^16 + 90000000*x^2 - 180000000)*e^(-16)/x^6

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maple [A]  time = 0.04, size = 21, normalized size = 0.95

method	result	size
risch	\(\frac {{\mathrm e}^{-16} \left (-30 x^{4} {\mathrm e}^{16}-2700000000 x^{2}[phone]\right )}{x^{6}}\)	\(21\)
gosper	\(-\frac {30 \left (x^{4} {\mathrm e}^{16}[phone] x^{2}-180000000\right ) {\mathrm e}^{-16}}{x^{6}}\)	\(25\)
default	\(60 \,{\mathrm e}^{-16} \left (-\frac {{\mathrm e}^{16}}{2 x^{2}}-\frac {45000000}{x^{4}}+\frac {90000000}{x^{6}}\right )\)	\(25\)
norman	\(\frac {\left (5400000000 \,{\mathrm e}^{-4}-2700000000 x^{2} {\mathrm e}^{-4}-30 x^{4} {\mathrm e}^{12}\right ) {\mathrm e}^{-12}}{x^{6}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((60*x^4*exp(4)^4+10800000000*x^2-32400000000)/x^7/exp(4)^4,x,method=_RETURNVERBOSE)

[Out]

exp(-16)*(-30*x^4*exp(16)-2700000000*x^2+5400000000)/x^6

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maxima [A]  time = 0.41, size = 20, normalized size = 0.91 \begin {gather*} -\frac {30 \, {\left (x^{4} e^{16} + 90000000 \, x^{2} - 180000000\right )} e^{\left (-16\right )}}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((60*x^4*exp(4)^4+10800000000*x^2-32400000000)/x^7/exp(4)^4,x, algorithm="maxima")

[Out]

-30*(x^4*e^16 + 90000000*x^2 - 180000000)*e^(-16)/x^6

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mupad [B]  time = 0.05, size = 21, normalized size = 0.95 \begin {gather*} -\frac {{\mathrm {e}}^{-16}\,\left (30\,{\mathrm {e}}^{16}\,x^4+2700000000\,x^2-5400000000\right )}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-16)*(60*x^4*exp(16) + 10800000000*x^2 - 32400000000))/x^7,x)

[Out]

-(exp(-16)*(30*x^4*exp(16) + 2700000000*x^2 - 5400000000))/x^6

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sympy [A]  time = 0.24, size = 20, normalized size = 0.91 \begin {gather*} \frac {- 30 x^{4} e^{16} - 2700000000 x^{2} + 5400000000}{x^{6} e^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((60*x**4*exp(4)**4+10800000000*x**2-32400000000)/x**7/exp(4)**4,x)

[Out]

(-30*x**4*exp(16) - 2700000000*x**2 + 5400000000)*exp(-16)/x**6

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