3.32.94 \(\int \frac {256 e^8+48 x^5-72 x^6+e^4 (-128 x^2+96 x^3)}{9 x^5} \, dx\)

Optimal. Leaf size=23 \[ 1-4 \left (-2+x+\frac {4 \left (e^4+x^2\right )}{3 x^2}\right )^2 \]

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Rubi [A]  time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.78, number of steps used = 3, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {12, 14} \begin {gather*} -\frac {64 e^8}{9 x^4}-4 x^2+\frac {64 e^4}{9 x^2}+\frac {16 x}{3}-\frac {32 e^4}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(256*E^8 + 48*x^5 - 72*x^6 + E^4*(-128*x^2 + 96*x^3))/(9*x^5),x]

[Out]

(-64*E^8)/(9*x^4) + (64*E^4)/(9*x^2) - (32*E^4)/(3*x) + (16*x)/3 - 4*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 {1}{9} \int \frac {256 e^8+48 x^5-72 x^6+e^4 \left (-128 x^2+96 x^3\right )}{x^5} \, dx\\ &=\frac {1}{9} \int \left (48+\frac {256 e^8}{x^5}-\frac {128 e^4}{x^3}+\frac {96 e^4}{x^2}-72 x\right ) \, dx\\ &=-\frac {64 e^8}{9 x^4}+\frac {64 e^4}{9 x^2}-\frac {32 e^4}{3 x}+\frac {16 x}{3}-4 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 39, normalized size = 1.70 \begin {gather*} \frac {8}{9} \left (-\frac {8 e^8}{x^4}+\frac {8 e^4}{x^2}-\frac {12 e^4}{x}+6 x-\frac {9 x^2}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(256*E^8 + 48*x^5 - 72*x^6 + E^4*(-128*x^2 + 96*x^3))/(9*x^5),x]

[Out]

(8*((-8*E^8)/x^4 + (8*E^4)/x^2 - (12*E^4)/x + 6*x - (9*x^2)/2))/9

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fricas [A]  time = 0.56, size = 35, normalized size = 1.52 \begin {gather*} -\frac {4 \, {\left (9 \, x^{6} - 12 \, x^{5} + 8 \, {\left (3 \, x^{3} - 2 \, x^{2}\right )} e^{4} + 16 \, e^{8}\right )}}{9 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(256*exp(4)^2+(96*x^3-128*x^2)*exp(4)-72*x^6+48*x^5)/x^5,x, algorithm="fricas")

[Out]

-4/9*(9*x^6 - 12*x^5 + 8*(3*x^3 - 2*x^2)*e^4 + 16*e^8)/x^4

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giac [A]  time = 0.14, size = 33, normalized size = 1.43 \begin {gather*} -4 \, x^{2} + \frac {16}{3} \, x - \frac {32 \, {\left (3 \, x^{3} e^{4} - 2 \, x^{2} e^{4} + 2 \, e^{8}\right )}}{9 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(256*exp(4)^2+(96*x^3-128*x^2)*exp(4)-72*x^6+48*x^5)/x^5,x, algorithm="giac")

[Out]

-4*x^2 + 16/3*x - 32/9*(3*x^3*e^4 - 2*x^2*e^4 + 2*e^8)/x^4

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maple [A]  time = 0.04, size = 31, normalized size = 1.35

method	result	size
default	\(\frac {16 x}{3}-4 x^{2}-\frac {32 \,{\mathrm e}^{4}}{3 x}+\frac {64 \,{\mathrm e}^{4}}{9 x^{2}}-\frac {64 \,{\mathrm e}^{8}}{9 x^{4}}\)	\(31\)
risch	\(-4 x^{2}+\frac {16 x}{3}+\frac {-96 x^{3} {\mathrm e}^{4}+64 x^{2} {\mathrm e}^{4}-64 \,{\mathrm e}^{8}}{9 x^{4}}\)	\(34\)
norman	\(\frac {\frac {16 x^{5}}{3}-4 x^{6}-\frac {64 \,{\mathrm e}^{8}}{9}+\frac {64 x^{2} {\mathrm e}^{4}}{9}-\frac {32 x^{3} {\mathrm e}^{4}}{3}}{x^{4}}\)	\(36\)
gosper	\(-\frac {4 \left (9 x^{6}-12 x^{5}+24 x^{3} {\mathrm e}^{4}-16 x^{2} {\mathrm e}^{4}+16 \,{\mathrm e}^{8}\right )}{9 x^{4}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*(256*exp(4)^2+(96*x^3-128*x^2)*exp(4)-72*x^6+48*x^5)/x^5,x,method=_RETURNVERBOSE)

[Out]

16/3*x-4*x^2-32/3*exp(4)/x+64/9*exp(4)/x^2-64/9*exp(8)/x^4

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maxima [A]  time = 0.42, size = 33, normalized size = 1.43 \begin {gather*} -4 \, x^{2} + \frac {16}{3} \, x - \frac {32 \, {\left (3 \, x^{3} e^{4} - 2 \, x^{2} e^{4} + 2 \, e^{8}\right )}}{9 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(256*exp(4)^2+(96*x^3-128*x^2)*exp(4)-72*x^6+48*x^5)/x^5,x, algorithm="maxima")

[Out]

-4*x^2 + 16/3*x - 32/9*(3*x^3*e^4 - 2*x^2*e^4 + 2*e^8)/x^4

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mupad [B]  time = 1.89, size = 34, normalized size = 1.48 \begin {gather*} -\frac {4\,\left (9\,x^6-12\,x^5+24\,{\mathrm {e}}^4\,x^3-16\,{\mathrm {e}}^4\,x^2+16\,{\mathrm {e}}^8\right )}{9\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((256*exp(8))/9 - (exp(4)*(128*x^2 - 96*x^3))/9 + (16*x^5)/3 - 8*x^6)/x^5,x)

[Out]

-(4*(16*exp(8) - 16*x^2*exp(4) + 24*x^3*exp(4) - 12*x^5 + 9*x^6))/(9*x^4)

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sympy [A]  time = 0.23, size = 36, normalized size = 1.57 \begin {gather*} - 4 x^{2} + \frac {16 x}{3} - \frac {96 x^{3} e^{4} - 64 x^{2} e^{4} + 64 e^{8}}{9 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(256*exp(4)**2+(96*x**3-128*x**2)*exp(4)-72*x**6+48*x**5)/x**5,x)

[Out]

-4*x**2 + 16*x/3 - (96*x**3*exp(4) - 64*x**2*exp(4) + 64*exp(8))/(9*x**4)

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