3.31.4 \(\int \frac {3072+6 x^5+e^x (1024+3 x^5)}{3 x^5+e^x x^5} \, dx\)

Optimal. Leaf size=16 \[ -6-\frac {256}{x^4}+2 x+\log \left (3+e^x\right ) \]

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Rubi [A]  time = 0.10, antiderivative size = 15, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6742, 2282, 36, 29, 31, 14} \begin {gather*} -\frac {256}{x^4}+2 x+\log \left (e^x+3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3072 + 6*x^5 + E^x*(1024 + 3*x^5))/(3*x^5 + E^x*x^5),x]

[Out]

-256/x^4 + 2*x + Log[3 + E^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]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {3}{3+e^x}+\frac {1024+3 x^5}{x^5}\right ) \, dx\\ &=-\left (3 \int \frac {1}{3+e^x} \, dx\right )+\int \frac {1024+3 x^5}{x^5} \, dx\\ &=-\left (3 \operatorname {Subst}\left (\int \frac {1}{x (3+x)} \, dx,x,e^x\right )\right )+\int \left (3+\frac {1024}{x^5}\right ) \, dx\\ &=-\frac {256}{x^4}+3 x-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )+\operatorname {Subst}\left (\int \frac {1}{3+x} \, dx,x,e^x\right )\\ &=-\frac {256}{x^4}+2 x+\log \left (3+e^x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 15, normalized size = 0.94 \begin {gather*} -\frac {256}{x^4}+2 x+\log \left (3+e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3072 + 6*x^5 + E^x*(1024 + 3*x^5))/(3*x^5 + E^x*x^5),x]

[Out]

-256/x^4 + 2*x + Log[3 + E^x]

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fricas [A]  time = 0.62, size = 20, normalized size = 1.25 \begin {gather*} \frac {2 \, x^{5} + x^{4} \log \left (e^{x} + 3\right ) - 256}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^5+1024)*exp(x)+6*x^5+3072)/(x^5*exp(x)+3*x^5),x, algorithm="fricas")

[Out]

(2*x^5 + x^4*log(e^x + 3) - 256)/x^4

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giac [A]  time = 0.35, size = 20, normalized size = 1.25 \begin {gather*} \frac {2 \, x^{5} + x^{4} \log \left (e^{x} + 3\right ) - 256}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^5+1024)*exp(x)+6*x^5+3072)/(x^5*exp(x)+3*x^5),x, algorithm="giac")

[Out]

(2*x^5 + x^4*log(e^x + 3) - 256)/x^4

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maple [A]  time = 0.05, size = 15, normalized size = 0.94

method	result	size
risch	\(2 x -\frac {256}{x^{4}}+\ln \left (3+{\mathrm e}^{x}\right )\)	\(15\)
norman	\(\frac {2 x^{5}-256}{x^{4}}+\ln \left (3+{\mathrm e}^{x}\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^5+1024)*exp(x)+6*x^5+3072)/(x^5*exp(x)+3*x^5),x,method=_RETURNVERBOSE)

[Out]

2*x-256/x^4+ln(3+exp(x))

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maxima [A]  time = 0.42, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (x^{5} - 128\right )}}{x^{4}} + \log \left (e^{x} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^5+1024)*exp(x)+6*x^5+3072)/(x^5*exp(x)+3*x^5),x, algorithm="maxima")

[Out]

2*(x^5 - 128)/x^4 + log(e^x + 3)

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mupad [B]  time = 1.74, size = 17, normalized size = 1.06 \begin {gather*} \ln \left ({\mathrm {e}}^x+3\right )+\frac {2\,x^5-256}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(3*x^5 + 1024) + 6*x^5 + 3072)/(x^5*exp(x) + 3*x^5),x)

[Out]

log(exp(x) + 3) + (2*x^5 - 256)/x^4

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sympy [A]  time = 0.11, size = 14, normalized size = 0.88 \begin {gather*} 2 x + \log {\left (e^{x} + 3 \right )} - \frac {256}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**5+1024)*exp(x)+6*x**5+3072)/(x**5*exp(x)+3*x**5),x)

[Out]

2*x + log(exp(x) + 3) - 256/x**4

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