3.58.55 \(\int \frac {3+e^x (8 x^3+4 x^4)}{4 x^3} \, dx\)

Optimal. Leaf size=17 \[ 5+e^x-\frac {3}{8 x^2}+e^x x \]

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 14, 2176, 2194} \begin {gather*} -\frac {3}{8 x^2}+e^x (x+2)-e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + E^x*(8*x^3 + 4*x^4))/(4*x^3),x]

[Out]

-E^x - 3/(8*x^2) + E^x*(2 + x)

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]]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {3+e^x \left (8 x^3+4 x^4\right )}{x^3} \, dx\\ &=\frac {1}{4} \int \left (\frac {3}{x^3}+4 e^x (2+x)\right ) \, dx\\ &=-\frac {3}{8 x^2}+\int e^x (2+x) \, dx\\ &=-\frac {3}{8 x^2}+e^x (2+x)-\int e^x \, dx\\ &=-e^x-\frac {3}{8 x^2}+e^x (2+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 15, normalized size = 0.88 \begin {gather*} -\frac {3}{8 x^2}+e^x (1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + E^x*(8*x^3 + 4*x^4))/(4*x^3),x]

[Out]

-3/(8*x^2) + E^x*(1 + x)

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fricas [A]  time = 0.82, size = 18, normalized size = 1.06 \begin {gather*} \frac {8 \, {\left (x^{3} + x^{2}\right )} e^{x} - 3}{8 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((4*x^4+8*x^3)*exp(x)+3)/x^3,x, algorithm="fricas")

[Out]

1/8*(8*(x^3 + x^2)*e^x - 3)/x^2

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giac [A]  time = 0.14, size = 21, normalized size = 1.24 \begin {gather*} \frac {8 \, x^{3} e^{x} + 8 \, x^{2} e^{x} - 3}{8 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((4*x^4+8*x^3)*exp(x)+3)/x^3,x, algorithm="giac")

[Out]

1/8*(8*x^3*e^x + 8*x^2*e^x - 3)/x^2

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maple [A]  time = 0.02, size = 13, normalized size = 0.76

method	result	size
default	\(-\frac {3}{8 x^{2}}+{\mathrm e}^{x} x +{\mathrm e}^{x}\)	\(13\)
risch	\(-\frac {3}{8 x^{2}}+\frac {\left (4 x +4\right ) {\mathrm e}^{x}}{4}\)	\(16\)
norman	\(\frac {-\frac {3}{8}+{\mathrm e}^{x} x^{2}+{\mathrm e}^{x} x^{3}}{x^{2}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*((4*x^4+8*x^3)*exp(x)+3)/x^3,x,method=_RETURNVERBOSE)

[Out]

-3/8/x^2+exp(x)*x+exp(x)

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maxima [A]  time = 0.36, size = 16, normalized size = 0.94 \begin {gather*} {\left (x - 1\right )} e^{x} - \frac {3}{8 \, x^{2}} + 2 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((4*x^4+8*x^3)*exp(x)+3)/x^3,x, algorithm="maxima")

[Out]

(x - 1)*e^x - 3/8/x^2 + 2*e^x

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mupad [B]  time = 0.06, size = 12, normalized size = 0.71 \begin {gather*} {\mathrm {e}}^x+x\,{\mathrm {e}}^x-\frac {3}{8\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(x)*(8*x^3 + 4*x^4))/4 + 3/4)/x^3,x)

[Out]

exp(x) + x*exp(x) - 3/(8*x^2)

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sympy [A]  time = 0.09, size = 12, normalized size = 0.71 \begin {gather*} \left (x + 1\right ) e^{x} - \frac {3}{8 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((4*x**4+8*x**3)*exp(x)+3)/x**3,x)

[Out]

(x + 1)*exp(x) - 3/(8*x**2)

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