3.22.51 \(\int \frac {8+5 x^3-24 x^4-12 x^5+e^x (-8 x^4-4 x^5)}{x^3} \, dx\)

Optimal. Leaf size=22 \[ 5+x-4 \left (\frac {1}{x^2}-x+x^2 \left (3+e^x+x\right )\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 27, normalized size of antiderivative = 1.23, number of steps used = 11, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {14, 2196, 2176, 2194} \begin {gather*} -4 x^3-4 e^x x^2-12 x^2-\frac {4}{x^2}+5 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8 + 5*x^3 - 24*x^4 - 12*x^5 + E^x*(-8*x^4 - 4*x^5))/x^3,x]

[Out]

-4/x^2 + 5*x - 12*x^2 - 4*E^x*x^2 - 4*x^3

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]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-4 e^x x (2+x)+\frac {8+5 x^3-24 x^4-12 x^5}{x^3}\right ) \, dx\\ &=-\left (4 \int e^x x (2+x) \, dx\right )+\int \frac {8+5 x^3-24 x^4-12 x^5}{x^3} \, dx\\ &=-\left (4 \int \left (2 e^x x+e^x x^2\right ) \, dx\right )+\int \left (5+\frac {8}{x^3}-24 x-12 x^2\right ) \, dx\\ &=-\frac {4}{x^2}+5 x-12 x^2-4 x^3-4 \int e^x x^2 \, dx-8 \int e^x x \, dx\\ &=-\frac {4}{x^2}+5 x-8 e^x x-12 x^2-4 e^x x^2-4 x^3+8 \int e^x \, dx+8 \int e^x x \, dx\\ &=8 e^x-\frac {4}{x^2}+5 x-12 x^2-4 e^x x^2-4 x^3-8 \int e^x \, dx\\ &=-\frac {4}{x^2}+5 x-12 x^2-4 e^x x^2-4 x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 24, normalized size = 1.09 \begin {gather*} -\frac {4}{x^2}+5 x-4 \left (3+e^x\right ) x^2-4 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 + 5*x^3 - 24*x^4 - 12*x^5 + E^x*(-8*x^4 - 4*x^5))/x^3,x]

[Out]

-4/x^2 + 5*x - 4*(3 + E^x)*x^2 - 4*x^3

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fricas [A]  time = 0.70, size = 29, normalized size = 1.32 \begin {gather*} -\frac {4 \, x^{5} + 4 \, x^{4} e^{x} + 12 \, x^{4} - 5 \, x^{3} + 4}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^5-8*x^4)*exp(x)-12*x^5-24*x^4+5*x^3+8)/x^3,x, algorithm="fricas")

[Out]

-(4*x^5 + 4*x^4*e^x + 12*x^4 - 5*x^3 + 4)/x^2

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giac [A]  time = 0.19, size = 29, normalized size = 1.32 \begin {gather*} -\frac {4 \, x^{5} + 4 \, x^{4} e^{x} + 12 \, x^{4} - 5 \, x^{3} + 4}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^5-8*x^4)*exp(x)-12*x^5-24*x^4+5*x^3+8)/x^3,x, algorithm="giac")

[Out]

-(4*x^5 + 4*x^4*e^x + 12*x^4 - 5*x^3 + 4)/x^2

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maple [A]  time = 0.03, size = 27, normalized size = 1.23

method	result	size
default	\(-12 x^{2}+5 x -\frac {4}{x^{2}}-4 x^{3}-4 \,{\mathrm e}^{x} x^{2}\)	\(27\)
risch	\(-12 x^{2}+5 x -\frac {4}{x^{2}}-4 x^{3}-4 \,{\mathrm e}^{x} x^{2}\)	\(27\)
norman	\(\frac {-4+5 x^{3}-12 x^{4}-4 x^{5}-4 \,{\mathrm e}^{x} x^{4}}{x^{2}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^5-8*x^4)*exp(x)-12*x^5-24*x^4+5*x^3+8)/x^3,x,method=_RETURNVERBOSE)

[Out]

-12*x^2+5*x-4/x^2-4*x^3-4*exp(x)*x^2

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maxima [A]  time = 0.36, size = 38, normalized size = 1.73 \begin {gather*} -4 \, x^{3} - 12 \, x^{2} - 4 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} - 8 \, {\left (x - 1\right )} e^{x} + 5 \, x - \frac {4}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^5-8*x^4)*exp(x)-12*x^5-24*x^4+5*x^3+8)/x^3,x, algorithm="maxima")

[Out]

-4*x^3 - 12*x^2 - 4*(x^2 - 2*x + 2)*e^x - 8*(x - 1)*e^x + 5*x - 4/x^2

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mupad [B]  time = 1.18, size = 25, normalized size = 1.14 \begin {gather*} 5\,x-x^2\,\left (4\,{\mathrm {e}}^x+12\right )-\frac {4}{x^2}-4\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(8*x^4 + 4*x^5) - 5*x^3 + 24*x^4 + 12*x^5 - 8)/x^3,x)

[Out]

5*x - x^2*(4*exp(x) + 12) - 4/x^2 - 4*x^3

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sympy [A]  time = 0.11, size = 26, normalized size = 1.18 \begin {gather*} - 4 x^{3} - 4 x^{2} e^{x} - 12 x^{2} + 5 x - \frac {4}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**5-8*x**4)*exp(x)-12*x**5-24*x**4+5*x**3+8)/x**3,x)

[Out]

-4*x**3 - 4*x**2*exp(x) - 12*x**2 + 5*x - 4/x**2

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