3.78.30 \(\int \frac {20 e^{4/x}-30 x^4-4 x^5}{x^2} \, dx\)

Optimal. Leaf size=18 \[ -5 e^{4/x}-x^3 (10+x) \]

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Rubi [A]  time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {14, 2209, 43} \begin {gather*} -x^4-10 x^3-5 e^{4/x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(20*E^(4/x) - 30*x^4 - 4*x^5)/x^2,x]

[Out]

-5*E^(4/x) - 10*x^3 - x^4

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {20 e^{4/x}}{x^2}-2 x^2 (15+2 x)\right ) \, dx\\ &=-\left (2 \int x^2 (15+2 x) \, dx\right )+20 \int \frac {e^{4/x}}{x^2} \, dx\\ &=-5 e^{4/x}-2 \int \left (15 x^2+2 x^3\right ) \, dx\\ &=-5 e^{4/x}-10 x^3-x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.11 \begin {gather*} -5 e^{4/x}-10 x^3-x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20*E^(4/x) - 30*x^4 - 4*x^5)/x^2,x]

[Out]

-5*E^(4/x) - 10*x^3 - x^4

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fricas [A]  time = 0.66, size = 19, normalized size = 1.06 \begin {gather*} -x^{4} - 10 \, x^{3} - 5 \, e^{\frac {4}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*exp(4/x)-4*x^5-30*x^4)/x^2,x, algorithm="fricas")

[Out]

-x^4 - 10*x^3 - 5*e^(4/x)

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giac [A]  time = 0.19, size = 23, normalized size = 1.28 \begin {gather*} -x^{4} {\left (\frac {10}{x} + \frac {5 \, e^{\frac {4}{x}}}{x^{4}} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*exp(4/x)-4*x^5-30*x^4)/x^2,x, algorithm="giac")

[Out]

-x^4*(10/x + 5*e^(4/x)/x^4 + 1)

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maple [A]  time = 0.03, size = 20, normalized size = 1.11

method	result	size
derivativedivides	\(-x^{4}-10 x^{3}-5 \,{\mathrm e}^{\frac {4}{x}}\)	\(20\)
default	\(-x^{4}-10 x^{3}-5 \,{\mathrm e}^{\frac {4}{x}}\)	\(20\)
risch	\(-x^{4}-10 x^{3}-5 \,{\mathrm e}^{\frac {4}{x}}\)	\(20\)
norman	\(\frac {-10 x^{4}-x^{5}-5 x \,{\mathrm e}^{\frac {4}{x}}}{x}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*exp(4/x)-4*x^5-30*x^4)/x^2,x,method=_RETURNVERBOSE)

[Out]

-x^4-10*x^3-5*exp(4/x)

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maxima [A]  time = 0.36, size = 19, normalized size = 1.06 \begin {gather*} -x^{4} - 10 \, x^{3} - 5 \, e^{\frac {4}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*exp(4/x)-4*x^5-30*x^4)/x^2,x, algorithm="maxima")

[Out]

-x^4 - 10*x^3 - 5*e^(4/x)

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mupad [B]  time = 5.14, size = 19, normalized size = 1.06 \begin {gather*} -5\,{\mathrm {e}}^{4/x}-10\,x^3-x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(30*x^4 - 20*exp(4/x) + 4*x^5)/x^2,x)

[Out]

- 5*exp(4/x) - 10*x^3 - x^4

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sympy [A]  time = 0.12, size = 15, normalized size = 0.83 \begin {gather*} - x^{4} - 10 x^{3} - 5 e^{\frac {4}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*exp(4/x)-4*x**5-30*x**4)/x**2,x)

[Out]

-x**4 - 10*x**3 - 5*exp(4/x)

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