3.99.99 \(\int \frac {-1+16 x+8 e^3 x+e^x x-48 x^2}{x} \, dx\)

Optimal. Leaf size=27 \[ e^x+8 x \left (e^3+x+x \left (-5+\frac {2+x}{x}\right )\right )-\log (x) \]

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Rubi [A]  time = 0.02, antiderivative size = 21, normalized size of antiderivative = 0.78, number of steps used = 6, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6, 14, 2194} \begin {gather*} -24 x^2+8 \left (2+e^3\right ) x+e^x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 16*x + 8*E^3*x + E^x*x - 48*x^2)/x,x]

[Out]

E^x + 8*(2 + E^3)*x - 24*x^2 - Log[x]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 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} &=\int \frac {-1+e^x x+\left (16+8 e^3\right ) x-48 x^2}{x} \, dx\\ &=\int \left (e^x+\frac {-1+8 \left (2+e^3\right ) x-48 x^2}{x}\right ) \, dx\\ &=\int e^x \, dx+\int \frac {-1+8 \left (2+e^3\right ) x-48 x^2}{x} \, dx\\ &=e^x+\int \left (8 \left (2+e^3\right )-\frac {1}{x}-48 x\right ) \, dx\\ &=e^x+8 \left (2+e^3\right ) x-24 x^2-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 0.81 \begin {gather*} e^x+16 x+8 e^3 x-24 x^2-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 16*x + 8*E^3*x + E^x*x - 48*x^2)/x,x]

[Out]

E^x + 16*x + 8*E^3*x - 24*x^2 - Log[x]

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fricas [A]  time = 0.98, size = 20, normalized size = 0.74 \begin {gather*} -24 \, x^{2} + 8 \, x e^{3} + 16 \, x + e^{x} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x+8*x*exp(3)-48*x^2+16*x-1)/x,x, algorithm="fricas")

[Out]

-24*x^2 + 8*x*e^3 + 16*x + e^x - log(x)

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giac [A]  time = 0.14, size = 20, normalized size = 0.74 \begin {gather*} -24 \, x^{2} + 8 \, x e^{3} + 16 \, x + e^{x} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x+8*x*exp(3)-48*x^2+16*x-1)/x,x, algorithm="giac")

[Out]

-24*x^2 + 8*x*e^3 + 16*x + e^x - log(x)

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maple [A]  time = 0.03, size = 21, normalized size = 0.78

method	result	size
default	\(-24 x^{2}+16 x -\ln \relax (x )+8 x \,{\mathrm e}^{3}+{\mathrm e}^{x}\)	\(21\)
norman	\(\left (8 \,{\mathrm e}^{3}+16\right ) x -24 x^{2}+{\mathrm e}^{x}-\ln \relax (x )\)	\(21\)
risch	\(-24 x^{2}+16 x -\ln \relax (x )+8 x \,{\mathrm e}^{3}+{\mathrm e}^{x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*x+8*x*exp(3)-48*x^2+16*x-1)/x,x,method=_RETURNVERBOSE)

[Out]

-24*x^2+16*x-ln(x)+8*x*exp(3)+exp(x)

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maxima [A]  time = 0.36, size = 20, normalized size = 0.74 \begin {gather*} -24 \, x^{2} + 8 \, x e^{3} + 16 \, x + e^{x} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x+8*x*exp(3)-48*x^2+16*x-1)/x,x, algorithm="maxima")

[Out]

-24*x^2 + 8*x*e^3 + 16*x + e^x - log(x)

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mupad [B]  time = 0.06, size = 20, normalized size = 0.74 \begin {gather*} {\mathrm {e}}^x-\ln \relax (x)-24\,x^2+x\,\left (8\,{\mathrm {e}}^3+16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x + 8*x*exp(3) + x*exp(x) - 48*x^2 - 1)/x,x)

[Out]

exp(x) - log(x) - 24*x^2 + x*(8*exp(3) + 16)

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sympy [A]  time = 0.13, size = 20, normalized size = 0.74 \begin {gather*} - 24 x^{2} - x \left (- 8 e^{3} - 16\right ) + e^{x} - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x+8*x*exp(3)-48*x**2+16*x-1)/x,x)

[Out]

-24*x**2 - x*(-8*exp(3) - 16) + exp(x) - log(x)

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