3.90.15 \(\int \frac {-5-48 e^{3 x} x^2-12 e^{6 x} x^2}{2 x^2} \, dx\)

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {12, 14, 2194} \begin {gather*} -8 e^{3 x}-e^{6 x}+\frac {5}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 - 48*E^(3*x)*x^2 - 12*E^(6*x)*x^2)/(2*x^2),x]

[Out]

-8*E^(3*x) - E^(6*x) + 5/(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 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}{2} \int \frac {-5-48 e^{3 x} x^2-12 e^{6 x} x^2}{x^2} \, dx\\ &=\frac {1}{2} \int \left (-48 e^{3 x}-12 e^{6 x}-\frac {5}{x^2}\right ) \, dx\\ &=\frac {5}{2 x}-6 \int e^{6 x} \, dx-24 \int e^{3 x} \, dx\\ &=-8 e^{3 x}-e^{6 x}+\frac {5}{2 x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} -8 e^{3 x}-e^{6 x}+\frac {5}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 - 48*E^(3*x)*x^2 - 12*E^(6*x)*x^2)/(2*x^2),x]

[Out]

-8*E^(3*x) - E^(6*x) + 5/(2*x)

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 21, normalized size = 0.95 \begin {gather*} -\frac {2 \, x e^{\left (6 \, x\right )} + 16 \, x e^{\left (3 \, x\right )} - 5}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-12*x^2*exp(3*x)^2-48*x^2*exp(3*x)-5)/x^2,x, algorithm="fricas")

[Out]

-1/2*(2*x*e^(6*x) + 16*x*e^(3*x) - 5)/x

________________________________________________________________________________________

giac [A]  time = 0.11, size = 21, normalized size = 0.95 \begin {gather*} -\frac {2 \, x e^{\left (6 \, x\right )} + 16 \, x e^{\left (3 \, x\right )} - 5}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-12*x^2*exp(3*x)^2-48*x^2*exp(3*x)-5)/x^2,x, algorithm="giac")

[Out]

-1/2*(2*x*e^(6*x) + 16*x*e^(3*x) - 5)/x

________________________________________________________________________________________

maple [A]  time = 0.05, size = 19, normalized size = 0.86

method	result	size
risch	\(\frac {5}{2 x}-{\mathrm e}^{6 x}-8 \,{\mathrm e}^{3 x}\)	\(19\)
derivativedivides	\(\frac {5}{2 x}-{\mathrm e}^{6 x}-8 \,{\mathrm e}^{3 x}\)	\(21\)
default	\(\frac {5}{2 x}-{\mathrm e}^{6 x}-8 \,{\mathrm e}^{3 x}\)	\(21\)
norman	\(\frac {\frac {5}{2}-8 x \,{\mathrm e}^{3 x}-x \,{\mathrm e}^{6 x}}{x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(-12*x^2*exp(3*x)^2-48*x^2*exp(3*x)-5)/x^2,x,method=_RETURNVERBOSE)

[Out]

5/2/x-exp(6*x)-8*exp(3*x)

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 18, normalized size = 0.82 \begin {gather*} \frac {5}{2 \, x} - e^{\left (6 \, x\right )} - 8 \, e^{\left (3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-12*x^2*exp(3*x)^2-48*x^2*exp(3*x)-5)/x^2,x, algorithm="maxima")

[Out]

5/2/x - e^(6*x) - 8*e^(3*x)

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 18, normalized size = 0.82 \begin {gather*} \frac {5}{2\,x}-{\mathrm {e}}^{6\,x}-8\,{\mathrm {e}}^{3\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(24*x^2*exp(3*x) + 6*x^2*exp(6*x) + 5/2)/x^2,x)

[Out]

5/(2*x) - exp(6*x) - 8*exp(3*x)

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 15, normalized size = 0.68 \begin {gather*} - e^{6 x} - 8 e^{3 x} + \frac {5}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-12*x**2*exp(3*x)**2-48*x**2*exp(3*x)-5)/x**2,x)

[Out]

-exp(6*x) - 8*exp(3*x) + 5/(2*x)

________________________________________________________________________________________