3.66.73 \(\int \frac {e^{16 x} (6 x+48 x^2)}{-2+3 e^{16 x} x^2} \, dx\)

Optimal. Leaf size=14 \[ \log \left (-\frac {2}{3}+e^{16 x} x^2\right ) \]

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Rubi [A]  time = 0.11, antiderivative size = 13, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1593, 6684} \begin {gather*} \log \left (2-3 e^{16 x} x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(16*x)*(6*x + 48*x^2))/(-2 + 3*E^(16*x)*x^2),x]

[Out]

Log[2 - 3*E^(16*x)*x^2]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{16 x} x (6+48 x)}{-2+3 e^{16 x} x^2} \, dx\\ &=\log \left (2-3 e^{16 x} x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 13, normalized size = 0.93 \begin {gather*} \log \left (-2+3 e^{16 x} x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(16*x)*(6*x + 48*x^2))/(-2 + 3*E^(16*x)*x^2),x]

[Out]

Log[-2 + 3*E^(16*x)*x^2]

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fricas [A]  time = 1.13, size = 21, normalized size = 1.50 \begin {gather*} 2 \, \log \relax (x) + \log \left (\frac {3 \, x^{2} e^{\left (16 \, x\right )} - 2}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x^2+6*x)*exp(x)^16/(3*x^2*exp(x)^16-2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.23, size = 12, normalized size = 0.86 \begin {gather*} \log \left (3 \, x^{2} e^{\left (16 \, x\right )} - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x^2+6*x)*exp(x)^16/(3*x^2*exp(x)^16-2),x, algorithm="giac")

[Out]

log(3*x^2*e^(16*x) - 2)

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maple [A]  time = 0.03, size = 17, normalized size = 1.21

method	result	size
risch	\(2 \ln \relax (x )+\ln \left ({\mathrm e}^{16 x}-\frac {2}{3 x^{2}}\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((48*x^2+6*x)*exp(x)^16/(3*x^2*exp(x)^16-2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.41, size = 22, normalized size = 1.57 \begin {gather*} 2 \, \log \relax (x) + \log \left (\frac {3 \, x^{2} e^{\left (16 \, x\right )} - 2}{3 \, x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x^2+6*x)*exp(x)^16/(3*x^2*exp(x)^16-2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.17, size = 12, normalized size = 0.86 \begin {gather*} \ln \left (3\,x^2\,{\mathrm {e}}^{16\,x}-2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(16*x)*(6*x + 48*x^2))/(3*x^2*exp(16*x) - 2),x)

[Out]

log(3*x^2*exp(16*x) - 2)

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sympy [A]  time = 0.16, size = 17, normalized size = 1.21 \begin {gather*} 2 \log {\relax (x )} + \log {\left (e^{16 x} - \frac {2}{3 x^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x**2+6*x)*exp(x)**16/(3*x**2*exp(x)**16-2),x)

[Out]

2*log(x) + log(exp(16*x) - 2/(3*x**2))

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