3.85.33 $\int \frac{e^{-3x}(36+72x-2e^{3x}{x}^2+(24+36x)\log (x))}{x^3}dx$

Optimal. Leaf size=22 $$4-\frac{12e^{-3x}(2+\log (x))}{x^2}-\log \left(x^2\right)$$

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Rubi [A]  time = 0.68, antiderivative size = 23, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, integrand size = 32, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.125, Rules used = {6741, 12, 6742, 2288} $$\begin{array}{c}-\frac{12e^{-3x}(2x+x\log (x))}{x^3}-2\log (x)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(36 + 72*x - 2*E^(3*x)*x^2 + (24 + 36*x)*Log[x])/(E^(3*x)*x^3),x]

[Out]

-2*Log[x] - (12*(2*x + x*Log[x]))/(E^(3*x)*x^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{2e^{-3x}(18+36x-e^{3x}{x}^2+12\log (x)+18x\log (x))}{x^3}dx\\ & =2\int \frac{e^{-3x}(18+36x-e^{3x}{x}^2+12\log (x)+18x\log (x))}{x^3}dx\\ & =2\int (-\frac{1}{x}+\frac{6e^{-3x}(3+6x+2\log (x)+3x\log (x))}{x^3})dx\\ & =-2\log (x)+12\int \frac{e^{-3x}(3+6x+2\log (x)+3x\log (x))}{x^3}dx\\ & =-2\log (x)-\frac{12e^{-3x}(2x+x\log (x))}{x^3}\end{array}\end{array}$$

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Mathematica [A]  time = 0.30, size = 19, normalized size = 0.86 $$\begin{array}{c}-2\log (x)-\frac{12e^{-3x}(2+\log (x))}{x^2}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(36 + 72*x - 2*E^(3*x)*x^2 + (24 + 36*x)*Log[x])/(E^(3*x)*x^3),x]

[Out]

-2*Log[x] - (12*(2 + Log[x]))/(E^(3*x)*x^2)

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fricas [A]  time = 0.80, size = 24, normalized size = 1.09 $$\begin{array}{c}-\frac{2((x^2{e}^{\left(3x\right)}+6)\log (x)+12)e^{(-3x)}}{x^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x+24)*log(x)-2*x^2*exp(x)^3+72*x+36)/x^3/exp(x)^3,x, algorithm="fricas")

[Out]

-2*((x^2*e^(3*x) + 6)*log(x) + 12)*e^(-3*x)/x^2

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giac [A]  time = 0.14, size = 26, normalized size = 1.18 $$\begin{array}{c}-\frac{2(x^2\log (x)+6e^{(-3x)}\log (x)+12e^{(-3x)})}{x^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x+24)*log(x)-2*x^2*exp(x)^3+72*x+36)/x^3/exp(x)^3,x, algorithm="giac")

[Out]

-2*(x^2*log(x) + 6*e^(-3*x)*log(x) + 12*e^(-3*x))/x^2

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maple [A]  time = 0.08, size = 34, normalized size = 1.55

method	result	size
risch	$-\frac{12{\mathrm{e}}^{-3x}\ln (x)}{x^2}-\frac{2(\ln (x)x^2{\mathrm{e}}^{3x}+12){\mathrm{e}}^{-3x}}{x^2}$	$34$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((36*x+24)*ln(x)-2*x^2*exp(x)^3+72*x+36)/x^3/exp(x)^3,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}-\frac{12e^{(-3x)}\log (x)}{x^2}-216\mathrm{\Gamma }(-1,3x)-324\mathrm{\Gamma }(-2,3x)+12\int \frac{e^{(-3x)}}{x^3}dx-2\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x+24)*log(x)-2*x^2*exp(x)^3+72*x+36)/x^3/exp(x)^3,x, algorithm="maxima")

[Out]

-12*e^(-3*x)*log(x)/x^2 - 216*gamma(-1, 3*x) - 324*gamma(-2, 3*x) + 12*integrate(e^(-3*x)/x^3, x) - 2*log(x)

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mupad [B]  time = 5.30, size = 20, normalized size = 0.91 $$\begin{array}{c}-2\ln (x)-\frac{2{\mathrm{e}}^{-3x}(6\ln (x)+12)}{x^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-3*x)*(72*x - 2*x^2*exp(3*x) + log(x)*(36*x + 24) + 36))/x^3,x)

[Out]

- 2*log(x) - (2*exp(-3*x)*(6*log(x) + 12))/x^2

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sympy [A]  time = 0.43, size = 20, normalized size = 0.91 $$\begin{array}{c}-2\log (x)+\frac{(-12\log (x)-24)e^{-3x}}{x^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x+24)*ln(x)-2*x**2*exp(x)**3+72*x+36)/x**3/exp(x)**3,x)

[Out]

-2*log(x) + (-12*log(x) - 24)*exp(-3*x)/x**2

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