3.89.32 $\int \frac{-e^{3}x+2x^2-12\log (5)-72\log (x)}{x}dx$

Optimal. Leaf size=24 $$\frac{23}{3}-e^{3}x+x^2-(\log (5)+6\log (x){)}^2$$

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 22, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 2, integrand size = 24, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.083, Rules used = {14, 2301} $$\begin{array}{c}{x}^2-e^{3}x-36{\log }^2(x)-12\log (5)\log (x)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-(E^3*x) + 2*x^2 - 12*Log[5] - 72*Log[x])/x,x]

[Out]

-(E^3*x) + x^2 - 12*Log[5]*Log[x] - 36*Log[x]^2

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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (\frac{-e^{3}x+2x^2-12\log (5)}{x}-\frac{72\log (x)}{x})dx\\ & =-(72\int \frac{\log (x)}{x}dx)+\int \frac{-e^{3}x+2x^2-12\log (5)}{x}dx\\ & =-36{\log }^2(x)+\int (-e^3+2x-\frac{12\log (5)}{x})dx\\ & =-e^{3}x+x^2-12\log (5)\log (x)-36{\log }^2(x)\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 22, normalized size = 0.92 $$\begin{array}{c}-e^{3}x+x^2-12\log (5)\log (x)-36{\log }^2(x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-(E^3*x) + 2*x^2 - 12*Log[5] - 72*Log[x])/x,x]

[Out]

-(E^3*x) + x^2 - 12*Log[5]*Log[x] - 36*Log[x]^2

________________________________________________________________________________________

fricas [A]  time = 0.53, size = 21, normalized size = 0.88 $$\begin{array}{c}{x}^2-x{e}^3-12\log (5)\log (x)-36\log (x{)}^2\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-72*log(x)-12*log(5)-x*exp(3)+2*x^2)/x,x, algorithm="fricas")

[Out]

x^2 - x*e^3 - 12*log(5)*log(x) - 36*log(x)^2

________________________________________________________________________________________

giac [A]  time = 0.19, size = 21, normalized size = 0.88 $$\begin{array}{c}{x}^2-x{e}^3-12\log (5)\log (x)-36\log (x{)}^2\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-72*log(x)-12*log(5)-x*exp(3)+2*x^2)/x,x, algorithm="giac")

[Out]

x^2 - x*e^3 - 12*log(5)*log(x) - 36*log(x)^2

________________________________________________________________________________________

maple [A]  time = 0.04, size = 22, normalized size = 0.92

method	result	size
default	$-x{\mathrm{e}}^3+x^2-36\ln (x{)}^2-12\ln (5)\ln (x)$	$22$
norman	$-x{\mathrm{e}}^3+x^2-36\ln (x{)}^2-12\ln (5)\ln (x)$	$22$
risch	$-x{\mathrm{e}}^3+x^2-36\ln (x{)}^2-12\ln (5)\ln (x)$	$22$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-72*ln(x)-12*ln(5)-x*exp(3)+2*x^2)/x,x,method=_RETURNVERBOSE)

[Out]

-x*exp(3)+x^2-36*ln(x)^2-12*ln(5)*ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 21, normalized size = 0.88 $$\begin{array}{c}{x}^2-x{e}^3-12\log (5)\log (x)-36\log (x{)}^2\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-72*log(x)-12*log(5)-x*exp(3)+2*x^2)/x,x, algorithm="maxima")

[Out]

x^2 - x*e^3 - 12*log(5)*log(x) - 36*log(x)^2

________________________________________________________________________________________

mupad [B]  time = 5.30, size = 21, normalized size = 0.88 $$\begin{array}{c}{x}^2-{\mathrm{e}}^{3}x-36{\ln (x)}^2-12\ln (5)\ln (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(12*log(5) + 72*log(x) + x*exp(3) - 2*x^2)/x,x)

[Out]

x^2 - 36*log(x)^2 - 12*log(5)*log(x) - x*exp(3)

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 22, normalized size = 0.92 $$\begin{array}{c}{x}^2-x{e}^3-36\log {(x)}^2-12\log (5)\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-72*ln(x)-12*ln(5)-x*exp(3)+2*x**2)/x,x)

[Out]

x**2 - x*exp(3) - 36*log(x)**2 - 12*log(5)*log(x)

________________________________________________________________________________________