3.33.19 $\int \frac{7e^4+e^{2x}(6+12x)+e^{2+x}(12+12x)}{6e^{2x}x+12e^{2+x}x+e^4(22+7x)}dx$

Optimal. Leaf size=23 $$\log (-2+x+6(4+\frac{{(x+e^{-2+x}x)}^2}{x}))$$

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Rubi [A]  time = 0.10, antiderivative size = 27, normalized size of antiderivative = 1.17, number of steps used = 1, number of rules used = 1, integrand size = 57, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.018, Rules used = {6684} $$\begin{array}{c}\log (6e^{2x}x+12e^{x+2}x+e^4(7x+22))\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(7*E^4 + E^(2*x)*(6 + 12*x) + E^(2 + x)*(12 + 12*x))/(6*E^(2*x)*x + 12*E^(2 + x)*x + E^4*(22 + 7*x)),x]

[Out]

Log[6*E^(2*x)*x + 12*E^(2 + x)*x + E^4*(22 + 7*x)]

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\log (6e^{2x}x+12e^{2+x}x+e^4(22+7x))\end{array}\end{array}$$

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Mathematica [A]  time = 0.41, size = 29, normalized size = 1.26 $$\begin{array}{c}\log (22e^4+7e^{4}x+6e^{2x}x+12e^{2+x}x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(7*E^4 + E^(2*x)*(6 + 12*x) + E^(2 + x)*(12 + 12*x))/(6*E^(2*x)*x + 12*E^(2 + x)*x + E^4*(22 + 7*x))
,x]

[Out]

Log[22*E^4 + 7*E^4*x + 6*E^(2*x)*x + 12*E^(2 + x)*x]

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fricas [A]  time = 0.52, size = 33, normalized size = 1.43 $$\begin{array}{c}\log (x)+\log \left(\frac{(7x+22)e^8+6x{e}^{(2x+4)}+12x{e}^{(x+6)}}{x}\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x+6)*exp(x)^2+(12*x+12)*exp(2)*exp(x)+7*exp(2)^2)/(6*x*exp(x)^2+12*x*exp(2)*exp(x)+(7*x+22)*exp
(2)^2),x, algorithm="fricas")

[Out]

log(x) + log(((7*x + 22)*e^8 + 6*x*e^(2*x + 4) + 12*x*e^(x + 6))/x)

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giac [A]  time = 0.26, size = 25, normalized size = 1.09 $$\begin{array}{c}\log (7x{e}^4+6x{e}^{\left(2x\right)}+12x{e}^{(x+2)}+22e^4)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x+6)*exp(x)^2+(12*x+12)*exp(2)*exp(x)+7*exp(2)^2)/(6*x*exp(x)^2+12*x*exp(2)*exp(x)+(7*x+22)*exp
(2)^2),x, algorithm="giac")

[Out]

log(7*x*e^4 + 6*x*e^(2*x) + 12*x*e^(x + 2) + 22*e^4)

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maple [A]  time = 0.06, size = 28, normalized size = 1.22

method	result	size
risch	$\ln (x)+\ln ({\mathrm{e}}^{2x}+2{\mathrm{e}}^{2+x}+\frac{(7x+22){\mathrm{e}}^4}{6x})$	$28$
norman	$\ln (7x{\mathrm{e}}^4+12x{\mathrm{e}}^2{\mathrm{e}}^x+6x{\mathrm{e}}^{2x}+22{\mathrm{e}}^4)$	$30$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x+6)*exp(x)^2+(12*x+12)*exp(2)*exp(x)+7*exp(2)^2)/(6*x*exp(x)^2+12*x*exp(2)*exp(x)+(7*x+22)*exp(2)^2)
,x,method=_RETURNVERBOSE)

[Out]

ln(x)+ln(exp(2*x)+2*exp(2+x)+1/6*(7*x+22)*exp(4)/x)

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maxima [A]  time = 0.48, size = 33, normalized size = 1.43 $$\begin{array}{c}\log (x)+\log \left(\frac{7x{e}^4+6x{e}^{\left(2x\right)}+12x{e}^{(x+2)}+22e^4}{6x}\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x+6)*exp(x)^2+(12*x+12)*exp(2)*exp(x)+7*exp(2)^2)/(6*x*exp(x)^2+12*x*exp(2)*exp(x)+(7*x+22)*exp
(2)^2),x, algorithm="maxima")

[Out]

log(x) + log(1/6*(7*x*e^4 + 6*x*e^(2*x) + 12*x*e^(x + 2) + 22*e^4)/x)

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mupad [B]  time = 2.05, size = 27, normalized size = 1.17 $$\begin{array}{c}\ln (22{\mathrm{e}}^8+12x{\mathrm{e}}^{x+6}+7x{\mathrm{e}}^8+6x{\mathrm{e}}^{2x+4})\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7*exp(4) + exp(2*x)*(12*x + 6) + exp(2)*exp(x)*(12*x + 12))/(6*x*exp(2*x) + exp(4)*(7*x + 22) + 12*x*exp(
2)*exp(x)),x)

[Out]

log(22*exp(8) + 12*x*exp(x + 6) + 7*x*exp(8) + 6*x*exp(2*x + 4))

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sympy [A]  time = 0.29, size = 32, normalized size = 1.39 $$\begin{array}{c}\log (x)+\log (e^{2x}+2e^2{e}^x+\frac{7x{e}^4+22e^4}{6x})\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x+6)*exp(x)**2+(12*x+12)*exp(2)*exp(x)+7*exp(2)**2)/(6*x*exp(x)**2+12*x*exp(2)*exp(x)+(7*x+22)*
exp(2)**2),x)

[Out]

log(x) + log(exp(2*x) + 2*exp(2)*exp(x) + (7*x*exp(4) + 22*exp(4))/(6*x))

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