3.73.82 $\int \frac{-2592x^2+144x^3+576x^5+e^4(-18+4x^3)+e^2(432x-96x^4)}{3e^4{x}^3-72e^2{x}^4+432x^5}dx$

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

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Rubi [A]  time = 0.09, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 65, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.062, Rules used = {1594, 27, 12, 1620} $$\begin{array}{c}\frac{3}{x^2}+\frac{4x}{3}+\frac{4}{e^2-12x}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-2592*x^2 + 144*x^3 + 576*x^5 + E^4*(-18 + 4*x^3) + E^2*(432*x - 96*x^4))/(3*E^4*x^3 - 72*E^2*x^4 + 432*x
^5),x]

[Out]

4/(E^2 - 12*x) + 3/x^2 + (4*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 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1594

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

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-2592x^2+144x^3+576x^5+e^4(-18+4x^3)+e^2(432x-96x^4)}{x^3(3e^4-72e^{2}x+432x^2)}dx\\ & =\int \frac{-2592x^2+144x^3+576x^5+e^4(-18+4x^3)+e^2(432x-96x^4)}{3{(e^2-12x)}^2{x}^3}dx\\ & =\frac{1}{3}\int \frac{-2592x^2+144x^3+576x^5+e^4(-18+4x^3)+e^2(432x-96x^4)}{{(e^2-12x)}^2{x}^3}dx\\ & =\frac{1}{3}\int (4+\frac{144}{{(e^2-12x)}^2}-\frac{18}{x^3})dx\\ & =\frac{4}{e^2-12x}+\frac{3}{x^2}+\frac{4x}{3}\end{array}\end{array}$$

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Mathematica [A]  time = 0.02, size = 28, normalized size = 1.27 $$\begin{array}{c}\frac{2}{3}(\frac{9}{2x^2}+2x-\frac{6}{-e^2+12x})\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-2592*x^2 + 144*x^3 + 576*x^5 + E^4*(-18 + 4*x^3) + E^2*(432*x - 96*x^4))/(3*E^4*x^3 - 72*E^2*x^4 +
432*x^5),x]

[Out]

(2*(9/(2*x^2) + 2*x - 6/(-E^2 + 12*x)))/3

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fricas [A]  time = 0.52, size = 42, normalized size = 1.91 $$\begin{array}{c}\frac{48x^4-12x^2-(4x^3+9)e^2+108x}{3(12x^3-x^2{e}^2)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-18)*exp(2)^2+(-96*x^4+432*x)*exp(2)+576*x^5+144*x^3-2592*x^2)/(3*x^3*exp(2)^2-72*x^4*exp(2)+
432*x^5),x, algorithm="fricas")

[Out]

1/3*(48*x^4 - 12*x^2 - (4*x^3 + 9)*e^2 + 108*x)/(12*x^3 - x^2*e^2)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\text{Exception raised: NotImplementedError}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-18)*exp(2)^2+(-96*x^4+432*x)*exp(2)+576*x^5+144*x^3-2592*x^2)/(3*x^3*exp(2)^2-72*x^4*exp(2)+
432*x^5),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2/3*(288*sageVARx/144+9*1/2/sageVARx^2+7
2*1/24/sqrt(exp(2)^2-exp(4))*ln(sqrt((288*sageVARx-24*exp(2))^2+(-24*sqrt(-exp(2)^2+exp(4)))^2)/sqrt((288*sage
VARx-24*exp(2))^2+(24

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maple [A]  time = 0.07, size = 30, normalized size = 1.36

method	result	size
risch	$\frac{4x}{3}+\frac{4x^2+3{\mathrm{e}}^2-36x}{x^2(-12x+{\mathrm{e}}^2)}$	$30$
norman	$\frac{(\frac{{\mathrm{e}}^4}{9}+4)x^2-36x-16x^4+3{\mathrm{e}}^2}{x^2(-12x+{\mathrm{e}}^2)}$	$38$
gosper	$\frac{x^2{\mathrm{e}}^4-144x^4+36x^2+27{\mathrm{e}}^2-324x}{9x^2(-12x+{\mathrm{e}}^2)}$	$40$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3-18)*exp(2)^2+(-96*x^4+432*x)*exp(2)+576*x^5+144*x^3-2592*x^2)/(3*x^3*exp(2)^2-72*x^4*exp(2)+432*x^
5),x,method=_RETURNVERBOSE)

[Out]

4/3*x+(4*x^2+3*exp(2)-36*x)/x^2/(-12*x+exp(2))

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maxima [A]  time = 0.36, size = 34, normalized size = 1.55 $$\begin{array}{c}\frac{4}{3}x-\frac{4x^2-36x+3e^2}{12x^3-x^2{e}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-18)*exp(2)^2+(-96*x^4+432*x)*exp(2)+576*x^5+144*x^3-2592*x^2)/(3*x^3*exp(2)^2-72*x^4*exp(2)+
432*x^5),x, algorithm="maxima")

[Out]

4/3*x - (4*x^2 - 36*x + 3*e^2)/(12*x^3 - x^2*e^2)

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mupad [B]  time = 4.28, size = 32, normalized size = 1.45 $$\begin{array}{c}\frac{4x}{3}-\frac{4x^2-36x+3{\mathrm{e}}^2}{x^2(12x-{\mathrm{e}}^2)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2)*(432*x - 96*x^4) + exp(4)*(4*x^3 - 18) - 2592*x^2 + 144*x^3 + 576*x^5)/(3*x^3*exp(4) - 72*x^4*exp(
2) + 432*x^5),x)

[Out]

(4*x)/3 - (3*exp(2) - 36*x + 4*x^2)/(x^2*(12*x - exp(2)))

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sympy [A]  time = 0.50, size = 29, normalized size = 1.32 $$\begin{array}{c}\frac{4x}{3}+\frac{-4x^2+36x-3e^2}{12x^3-x^2{e}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3-18)*exp(2)**2+(-96*x**4+432*x)*exp(2)+576*x**5+144*x**3-2592*x**2)/(3*x**3*exp(2)**2-72*x**
4*exp(2)+432*x**5),x)

[Out]

4*x/3 + (-4*x**2 + 36*x - 3*exp(2))/(12*x**3 - x**2*exp(2))

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