3.17.66 $\int \frac{135x+40x^6+15x^7+e^4(40x^4+15x^5)+e^2(90+80x^5+30x^6)}{e^4{x}^3+2e^2{x}^4+x^5}dx$

Optimal. Leaf size=27 $$5x^2(4+x)-4(5+\frac{45}{4x^2(e^2+x)})$$

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Rubi [A]  time = 0.11, antiderivative size = 39, normalized size of antiderivative = 1.44, number of steps used = 4, number of rules used = 3, integrand size = 67, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.045, Rules used = {1594, 27, 1620} $$\begin{array}{c}5x^3+20x^2-\frac{45}{e^2{x}^2}-\frac{45}{e^4(x+e^2)}+\frac{45}{e^{4}x}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(135*x + 40*x^6 + 15*x^7 + E^4*(40*x^4 + 15*x^5) + E^2*(90 + 80*x^5 + 30*x^6))/(E^4*x^3 + 2*E^2*x^4 + x^5)
,x]

[Out]

-45/(E^2*x^2) + 45/(E^4*x) + 20*x^2 + 5*x^3 - 45/(E^4*(E^2 + 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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{135x+40x^6+15x^7+e^4(40x^4+15x^5)+e^2(90+80x^5+30x^6)}{x^3(e^4+2e^{2}x+x^2)}dx\\ & =\int \frac{135x+40x^6+15x^7+e^4(40x^4+15x^5)+e^2(90+80x^5+30x^6)}{x^3{(e^2+x)}^2}dx\\ & =\int (\frac{90}{e^2{x}^3}-\frac{45}{e^4{x}^2}+40x+15x^2+\frac{45}{e^4{(e^2+x)}^2})dx\\ & =-\frac{45}{e^2{x}^2}+\frac{45}{e^{4}x}+20x^2+5x^3-\frac{45}{e^4(e^2+x)}\end{array}\end{array}$$

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Mathematica [A]  time = 0.02, size = 32, normalized size = 1.19 $$\begin{array}{c}\frac{5(-9+4x^5+x^6+e^2{x}^4(4+x))}{x^2(e^2+x)}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(135*x + 40*x^6 + 15*x^7 + E^4*(40*x^4 + 15*x^5) + E^2*(90 + 80*x^5 + 30*x^6))/(E^4*x^3 + 2*E^2*x^4
+ x^5),x]

[Out]

(5*(-9 + 4*x^5 + x^6 + E^2*x^4*(4 + x)))/(x^2*(E^2 + x))

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fricas [A]  time = 0.78, size = 36, normalized size = 1.33 $$\begin{array}{c}\frac{5(x^6+4x^5+(x^5+4x^4)e^2-9)}{x^3+x^2{e}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((15*x^5+40*x^4)*exp(2)^2+(30*x^6+80*x^5+90)*exp(2)+15*x^7+40*x^6+135*x)/(x^3*exp(2)^2+2*x^4*exp(2)+
x^5),x, algorithm="fricas")

[Out]

5*(x^6 + 4*x^5 + (x^5 + 4*x^4)*e^2 - 9)/(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(((15*x^5+40*x^4)*exp(2)^2+(30*x^6+80*x^5+90)*exp(2)+15*x^7+40*x^6+135*x)/(x^3*exp(2)^2+2*x^4*exp(2)+
x^5),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 5*(sageVARx^3+4*sageVARx^2+((-27*exp(4)^
2+36*exp(4)*exp(2)^2)*sageVARx-9*exp(4)^2*exp(2))/exp(4)^3/sageVARx^2+(36*exp(4)*exp(2)-36*exp(2)^3)/exp(4)^3*
ln(sageVARx^2+2*sageV

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maple [A]  time = 0.08, size = 23, normalized size = 0.85

method	result	size
risch	$5x^3+20x^2-\frac{45}{x^2(x+{\mathrm{e}}^2)}$	$23$
gosper	$\frac{5{\mathrm{e}}^2{x}^5+5x^6+20x^4{\mathrm{e}}^2+20x^5-45}{x^2(x+{\mathrm{e}}^2)}$	$35$
norman	$\frac{-45+(20+5{\mathrm{e}}^2)x^5+5x^6+20x^4{\mathrm{e}}^2}{x^2(x+{\mathrm{e}}^2)}$	$35$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((15*x^5+40*x^4)*exp(2)^2+(30*x^6+80*x^5+90)*exp(2)+15*x^7+40*x^6+135*x)/(x^3*exp(2)^2+2*x^4*exp(2)+x^5),x
,method=_RETURNVERBOSE)

[Out]

5*x^3+20*x^2-45/x^2/(x+exp(2))

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maxima [A]  time = 0.43, size = 25, normalized size = 0.93 $$\begin{array}{c}5x^3+20x^2-\frac{45}{x^3+x^2{e}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((15*x^5+40*x^4)*exp(2)^2+(30*x^6+80*x^5+90)*exp(2)+15*x^7+40*x^6+135*x)/(x^3*exp(2)^2+2*x^4*exp(2)+
x^5),x, algorithm="maxima")

[Out]

5*x^3 + 20*x^2 - 45/(x^3 + x^2*e^2)

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mupad [B]  time = 1.14, size = 47, normalized size = 1.74 $$\begin{array}{c}20x^2-x(80{\mathrm{e}}^2+15{\mathrm{e}}^4-5{\mathrm{e}}^2(3{\mathrm{e}}^2+16))-\frac{45}{x^3+{\mathrm{e}}^2{x}^2}+5x^3\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((135*x + exp(4)*(40*x^4 + 15*x^5) + exp(2)*(80*x^5 + 30*x^6 + 90) + 40*x^6 + 15*x^7)/(2*x^4*exp(2) + x^3*e
xp(4) + x^5),x)

[Out]

20*x^2 - x*(80*exp(2) + 15*exp(4) - 5*exp(2)*(3*exp(2) + 16)) - 45/(x^2*exp(2) + x^3) + 5*x^3

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sympy [A]  time = 0.19, size = 20, normalized size = 0.74 $$\begin{array}{c}5x^3+20x^2-\frac{45}{x^3+x^2{e}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((15*x**5+40*x**4)*exp(2)**2+(30*x**6+80*x**5+90)*exp(2)+15*x**7+40*x**6+135*x)/(x**3*exp(2)**2+2*x*
*4*exp(2)+x**5),x)

[Out]

5*x**3 + 20*x**2 - 45/(x**3 + x**2*exp(2))

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