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

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

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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 {gather*} 5 x^3+20 x^2-\frac {45}{e^2 x^2}-\frac {45}{e^4 \left (x+e^2\right )}+\frac {45}{e^4 x} \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=\int \frac {135 x+40 x^6+15 x^7+e^4 \left (40 x^4+15 x^5\right )+e^2 \left (90+80 x^5+30 x^6\right )}{x^3 \left (e^4+2 e^2 x+x^2\right )} \, dx\\ &=\int \frac {135 x+40 x^6+15 x^7+e^4 \left (40 x^4+15 x^5\right )+e^2 \left (90+80 x^5+30 x^6\right )}{x^3 \left (e^2+x\right )^2} \, dx\\ &=\int \left (\frac {90}{e^2 x^3}-\frac {45}{e^4 x^2}+40 x+15 x^2+\frac {45}{e^4 \left (e^2+x\right )^2}\right ) \, dx\\ &=-\frac {45}{e^2 x^2}+\frac {45}{e^4 x}+20 x^2+5 x^3-\frac {45}{e^4 \left (e^2+x\right )}\\ \end {aligned} \end {gather*}

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

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 {gather*} \frac {5 \, {\left (x^{6} + 4 \, x^{5} + {\left (x^{5} + 4 \, x^{4}\right )} e^{2} - 9\right )}}{x^{3} + x^{2} e^{2}} \end {gather*}

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 {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

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	\(5 x^{3}+20 x^{2}-\frac {45}{x^{2} \left (x +{\mathrm e}^{2}\right )}\)	\(23\)
gosper	\(\frac {5 \,{\mathrm e}^{2} x^{5}+5 x^{6}+20 x^{4} {\mathrm e}^{2}+20 x^{5}-45}{x^{2} \left (x +{\mathrm e}^{2}\right )}\)	\(35\)
norman	\(\frac {-45+\left (20+5 \,{\mathrm e}^{2}\right ) x^{5}+5 x^{6}+20 x^{4} {\mathrm e}^{2}}{x^{2} \left (x +{\mathrm e}^{2}\right )}\)	\(35\)

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 {gather*} 5 \, x^{3} + 20 \, x^{2} - \frac {45}{x^{3} + x^{2} e^{2}} \end {gather*}

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 {gather*} 20\,x^2-x\,\left (80\,{\mathrm {e}}^2+15\,{\mathrm {e}}^4-5\,{\mathrm {e}}^2\,\left (3\,{\mathrm {e}}^2+16\right )\right )-\frac {45}{x^3+{\mathrm {e}}^2\,x^2}+5\,x^3 \end {gather*}

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 {gather*} 5 x^{3} + 20 x^{2} - \frac {45}{x^{3} + x^{2} e^{2}} \end {gather*}

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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