3.17.9 \(\int \frac {e (32 x+8 x^2-50 x^5)}{768+768 x+960 x^2+384 x^3+2592 x^4+1200 x^5+1200 x^6+1875 x^8} \, dx\)

Optimal. Leaf size=23 \[ \frac {e}{3 \left (7+25 x^2+\frac {(4+x)^2}{x^2}\right )} \]

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Rubi [A]  time = 0.19, antiderivative size = 25, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {12, 1594, 6688, 1588} \begin {gather*} \frac {e x^2}{3 \left (25 x^4+8 x^2+8 x+16\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E*(32*x + 8*x^2 - 50*x^5))/(768 + 768*x + 960*x^2 + 384*x^3 + 2592*x^4 + 1200*x^5 + 1200*x^6 + 1875*x^8),
x]

[Out]

(E*x^2)/(3*(16 + 8*x + 8*x^2 + 25*x^4))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e \int \frac {32 x+8 x^2-50 x^5}{768+768 x+960 x^2+384 x^3+2592 x^4+1200 x^5+1200 x^6+1875 x^8} \, dx\\ &=e \int \frac {x \left (32+8 x-50 x^4\right )}{768+768 x+960 x^2+384 x^3+2592 x^4+1200 x^5+1200 x^6+1875 x^8} \, dx\\ &=e \int \frac {2 x \left (16+4 x-25 x^4\right )}{3 \left (16+8 x+8 x^2+25 x^4\right )^2} \, dx\\ &=\frac {1}{3} (2 e) \int \frac {x \left (16+4 x-25 x^4\right )}{\left (16+8 x+8 x^2+25 x^4\right )^2} \, dx\\ &=\frac {e x^2}{3 \left (16+8 x+8 x^2+25 x^4\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.09 \begin {gather*} \frac {2 e x^2}{3 \left (32+16 x+16 x^2+50 x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E*(32*x + 8*x^2 - 50*x^5))/(768 + 768*x + 960*x^2 + 384*x^3 + 2592*x^4 + 1200*x^5 + 1200*x^6 + 1875
*x^8),x]

[Out]

(2*E*x^2)/(3*(32 + 16*x + 16*x^2 + 50*x^4))

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fricas [A]  time = 0.80, size = 24, normalized size = 1.04 \begin {gather*} \frac {x^{2} e}{3 \, {\left (25 \, x^{4} + 8 \, x^{2} + 8 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-50*x^5+8*x^2+32*x)*exp(1)/(1875*x^8+1200*x^6+1200*x^5+2592*x^4+384*x^3+960*x^2+768*x+768),x, algor
ithm="fricas")

[Out]

1/3*x^2*e/(25*x^4 + 8*x^2 + 8*x + 16)

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giac [A]  time = 0.19, size = 24, normalized size = 1.04 \begin {gather*} \frac {x^{2} e}{3 \, {\left (25 \, x^{4} + 8 \, x^{2} + 8 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-50*x^5+8*x^2+32*x)*exp(1)/(1875*x^8+1200*x^6+1200*x^5+2592*x^4+384*x^3+960*x^2+768*x+768),x, algor
ithm="giac")

[Out]

1/3*x^2*e/(25*x^4 + 8*x^2 + 8*x + 16)

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maple [A]  time = 0.04, size = 23, normalized size = 1.00

method	result	size
default	\(\frac {x^{2} {\mathrm e}}{75 x^{4}+24 x^{2}+24 x +48}\)	\(23\)
risch	\(\frac {x^{2} {\mathrm e}}{75 x^{4}+24 x^{2}+24 x +48}\)	\(23\)
gosper	\(\frac {x^{2} {\mathrm e}}{75 x^{4}+24 x^{2}+24 x +48}\)	\(25\)
norman	\(\frac {-\frac {x \,{\mathrm e}}{3}-\frac {25 x^{4} {\mathrm e}}{24}-\frac {2 \,{\mathrm e}}{3}}{25 x^{4}+8 x^{2}+8 x +16}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-50*x^5+8*x^2+32*x)*exp(1)/(1875*x^8+1200*x^6+1200*x^5+2592*x^4+384*x^3+960*x^2+768*x+768),x,method=_RETU
RNVERBOSE)

[Out]

1/75*exp(1)*x^2/(x^4+8/25*x^2+8/25*x+16/25)

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maxima [A]  time = 0.39, size = 24, normalized size = 1.04 \begin {gather*} \frac {x^{2} e}{3 \, {\left (25 \, x^{4} + 8 \, x^{2} + 8 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-50*x^5+8*x^2+32*x)*exp(1)/(1875*x^8+1200*x^6+1200*x^5+2592*x^4+384*x^3+960*x^2+768*x+768),x, algor
ithm="maxima")

[Out]

1/3*x^2*e/(25*x^4 + 8*x^2 + 8*x + 16)

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mupad [B]  time = 0.07, size = 23, normalized size = 1.00 \begin {gather*} \frac {x^2\,\mathrm {e}}{3\,\left (25\,x^4+8\,x^2+8\,x+16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1)*(32*x + 8*x^2 - 50*x^5))/(768*x + 960*x^2 + 384*x^3 + 2592*x^4 + 1200*x^5 + 1200*x^6 + 1875*x^8 +
768),x)

[Out]

(x^2*exp(1))/(3*(8*x + 8*x^2 + 25*x^4 + 16))

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sympy [A]  time = 0.33, size = 20, normalized size = 0.87 \begin {gather*} \frac {e x^{2}}{75 x^{4} + 24 x^{2} + 24 x + 48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-50*x**5+8*x**2+32*x)*exp(1)/(1875*x**8+1200*x**6+1200*x**5+2592*x**4+384*x**3+960*x**2+768*x+768),
x)

[Out]

E*x**2/(75*x**4 + 24*x**2 + 24*x + 48)

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