3.63.78 \(\int \frac {14 e^4 x-324 x^4+27 x^6+10206 x^9+e^2 (-4+3 x^2-756 x^5)}{e^4-54 e^2 x^4+729 x^8} \, dx\)

Optimal. Leaf size=25 \[ \left (4-x^2\right ) \left (-7+\frac {x}{-e^2+27 x^4}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {28, 1858, 1586, 12, 30} \begin {gather*} 7 x^2-\frac {x \left (4-x^2\right )}{e^2-27 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(14*E^4*x - 324*x^4 + 27*x^6 + 10206*x^9 + E^2*(-4 + 3*x^2 - 756*x^5))/(E^4 - 54*E^2*x^4 + 729*x^8),x]

[Out]

7*x^2 - (x*(4 - x^2))/(E^2 - 27*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 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1858

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=729 \int \frac {14 e^4 x-324 x^4+27 x^6+10206 x^9+e^2 \left (-4+3 x^2-756 x^5\right )}{\left (-27 e^2+729 x^4\right )^2} \, dx\\ &=-\frac {x \left (4-x^2\right )}{e^2-27 x^4}+\frac {\int \frac {-21695547384 e^4 x+585779779368 e^2 x^5}{-27 e^2+729 x^4} \, dx}{57395628 e^2}\\ &=-\frac {x \left (4-x^2\right )}{e^2-27 x^4}+\frac {\int 803538792 e^2 x \, dx}{57395628 e^2}\\ &=-\frac {x \left (4-x^2\right )}{e^2-27 x^4}+14 \int x \, dx\\ &=7 x^2-\frac {x \left (4-x^2\right )}{e^2-27 x^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 23, normalized size = 0.92 \begin {gather*} x \left (7 x+\frac {-4+x^2}{e^2-27 x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(14*E^4*x - 324*x^4 + 27*x^6 + 10206*x^9 + E^2*(-4 + 3*x^2 - 756*x^5))/(E^4 - 54*E^2*x^4 + 729*x^8),
x]

[Out]

x*(7*x + (-4 + x^2)/(E^2 - 27*x^4))

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fricas [A]  time = 0.92, size = 34, normalized size = 1.36 \begin {gather*} \frac {189 \, x^{6} - x^{3} - 7 \, x^{2} e^{2} + 4 \, x}{27 \, x^{4} - e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((14*x*exp(1)^4+(-756*x^5+3*x^2-4)*exp(1)^2+10206*x^9+27*x^6-324*x^4)/(exp(1)^4-54*x^4*exp(1)^2+729*x
^8),x, algorithm="fricas")

[Out]

(189*x^6 - x^3 - 7*x^2*e^2 + 4*x)/(27*x^4 - e^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {10206 \, x^{9} + 27 \, x^{6} - 324 \, x^{4} + 14 \, x e^{4} - {\left (756 \, x^{5} - 3 \, x^{2} + 4\right )} e^{2}}{729 \, x^{8} - 54 \, x^{4} e^{2} + e^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((14*x*exp(1)^4+(-756*x^5+3*x^2-4)*exp(1)^2+10206*x^9+27*x^6-324*x^4)/(exp(1)^4-54*x^4*exp(1)^2+729*x
^8),x, algorithm="giac")

[Out]

integrate((10206*x^9 + 27*x^6 - 324*x^4 + 14*x*e^4 - (756*x^5 - 3*x^2 + 4)*e^2)/(729*x^8 - 54*x^4*e^2 + e^4),
x)

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maple [A]  time = 0.10, size = 25, normalized size = 1.00

method	result	size
risch	\(7 x^{2}+\frac {x^{3}-4 x}{-27 x^{4}+{\mathrm e}^{2}}\)	\(25\)
gosper	\(\frac {x \left (-189 x^{5}+7 \,{\mathrm e}^{2} x +x^{2}-4\right )}{-27 x^{4}+{\mathrm e}^{2}}\)	\(32\)
norman	\(\frac {-189 x^{6}+x^{3}+7 x^{2} {\mathrm e}^{2}-4 x}{-27 x^{4}+{\mathrm e}^{2}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((14*x*exp(1)^4+(-756*x^5+3*x^2-4)*exp(1)^2+10206*x^9+27*x^6-324*x^4)/(exp(1)^4-54*x^4*exp(1)^2+729*x^8),x,
method=_RETURNVERBOSE)

[Out]

7*x^2+(x^3-4*x)/(-27*x^4+exp(2))

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maxima [A]  time = 0.35, size = 27, normalized size = 1.08 \begin {gather*} 7 \, x^{2} - \frac {x^{3} - 4 \, x}{27 \, x^{4} - e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((14*x*exp(1)^4+(-756*x^5+3*x^2-4)*exp(1)^2+10206*x^9+27*x^6-324*x^4)/(exp(1)^4-54*x^4*exp(1)^2+729*x
^8),x, algorithm="maxima")

[Out]

7*x^2 - (x^3 - 4*x)/(27*x^4 - e^2)

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mupad [B]  time = 4.07, size = 23, normalized size = 0.92 \begin {gather*} 7\,x^2+\frac {x\,\left (x^2-4\right )}{{\mathrm {e}}^2-27\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((14*x*exp(4) - exp(2)*(756*x^5 - 3*x^2 + 4) - 324*x^4 + 27*x^6 + 10206*x^9)/(exp(4) - 54*x^4*exp(2) + 729*
x^8),x)

[Out]

7*x^2 + (x*(x^2 - 4))/(exp(2) - 27*x^4)

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sympy [A]  time = 0.51, size = 19, normalized size = 0.76 \begin {gather*} 7 x^{2} + \frac {- x^{3} + 4 x}{27 x^{4} - e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((14*x*exp(1)**4+(-756*x**5+3*x**2-4)*exp(1)**2+10206*x**9+27*x**6-324*x**4)/(exp(1)**4-54*x**4*exp(1
)**2+729*x**8),x)

[Out]

7*x**2 + (-x**3 + 4*x)/(27*x**4 - exp(2))

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