3.42.24 \(\int -\frac {2250}{16 x^4-8 e x^4+e^2 x^4} \, dx\)

Optimal. Leaf size=12 \[ \frac {750}{(4-e)^2 x^3} \]

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6, 12, 30} \begin {gather*} \frac {750}{(4-e)^2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-2250/(16*x^4 - 8*E*x^4 + E^2*x^4),x]

[Out]

750/((4 - E)^2*x^3)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

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

Rule 30

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int -\frac {2250}{(16-8 e) x^4+e^2 x^4} \, dx\\ &=\int -\frac {2250}{\left (16-8 e+e^2\right ) x^4} \, dx\\ &=-\frac {2250 \int \frac {1}{x^4} \, dx}{(4-e)^2}\\ &=\frac {750}{(4-e)^2 x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.83 \begin {gather*} \frac {750}{(-4+e)^2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-2250/(16*x^4 - 8*E*x^4 + E^2*x^4),x]

[Out]

750/((-4 + E)^2*x^3)

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fricas [B]  time = 0.65, size = 23, normalized size = 1.92 \begin {gather*} \frac {750}{x^{3} e^{2} - 8 \, x^{3} e + 16 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2250/(x^4*exp(1)^2-8*x^4*exp(1)+16*x^4),x, algorithm="fricas")

[Out]

750/(x^3*e^2 - 8*x^3*e + 16*x^3)

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giac [A]  time = 0.13, size = 15, normalized size = 1.25 \begin {gather*} \frac {750}{x^{3} {\left (e^{2} - 8 \, e + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2250/(x^4*exp(1)^2-8*x^4*exp(1)+16*x^4),x, algorithm="giac")

[Out]

750/(x^3*(e^2 - 8*e + 16))

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maple [A]  time = 0.05, size = 12, normalized size = 1.00

method	result	size
norman	\(\frac {750}{\left ({\mathrm e}-4\right )^{2} x^{3}}\)	\(12\)
risch	\(\frac {750}{x^{3} \left ({\mathrm e}^{2}-8 \,{\mathrm e}+16\right )}\)	\(16\)
gosper	\(\frac {750}{x^{3} \left ({\mathrm e}^{2}-8 \,{\mathrm e}+16\right )}\)	\(18\)
default	\(\frac {750}{x^{3} \left ({\mathrm e}^{2}-8 \,{\mathrm e}+16\right )}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2250/(x^4*exp(1)^2-8*x^4*exp(1)+16*x^4),x,method=_RETURNVERBOSE)

[Out]

750/(exp(1)-4)^2/x^3

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maxima [A]  time = 0.35, size = 15, normalized size = 1.25 \begin {gather*} \frac {750}{x^{3} {\left (e^{2} - 8 \, e + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2250/(x^4*exp(1)^2-8*x^4*exp(1)+16*x^4),x, algorithm="maxima")

[Out]

750/(x^3*(e^2 - 8*e + 16))

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mupad [B]  time = 0.12, size = 11, normalized size = 0.92 \begin {gather*} \frac {750}{x^3\,{\left (\mathrm {e}-4\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2250/(x^4*exp(2) - 8*x^4*exp(1) + 16*x^4),x)

[Out]

750/(x^3*(exp(1) - 4)^2)

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sympy [A]  time = 0.06, size = 14, normalized size = 1.17 \begin {gather*} \frac {750}{x^{3} \left (- 8 e + e^{2} + 16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2250/(x**4*exp(1)**2-8*x**4*exp(1)+16*x**4),x)

[Out]

750/(x**3*(-8*E + exp(2) + 16))

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