3.102.42 \(\int \frac {-150000 e^{10} x^2+390625 x^3}{4096 e^{30}-96000 e^{20} x+750000 e^{10} x^2-1953125 x^3} \, dx\)

Optimal. Leaf size=17 \[ -\frac {5 x}{\left (-5+\frac {16 e^{10}}{25 x}\right )^2} \]

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Rubi [A]  time = 0.16, antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1593, 6688, 12, 74} \begin {gather*} -\frac {3125 x^3}{\left (16 e^{10}-125 x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-150000*E^10*x^2 + 390625*x^3)/(4096*E^30 - 96000*E^20*x + 750000*E^10*x^2 - 1953125*x^3),x]

[Out]

(-3125*x^3)/(16*E^10 - 125*x)^2

Rule 12

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

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - 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} &=\int \frac {x^2 \left (-150000 e^{10}+390625 x\right )}{4096 e^{30}-96000 e^{20} x+750000 e^{10} x^2-1953125 x^3} \, dx\\ &=\int \frac {3125 x^2 \left (-48 e^{10}+125 x\right )}{\left (16 e^{10}-125 x\right )^3} \, dx\\ &=3125 \int \frac {x^2 \left (-48 e^{10}+125 x\right )}{\left (16 e^{10}-125 x\right )^3} \, dx\\ &=-\frac {3125 x^3}{\left (16 e^{10}-125 x\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.02, size = 40, normalized size = 2.35 \begin {gather*} -\frac {-12288 e^{30}+192000 e^{20} x-750000 e^{10} x^2+1953125 x^3}{625 \left (16 e^{10}-125 x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-150000*E^10*x^2 + 390625*x^3)/(4096*E^30 - 96000*E^20*x + 750000*E^10*x^2 - 1953125*x^3),x]

[Out]

-1/625*(-12288*E^30 + 192000*E^20*x - 750000*E^10*x^2 + 1953125*x^3)/(16*E^10 - 125*x)^2

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fricas [B]  time = 0.46, size = 41, normalized size = 2.41 \begin {gather*} -\frac {1953125 \, x^{3} - 500000 \, x^{2} e^{10} + 128000 \, x e^{20} - 8192 \, e^{30}}{625 \, {\left (15625 \, x^{2} - 4000 \, x e^{10} + 256 \, e^{20}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-150000*x^2*exp(5)^2+390625*x^3)/(4096*exp(5)^6-96000*x*exp(5)^4+750000*x^2*exp(5)^2-1953125*x^3),x
, algorithm="fricas")

[Out]

-1/625*(1953125*x^3 - 500000*x^2*e^10 + 128000*x*e^20 - 8192*e^30)/(15625*x^2 - 4000*x*e^10 + 256*e^20)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {3125 \, {\left (125 \, x^{3} - 48 \, x^{2} e^{10}\right )}}{1953125 \, x^{3} - 750000 \, x^{2} e^{10} + 96000 \, x e^{20} - 4096 \, e^{30}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-150000*x^2*exp(5)^2+390625*x^3)/(4096*exp(5)^6-96000*x*exp(5)^4+750000*x^2*exp(5)^2-1953125*x^3),x
, algorithm="giac")

[Out]

integrate(-3125*(125*x^3 - 48*x^2*e^10)/(1953125*x^3 - 750000*x^2*e^10 + 96000*x*e^20 - 4096*e^30), x)

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maple [A]  time = 0.06, size = 18, normalized size = 1.06

method	result	size
norman	\(-\frac {3125 x^{3}}{\left (16 \,{\mathrm e}^{10}-125 x \right )^{2}}\)	\(18\)
gosper	\(-\frac {3125 x^{3}}{256 \,{\mathrm e}^{20}-4000 x \,{\mathrm e}^{10}+15625 x^{2}}\)	\(27\)
risch	\(-\frac {x}{5}+\frac {\frac {32 \,{\mathrm e}^{30}}{625}-\frac {3 x \,{\mathrm e}^{20}}{5}}{{\mathrm e}^{20}-\frac {125 x \,{\mathrm e}^{10}}{8}+\frac {15625 x^{2}}{256}}\)	\(31\)
default	\(-\frac {x}{5}-\frac {256 \left (\munderset {\textit {\_R} =\RootOf \left (-4096 \,{\mathrm e}^{30}+96000 \textit {\_Z} \,{\mathrm e}^{20}-750000 \textit {\_Z}^{2} {\mathrm e}^{10}+1953125 \textit {\_Z}^{3}\right )}{\sum }\frac {\left (-375 \textit {\_R} \,{\mathrm e}^{20}+16 \,{\mathrm e}^{30}\right ) \ln \left (x -\textit {\_R} \right )}{256 \,{\mathrm e}^{20}-4000 \textit {\_R} \,{\mathrm e}^{10}+15625 \textit {\_R}^{2}}\right )}{1875}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-150000*x^2*exp(5)^2+390625*x^3)/(4096*exp(5)^6-96000*x*exp(5)^4+750000*x^2*exp(5)^2-1953125*x^3),x,metho
d=_RETURNVERBOSE)

[Out]

-3125*x^3/(16*exp(5)^2-125*x)^2

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maxima [B]  time = 0.34, size = 33, normalized size = 1.94 \begin {gather*} -\frac {1}{5} \, x - \frac {256 \, {\left (375 \, x e^{20} - 32 \, e^{30}\right )}}{625 \, {\left (15625 \, x^{2} - 4000 \, x e^{10} + 256 \, e^{20}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-150000*x^2*exp(5)^2+390625*x^3)/(4096*exp(5)^6-96000*x*exp(5)^4+750000*x^2*exp(5)^2-1953125*x^3),x
, algorithm="maxima")

[Out]

-1/5*x - 256/625*(375*x*e^20 - 32*e^30)/(15625*x^2 - 4000*x*e^10 + 256*e^20)

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mupad [B]  time = 7.07, size = 25, normalized size = 1.47 \begin {gather*} \frac {\frac {8192\,{\mathrm {e}}^{30}}{625}-\frac {768\,x\,{\mathrm {e}}^{20}}{5}}{{\left (125\,x-16\,{\mathrm {e}}^{10}\right )}^2}-\frac {x}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(150000*x^2*exp(10) - 390625*x^3)/(4096*exp(30) - 96000*x*exp(20) + 750000*x^2*exp(10) - 1953125*x^3),x)

[Out]

((8192*exp(30))/625 - (768*x*exp(20))/5)/(125*x - 16*exp(10))^2 - x/5

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sympy [B]  time = 0.19, size = 32, normalized size = 1.88 \begin {gather*} - \frac {x}{5} - \frac {96000 x e^{20} - 8192 e^{30}}{9765625 x^{2} - 2500000 x e^{10} + 160000 e^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-150000*x**2*exp(5)**2+390625*x**3)/(4096*exp(5)**6-96000*x*exp(5)**4+750000*x**2*exp(5)**2-1953125
*x**3),x)

[Out]

-x/5 - (96000*x*exp(20) - 8192*exp(30))/(9765625*x**2 - 2500000*x*exp(10) + 160000*exp(20))

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