3.25.40 \(\int \frac {40 e^5+550 x-200 x^2}{-33275 x^4+18150 x^5-3300 x^6+200 x^7+e^{10} (-5324+2904 x-528 x^2+32 x^3)+e^5 (26620 x^2-14520 x^3+2640 x^4-160 x^5)} \, dx\)

Optimal. Leaf size=26 \[ \frac {1}{(-1+3 (4-x)+x)^2 \left (-\frac {2 e^5}{5}+x^2\right )} \]

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Rubi [B]  time = 0.18, antiderivative size = 71, normalized size of antiderivative = 2.73, number of steps used = 5, number of rules used = 3, integrand size = 82, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2074, 639, 206} \begin {gather*} -\frac {25 \left (220 x+8 e^5+605\right )}{\left (605-8 e^5\right )^2 \left (2 e^5-5 x^2\right )}+\frac {2200}{\left (605-8 e^5\right )^2 (11-2 x)}+\frac {20}{\left (605-8 e^5\right ) (11-2 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(40*E^5 + 550*x - 200*x^2)/(-33275*x^4 + 18150*x^5 - 3300*x^6 + 200*x^7 + E^10*(-5324 + 2904*x - 528*x^2 +
32*x^3) + E^5*(26620*x^2 - 14520*x^3 + 2640*x^4 - 160*x^5)),x]

[Out]

20/((605 - 8*E^5)*(11 - 2*x)^2) + 2200/((605 - 8*E^5)^2*(11 - 2*x)) - (25*(605 + 8*E^5 + 220*x))/((605 - 8*E^5
)^2*(2*E^5 - 5*x^2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {80}{\left (-605+8 e^5\right ) (-11+2 x)^3}+\frac {4400}{\left (-605+8 e^5\right )^2 (-11+2 x)^2}+\frac {250 \left (-88 e^5-\left (605+8 e^5\right ) x\right )}{\left (605-8 e^5\right )^2 \left (2 e^5-5 x^2\right )^2}+\frac {5500}{\left (-605+8 e^5\right )^2 \left (2 e^5-5 x^2\right )}\right ) \, dx\\ &=\frac {20}{\left (605-8 e^5\right ) (11-2 x)^2}+\frac {2200}{\left (605-8 e^5\right )^2 (11-2 x)}+\frac {250 \int \frac {-88 e^5-\left (605+8 e^5\right ) x}{\left (2 e^5-5 x^2\right )^2} \, dx}{\left (605-8 e^5\right )^2}+\frac {5500 \int \frac {1}{2 e^5-5 x^2} \, dx}{\left (605-8 e^5\right )^2}\\ &=\frac {20}{\left (605-8 e^5\right ) (11-2 x)^2}+\frac {2200}{\left (605-8 e^5\right )^2 (11-2 x)}-\frac {25 \left (605+8 e^5+220 x\right )}{\left (605-8 e^5\right )^2 \left (2 e^5-5 x^2\right )}+\frac {550 \sqrt {10} \tanh ^{-1}\left (\frac {\sqrt {\frac {5}{2}} x}{e^{5/2}}\right )}{e^{5/2} \left (605-8 e^5\right )^2}-\frac {5500 \int \frac {1}{2 e^5-5 x^2} \, dx}{\left (605-8 e^5\right )^2}\\ &=\frac {20}{\left (605-8 e^5\right ) (11-2 x)^2}+\frac {2200}{\left (605-8 e^5\right )^2 (11-2 x)}-\frac {25 \left (605+8 e^5+220 x\right )}{\left (605-8 e^5\right )^2 \left (2 e^5-5 x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 22, normalized size = 0.85 \begin {gather*} \frac {5}{(11-2 x)^2 \left (-2 e^5+5 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(40*E^5 + 550*x - 200*x^2)/(-33275*x^4 + 18150*x^5 - 3300*x^6 + 200*x^7 + E^10*(-5324 + 2904*x - 528
*x^2 + 32*x^3) + E^5*(26620*x^2 - 14520*x^3 + 2640*x^4 - 160*x^5)),x]

[Out]

5/((11 - 2*x)^2*(-2*E^5 + 5*x^2))

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fricas [A]  time = 0.55, size = 34, normalized size = 1.31 \begin {gather*} \frac {5}{20 \, x^{4} - 220 \, x^{3} + 605 \, x^{2} - 2 \, {\left (4 \, x^{2} - 44 \, x + 121\right )} e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*exp(5)-200*x^2+550*x)/((32*x^3-528*x^2+2904*x-5324)*exp(5)^2+(-160*x^5+2640*x^4-14520*x^3+26620*
x^2)*exp(5)+200*x^7-3300*x^6+18150*x^5-33275*x^4),x, algorithm="fricas")

[Out]

5/(20*x^4 - 220*x^3 + 605*x^2 - 2*(4*x^2 - 44*x + 121)*e^5)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*exp(5)-200*x^2+550*x)/((32*x^3-528*x^2+2904*x-5324)*exp(5)^2+(-160*x^5+2640*x^4-14520*x^3+26620*
x^2)*exp(5)+200*x^7-3300*x^6+18150*x^5-33275*x^4),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.23, size = 22, normalized size = 0.85

method	result	size
norman	\(-\frac {5}{\left (2 x -11\right )^{2} \left (-5 x^{2}+2 \,{\mathrm e}^{5}\right )}\)	\(22\)
risch	\(-\frac {5}{8 \left (-\frac {5 x^{4}}{2}+x^{2} {\mathrm e}^{5}+\frac {55 x^{3}}{2}-11 x \,{\mathrm e}^{5}-\frac {605 x^{2}}{8}+\frac {121 \,{\mathrm e}^{5}}{4}\right )}\)	\(36\)
gosper	\(-\frac {5}{-20 x^{4}+8 x^{2} {\mathrm e}^{5}+220 x^{3}-88 x \,{\mathrm e}^{5}-605 x^{2}+242 \,{\mathrm e}^{5}}\)	\(37\)
default	\(-\frac {10 \left (\frac {275 \left (7086244000 \,{\mathrm e}^{10}-2048 \,{\mathrm e}^{5} {\mathrm e}^{20}+619520 \,{\mathrm e}^{5} {\mathrm e}^{15}-70276800 \,{\mathrm e}^{5} {\mathrm e}^{10}-133974300625 \,{\mathrm e}^{5}+619520 \,{\mathrm e}^{20}-70276800 \,{\mathrm e}^{15}-2048 \,{\mathrm e}^{25}\right ) {\mathrm e}^{-5} x [phone] \,{\mathrm e}^{5}+9292800 \,{\mathrm e}^{20}-468512000 \,{\mathrm e}^{15}-40960 \,{\mathrm e}^{25}-35431220000 \,{\mathrm e}^{10}-\frac {405272259390625}{4}}{-\frac {5 x^{2}}{2}+{\mathrm e}^{5}}-880 \left (-128 \,{\mathrm e}^{5} {\mathrm e}^{20}+38720 \,{\mathrm e}^{5} {\mathrm e}^{15}-4392300 \,{\mathrm e}^{5} {\mathrm e}^{10}-38720 \,{\mathrm e}^{20}+4392300 \,{\mathrm e}^{15}+128 \,{\mathrm e}^{25}\right ) {\mathrm e}^{-5} \sqrt {10}\, {\mathrm e}^{-\frac {5}{2}} \arctanh \left (\frac {x \sqrt {10}\, {\mathrm e}^{-\frac {5}{2}}}{2}\right )\right )}{\left (-366025+9680 \,{\mathrm e}^{5}-64 \,{\mathrm e}^{10}\right )^{3}}-\frac {5 \left (3117947360000 \,{\mathrm e}^{5}-1802240 \,{\mathrm e}^{20}[phone] \,{\mathrm e}^{15}-61843584000 \,{\mathrm e}^{10}-58948692275000\right )}{\left (-366025+9680 \,{\mathrm e}^{5}-64 \,{\mathrm e}^{10}\right )^{3} \left (2 x -11\right )}-\frac {5 \left (-42871776200000 \,{\mathrm e}^{5}[phone] \,{\mathrm e}^{20}-14992384000 \,{\mathrm e}^{15}[phone] \,{\mathrm e}^{10}-262144 \,{\mathrm e}^{25}[phone]\right )}{2 \left (-366025+9680 \,{\mathrm e}^{5}-64 \,{\mathrm e}^{10}\right )^{3} \left (2 x -11\right )^{2}}\)	\(250\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((40*exp(5)-200*x^2+550*x)/((32*x^3-528*x^2+2904*x-5324)*exp(5)^2+(-160*x^5+2640*x^4-14520*x^3+26620*x^2)*e
xp(5)+200*x^7-3300*x^6+18150*x^5-33275*x^4),x,method=_RETURNVERBOSE)

[Out]

-5/(2*x-11)^2/(-5*x^2+2*exp(5))

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maxima [A]  time = 0.38, size = 35, normalized size = 1.35 \begin {gather*} \frac {5}{20 \, x^{4} - 220 \, x^{3} - x^{2} {\left (8 \, e^{5} - 605\right )} + 88 \, x e^{5} - 242 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*exp(5)-200*x^2+550*x)/((32*x^3-528*x^2+2904*x-5324)*exp(5)^2+(-160*x^5+2640*x^4-14520*x^3+26620*
x^2)*exp(5)+200*x^7-3300*x^6+18150*x^5-33275*x^4),x, algorithm="maxima")

[Out]

5/(20*x^4 - 220*x^3 - x^2*(8*e^5 - 605) + 88*x*e^5 - 242*e^5)

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mupad [B]  time = 0.18, size = 66, normalized size = 2.54 \begin {gather*} -\frac {20}{\left (8\,{\mathrm {e}}^5-605\right )\,{\left (2\,x-11\right )}^2}-\frac {2200}{{\left (8\,{\mathrm {e}}^5-605\right )}^2\,\left (2\,x-11\right )}-\frac {25\,\left (220\,x+8\,{\mathrm {e}}^5+605\right )}{{\left (8\,{\mathrm {e}}^5-605\right )}^2\,\left (2\,{\mathrm {e}}^5-5\,x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((550*x + 40*exp(5) - 200*x^2)/(exp(10)*(2904*x - 528*x^2 + 32*x^3 - 5324) - 33275*x^4 + 18150*x^5 - 3300*x
^6 + 200*x^7 + exp(5)*(26620*x^2 - 14520*x^3 + 2640*x^4 - 160*x^5)),x)

[Out]

- 20/((8*exp(5) - 605)*(2*x - 11)^2) - 2200/((8*exp(5) - 605)^2*(2*x - 11)) - (25*(220*x + 8*exp(5) + 605))/((
8*exp(5) - 605)^2*(2*exp(5) - 5*x^2))

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sympy [A]  time = 0.96, size = 32, normalized size = 1.23 \begin {gather*} \frac {5}{20 x^{4} - 220 x^{3} + x^{2} \left (605 - 8 e^{5}\right ) + 88 x e^{5} - 242 e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*exp(5)-200*x**2+550*x)/((32*x**3-528*x**2+2904*x-5324)*exp(5)**2+(-160*x**5+2640*x**4-14520*x**3
+26620*x**2)*exp(5)+200*x**7-3300*x**6+18150*x**5-33275*x**4),x)

[Out]

5/(20*x**4 - 220*x**3 + x**2*(605 - 8*exp(5)) + 88*x*exp(5) - 242*exp(5))

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