3.23.55 \(\int \frac {-12150+22680 x-17550 x^2+7200 x^3-1650 x^4+200 x^5-10 x^6+e^5 (-768-768 x+2592 x^2-2016 x^3-735 x^4+1993 x^5-1350 x^6+450 x^7-75 x^8+5 x^9)}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx\)

Optimal. Leaf size=28 \[ -5 (-5+x)^2+e^5 x \left (-\frac {16}{(3-x)^2}+x^2\right )^2 \]

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Rubi [B]  time = 0.15, antiderivative size = 71, normalized size of antiderivative = 2.54, number of steps used = 2, number of rules used = 1, integrand size = 105, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {2074} \begin {gather*} e^5 x^5-5 x^2+2 \left (25-16 e^5\right ) x+\frac {864 e^5}{3-x}-\frac {864 e^5}{(3-x)^2}-\frac {256 e^5}{(3-x)^3}+\frac {768 e^5}{(3-x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-12150 + 22680*x - 17550*x^2 + 7200*x^3 - 1650*x^4 + 200*x^5 - 10*x^6 + E^5*(-768 - 768*x + 2592*x^2 - 20
16*x^3 - 735*x^4 + 1993*x^5 - 1350*x^6 + 450*x^7 - 75*x^8 + 5*x^9))/(-243 + 405*x - 270*x^2 + 90*x^3 - 15*x^4
+ x^5),x]

[Out]

(768*E^5)/(3 - x)^4 - (256*E^5)/(3 - x)^3 - (864*E^5)/(3 - x)^2 + (864*E^5)/(3 - x) + 2*(25 - 16*E^5)*x - 5*x^
2 + E^5*x^5

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 (-2 \left (-25+16 e^5\right )-\frac {3072 e^5}{(-3+x)^5}-\frac {768 e^5}{(-3+x)^4}+\frac {1728 e^5}{(-3+x)^3}+\frac {864 e^5}{(-3+x)^2}-10 x+5 e^5 x^4\right ) \, dx\\ &=\frac {768 e^5}{(3-x)^4}-\frac {256 e^5}{(3-x)^3}-\frac {864 e^5}{(3-x)^2}+\frac {864 e^5}{3-x}+2 \left (25-16 e^5\right ) x-5 x^2+e^5 x^5\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.04, size = 59, normalized size = 2.11 \begin {gather*} -5 (-10+x) x+\frac {e^5 \left (3645-4604 x+2430 x^2-828 x^3+237 x^4+49 x^5-108 x^6+54 x^7-12 x^8+x^9\right )}{(-3+x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12150 + 22680*x - 17550*x^2 + 7200*x^3 - 1650*x^4 + 200*x^5 - 10*x^6 + E^5*(-768 - 768*x + 2592*x^
2 - 2016*x^3 - 735*x^4 + 1993*x^5 - 1350*x^6 + 450*x^7 - 75*x^8 + 5*x^9))/(-243 + 405*x - 270*x^2 + 90*x^3 - 1
5*x^4 + x^5),x]

[Out]

-5*(-10 + x)*x + (E^5*(3645 - 4604*x + 2430*x^2 - 828*x^3 + 237*x^4 + 49*x^5 - 108*x^6 + 54*x^7 - 12*x^8 + x^9
))/(-3 + x)^4

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fricas [B]  time = 0.81, size = 98, normalized size = 3.50 \begin {gather*} -\frac {5 \, x^{6} - 110 \, x^{5} + 870 \, x^{4} - 3240 \, x^{3} + 5805 \, x^{2} - {\left (x^{9} - 12 \, x^{8} + 54 \, x^{7} - 108 \, x^{6} + 49 \, x^{5} + 384 \, x^{4} - 2592 \, x^{3} + 10368 \, x^{2} - 20480 \, x + 15552\right )} e^{5} - 4050 \, x}{x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^9-75*x^8+450*x^7-1350*x^6+1993*x^5-735*x^4-2016*x^3+2592*x^2-768*x-768)*exp(5)-10*x^6+200*x^5-
1650*x^4+7200*x^3-17550*x^2+22680*x-12150)/(x^5-15*x^4+90*x^3-270*x^2+405*x-243),x, algorithm="fricas")

[Out]

-(5*x^6 - 110*x^5 + 870*x^4 - 3240*x^3 + 5805*x^2 - (x^9 - 12*x^8 + 54*x^7 - 108*x^6 + 49*x^5 + 384*x^4 - 2592
*x^3 + 10368*x^2 - 20480*x + 15552)*e^5 - 4050*x)/(x^4 - 12*x^3 + 54*x^2 - 108*x + 81)

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giac [B]  time = 0.19, size = 51, normalized size = 1.82 \begin {gather*} x^{5} e^{5} - 5 \, x^{2} - 32 \, x e^{5} + 50 \, x - \frac {32 \, {\left (27 \, x^{3} e^{5} - 216 \, x^{2} e^{5} + 559 \, x e^{5} - 486 \, e^{5}\right )}}{{\left (x - 3\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^9-75*x^8+450*x^7-1350*x^6+1993*x^5-735*x^4-2016*x^3+2592*x^2-768*x-768)*exp(5)-10*x^6+200*x^5-
1650*x^4+7200*x^3-17550*x^2+22680*x-12150)/(x^5-15*x^4+90*x^3-270*x^2+405*x-243),x, algorithm="giac")

[Out]

x^5*e^5 - 5*x^2 - 32*x*e^5 + 50*x - 32*(27*x^3*e^5 - 216*x^2*e^5 + 559*x*e^5 - 486*e^5)/(x - 3)^4

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maple [B]  time = 0.08, size = 57, normalized size = 2.04

method	result	size
default	\(x^{5} {\mathrm e}^{5}-5 x^{2}+50 x -32 x \,{\mathrm e}^{5}-\frac {864 \,{\mathrm e}^{5}}{\left (x -3\right )^{2}}-\frac {864 \,{\mathrm e}^{5}}{x -3}+\frac {768 \,{\mathrm e}^{5}}{\left (x -3\right )^{4}}+\frac {256 \,{\mathrm e}^{5}}{\left (x -3\right )^{3}}\)	\(57\)
risch	\(x^{5} {\mathrm e}^{5}-32 x \,{\mathrm e}^{5}-5 x^{2}+50 x +\frac {-864 x^{3} {\mathrm e}^{5}+6912 x^{2} {\mathrm e}^{5}-17888 x \,{\mathrm e}^{5}+15552 \,{\mathrm e}^{5}}{x^{4}-12 x^{3}+54 x^{2}-108 x +81}\)	\(66\)
norman	\(\frac {{\mathrm e}^{5} x^{9}-12 x^{8} {\mathrm e}^{5}+54 x^{7} {\mathrm e}^{5}+\left (-5-108 \,{\mathrm e}^{5}\right ) x^{6}+\left (110+49 \,{\mathrm e}^{5}\right ) x^{5}+70470+\left (-7200+2016 \,{\mathrm e}^{5}\right ) x^{3}+\left (41175-10368 \,{\mathrm e}^{5}\right ) x^{2}+\left (-89910+20992 \,{\mathrm e}^{5}\right ) x -15552 \,{\mathrm e}^{5}}{\left (x -3\right )^{4}}\)	\(81\)
gosper	\(\frac {{\mathrm e}^{5} x^{9}-12 x^{8} {\mathrm e}^{5}+54 x^{7} {\mathrm e}^{5}-108 x^{6} {\mathrm e}^{5}+49 x^{5} {\mathrm e}^{5}-5 x^{6}+110 x^{5}+2016 x^{3} {\mathrm e}^{5}-10368 x^{2} {\mathrm e}^{5}-7200 x^{3}+20992 x \,{\mathrm e}^{5}+41175 x^{2}-15552 \,{\mathrm e}^{5}-89910 x +70470}{x^{4}-12 x^{3}+54 x^{2}-108 x +81}\)	\(104\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^9-75*x^8+450*x^7-1350*x^6+1993*x^5-735*x^4-2016*x^3+2592*x^2-768*x-768)*exp(5)-10*x^6+200*x^5-1650*x
^4+7200*x^3-17550*x^2+22680*x-12150)/(x^5-15*x^4+90*x^3-270*x^2+405*x-243),x,method=_RETURNVERBOSE)

[Out]

x^5*exp(5)-5*x^2+50*x-32*x*exp(5)-864*exp(5)/(x-3)^2-864/(x-3)*exp(5)+768*exp(5)/(x-3)^4+256*exp(5)/(x-3)^3

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maxima [B]  time = 0.36, size = 67, normalized size = 2.39 \begin {gather*} x^{5} e^{5} - 5 \, x^{2} - 2 \, x {\left (16 \, e^{5} - 25\right )} - \frac {32 \, {\left (27 \, x^{3} e^{5} - 216 \, x^{2} e^{5} + 559 \, x e^{5} - 486 \, e^{5}\right )}}{x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^9-75*x^8+450*x^7-1350*x^6+1993*x^5-735*x^4-2016*x^3+2592*x^2-768*x-768)*exp(5)-10*x^6+200*x^5-
1650*x^4+7200*x^3-17550*x^2+22680*x-12150)/(x^5-15*x^4+90*x^3-270*x^2+405*x-243),x, algorithm="maxima")

[Out]

x^5*e^5 - 5*x^2 - 2*x*(16*e^5 - 25) - 32*(27*x^3*e^5 - 216*x^2*e^5 + 559*x*e^5 - 486*e^5)/(x^4 - 12*x^3 + 54*x
^2 - 108*x + 81)

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mupad [B]  time = 1.34, size = 66, normalized size = 2.36 \begin {gather*} x^5\,{\mathrm {e}}^5+\frac {-864\,{\mathrm {e}}^5\,x^3+6912\,{\mathrm {e}}^5\,x^2-17888\,{\mathrm {e}}^5\,x+15552\,{\mathrm {e}}^5}{x^4-12\,x^3+54\,x^2-108\,x+81}-5\,x^2-x\,\left (32\,{\mathrm {e}}^5-50\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(5)*(768*x - 2592*x^2 + 2016*x^3 + 735*x^4 - 1993*x^5 + 1350*x^6 - 450*x^7 + 75*x^8 - 5*x^9 + 768) -
22680*x + 17550*x^2 - 7200*x^3 + 1650*x^4 - 200*x^5 + 10*x^6 + 12150)/(405*x - 270*x^2 + 90*x^3 - 15*x^4 + x^5
- 243),x)

[Out]

x^5*exp(5) + (15552*exp(5) - 17888*x*exp(5) + 6912*x^2*exp(5) - 864*x^3*exp(5))/(54*x^2 - 108*x - 12*x^3 + x^4
+ 81) - 5*x^2 - x*(32*exp(5) - 50)

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sympy [B]  time = 0.32, size = 66, normalized size = 2.36 \begin {gather*} x^{5} e^{5} - 5 x^{2} + x \left (50 - 32 e^{5}\right ) + \frac {- 864 x^{3} e^{5} + 6912 x^{2} e^{5} - 17888 x e^{5} + 15552 e^{5}}{x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x**9-75*x**8+450*x**7-1350*x**6+1993*x**5-735*x**4-2016*x**3+2592*x**2-768*x-768)*exp(5)-10*x**6
+200*x**5-1650*x**4+7200*x**3-17550*x**2+22680*x-12150)/(x**5-15*x**4+90*x**3-270*x**2+405*x-243),x)

[Out]

x**5*exp(5) - 5*x**2 + x*(50 - 32*exp(5)) + (-864*x**3*exp(5) + 6912*x**2*exp(5) - 17888*x*exp(5) + 15552*exp(
5))/(x**4 - 12*x**3 + 54*x**2 - 108*x + 81)

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