3.101.59 \(\int \frac {-3125+20625 x-49125 x^2+49775 x^3-19650 x^4+3300 x^5-200 x^6+e^{10} (-5+33 x-10 x^2)}{125 x^2-825 x^3+1965 x^4-1991 x^5+786 x^6-132 x^7+8 x^8} \, dx\)

Optimal. Leaf size=27 \[ \frac {25+\frac {e^{10}}{(1-2 x)^2 (5-x)^2}-x}{x} \]

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Rubi [B]  time = 0.20, antiderivative size = 69, normalized size of antiderivative = 2.56, number of steps used = 2, number of rules used = 1, integrand size = 83, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {2074} \begin {gather*} \frac {29 e^{10}}{18225 (5-x)}+\frac {625+e^{10}}{25 x}+\frac {e^{10}}{405 (5-x)^2}+\frac {56 e^{10}}{729 (1-2 x)}+\frac {8 e^{10}}{81 (1-2 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3125 + 20625*x - 49125*x^2 + 49775*x^3 - 19650*x^4 + 3300*x^5 - 200*x^6 + E^10*(-5 + 33*x - 10*x^2))/(12
5*x^2 - 825*x^3 + 1965*x^4 - 1991*x^5 + 786*x^6 - 132*x^7 + 8*x^8),x]

[Out]

(8*E^10)/(81*(1 - 2*x)^2) + (56*E^10)/(729*(1 - 2*x)) + E^10/(405*(5 - x)^2) + (29*E^10)/(18225*(5 - x)) + (62
5 + E^10)/(25*x)

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 {2 e^{10}}{405 (-5+x)^3}+\frac {29 e^{10}}{18225 (-5+x)^2}+\frac {-625-e^{10}}{25 x^2}-\frac {32 e^{10}}{81 (-1+2 x)^3}+\frac {112 e^{10}}{729 (-1+2 x)^2}\right ) \, dx\\ &=\frac {8 e^{10}}{81 (1-2 x)^2}+\frac {56 e^{10}}{729 (1-2 x)}+\frac {e^{10}}{405 (5-x)^2}+\frac {29 e^{10}}{18225 (5-x)}+\frac {625+e^{10}}{25 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 34, normalized size = 1.26 \begin {gather*} \frac {e^{10}+25 \left (5-11 x+2 x^2\right )^2}{x \left (5-11 x+2 x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3125 + 20625*x - 49125*x^2 + 49775*x^3 - 19650*x^4 + 3300*x^5 - 200*x^6 + E^10*(-5 + 33*x - 10*x^2
))/(125*x^2 - 825*x^3 + 1965*x^4 - 1991*x^5 + 786*x^6 - 132*x^7 + 8*x^8),x]

[Out]

(E^10 + 25*(5 - 11*x + 2*x^2)^2)/(x*(5 - 11*x + 2*x^2)^2)

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fricas [B]  time = 2.18, size = 49, normalized size = 1.81 \begin {gather*} \frac {100 \, x^{4} - 1100 \, x^{3} + 3525 \, x^{2} - 2750 \, x + e^{10} + 625}{4 \, x^{5} - 44 \, x^{4} + 141 \, x^{3} - 110 \, x^{2} + 25 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^2+33*x-5)*exp(5)^2-200*x^6+3300*x^5-19650*x^4+49775*x^3-49125*x^2+20625*x-3125)/(8*x^8-132*x
^7+786*x^6-1991*x^5+1965*x^4-825*x^3+125*x^2),x, algorithm="fricas")

[Out]

(100*x^4 - 1100*x^3 + 3525*x^2 - 2750*x + e^10 + 625)/(4*x^5 - 44*x^4 + 141*x^3 - 110*x^2 + 25*x)

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giac [A]  time = 0.12, size = 48, normalized size = 1.78 \begin {gather*} \frac {e^{10} + 625}{25 \, x} - \frac {4 \, x^{3} e^{10} - 44 \, x^{2} e^{10} + 141 \, x e^{10} - 110 \, e^{10}}{25 \, {\left (2 \, x^{2} - 11 \, x + 5\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^2+33*x-5)*exp(5)^2-200*x^6+3300*x^5-19650*x^4+49775*x^3-49125*x^2+20625*x-3125)/(8*x^8-132*x
^7+786*x^6-1991*x^5+1965*x^4-825*x^3+125*x^2),x, algorithm="giac")

[Out]

1/25*(e^10 + 625)/x - 1/25*(4*x^3*e^10 - 44*x^2*e^10 + 141*x*e^10 - 110*e^10)/(2*x^2 - 11*x + 5)^2

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maple [A]  time = 0.08, size = 41, normalized size = 1.52

method	result	size
norman	\(\frac {100 x^{4}-1100 x^{3}+{\mathrm e}^{10}+3525 x^{2}-2750 x +625}{x \left (2 x^{2}-11 x +5\right )^{2}}\)	\(41\)
gosper	\(\frac {100 x^{4}-1100 x^{3}+{\mathrm e}^{10}+3525 x^{2}-2750 x +625}{x \left (4 x^{4}-44 x^{3}+141 x^{2}-110 x +25\right )}\)	\(51\)
risch	\(\frac {100 x^{4}-1100 x^{3}+{\mathrm e}^{10}+3525 x^{2}-2750 x +625}{x \left (4 x^{4}-44 x^{3}+141 x^{2}-110 x +25\right )}\)	\(52\)
default	\(\frac {8 \,{\mathrm e}^{10}}{81 \left (2 x -1\right )^{2}}-\frac {56 \,{\mathrm e}^{10}}{729 \left (2 x -1\right )}-\frac {-\frac {{\mathrm e}^{10}}{25}-25}{x}+\frac {{\mathrm e}^{10}}{405 \left (x -5\right )^{2}}-\frac {29 \,{\mathrm e}^{10}}{18225 \left (x -5\right )}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-10*x^2+33*x-5)*exp(5)^2-200*x^6+3300*x^5-19650*x^4+49775*x^3-49125*x^2+20625*x-3125)/(8*x^8-132*x^7+786
*x^6-1991*x^5+1965*x^4-825*x^3+125*x^2),x,method=_RETURNVERBOSE)

[Out]

(100*x^4-1100*x^3+exp(5)^2+3525*x^2-2750*x+625)/x/(2*x^2-11*x+5)^2

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maxima [B]  time = 0.36, size = 49, normalized size = 1.81 \begin {gather*} \frac {100 \, x^{4} - 1100 \, x^{3} + 3525 \, x^{2} - 2750 \, x + e^{10} + 625}{4 \, x^{5} - 44 \, x^{4} + 141 \, x^{3} - 110 \, x^{2} + 25 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^2+33*x-5)*exp(5)^2-200*x^6+3300*x^5-19650*x^4+49775*x^3-49125*x^2+20625*x-3125)/(8*x^8-132*x
^7+786*x^6-1991*x^5+1965*x^4-825*x^3+125*x^2),x, algorithm="maxima")

[Out]

(100*x^4 - 1100*x^3 + 3525*x^2 - 2750*x + e^10 + 625)/(4*x^5 - 44*x^4 + 141*x^3 - 110*x^2 + 25*x)

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mupad [B]  time = 6.39, size = 53, normalized size = 1.96 \begin {gather*} \frac {8\,{\mathrm {e}}^{10}}{81\,{\left (2\,x-1\right )}^2}-\frac {56\,{\mathrm {e}}^{10}}{729\,\left (2\,x-1\right )}+\frac {\frac {{\mathrm {e}}^{10}}{25}+25}{x}-\frac {29\,{\mathrm {e}}^{10}}{18225\,\left (x-5\right )}+\frac {{\mathrm {e}}^{10}}{405\,{\left (x-5\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(10)*(10*x^2 - 33*x + 5) - 20625*x + 49125*x^2 - 49775*x^3 + 19650*x^4 - 3300*x^5 + 200*x^6 + 3125)/(
125*x^2 - 825*x^3 + 1965*x^4 - 1991*x^5 + 786*x^6 - 132*x^7 + 8*x^8),x)

[Out]

(8*exp(10))/(81*(2*x - 1)^2) - (56*exp(10))/(729*(2*x - 1)) + (exp(10)/25 + 25)/x - (29*exp(10))/(18225*(x - 5
)) + exp(10)/(405*(x - 5)^2)

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sympy [B]  time = 0.74, size = 48, normalized size = 1.78 \begin {gather*} - \frac {- 100 x^{4} + 1100 x^{3} - 3525 x^{2} + 2750 x - e^{10} - 625}{4 x^{5} - 44 x^{4} + 141 x^{3} - 110 x^{2} + 25 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x**2+33*x-5)*exp(5)**2-200*x**6+3300*x**5-19650*x**4+49775*x**3-49125*x**2+20625*x-3125)/(8*x*
*8-132*x**7+786*x**6-1991*x**5+1965*x**4-825*x**3+125*x**2),x)

[Out]

-(-100*x**4 + 1100*x**3 - 3525*x**2 + 2750*x - exp(10) - 625)/(4*x**5 - 44*x**4 + 141*x**3 - 110*x**2 + 25*x)

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