3.23.80 \(\int \frac {97-25 x-18 x^2+50 x^3}{e (-54+1350 x-11250 x^2+31250 x^3)} \, dx\)

Optimal. Leaf size=27 \[ 3+\frac {\frac {-2+x}{2 x^2}+x}{e \left (-25+\frac {3}{x}\right )^2} \]

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Rubi [A]  time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.37, number of steps used = 3, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {12, 2074} \begin {gather*} \frac {x}{625 e}-\frac {679}{31250 e (3-25 x)}-\frac {29321}{31250 e (3-25 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(97 - 25*x - 18*x^2 + 50*x^3)/(E*(-54 + 1350*x - 11250*x^2 + 31250*x^3)),x]

[Out]

-29321/(31250*E*(3 - 25*x)^2) - 679/(31250*E*(3 - 25*x)) + x/(625*E)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {\int \frac {97-25 x-18 x^2+50 x^3}{-54+1350 x-11250 x^2+31250 x^3} \, dx}{e}\\ &=\frac {\int \left (\frac {1}{625}+\frac {29321}{625 (-3+25 x)^3}-\frac {679}{1250 (-3+25 x)^2}\right ) \, dx}{e}\\ &=-\frac {29321}{31250 e (3-25 x)^2}-\frac {679}{31250 e (3-25 x)}+\frac {x}{625 e}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.07 \begin {gather*} \frac {-31412+18325 x-11250 x^2+31250 x^3}{31250 e (3-25 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(97 - 25*x - 18*x^2 + 50*x^3)/(E*(-54 + 1350*x - 11250*x^2 + 31250*x^3)),x]

[Out]

(-31412 + 18325*x - 11250*x^2 + 31250*x^3)/(31250*E*(3 - 25*x)^2)

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fricas [A]  time = 1.05, size = 31, normalized size = 1.15 \begin {gather*} \frac {{\left (31250 \, x^{3} - 7500 \, x^{2} + 17425 \, x - 31358\right )} e^{\left (-1\right )}}{31250 \, {\left (625 \, x^{2} - 150 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x^3-18*x^2-25*x+97)/(31250*x^3-11250*x^2+1350*x-54)/exp(1),x, algorithm="fricas")

[Out]

1/31250*(31250*x^3 - 7500*x^2 + 17425*x - 31358)*e^(-1)/(625*x^2 - 150*x + 9)

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giac [A]  time = 0.31, size = 21, normalized size = 0.78 \begin {gather*} \frac {1}{31250} \, {\left (50 \, x + \frac {16975 \, x - 31358}{{\left (25 \, x - 3\right )}^{2}}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x^3-18*x^2-25*x+97)/(31250*x^3-11250*x^2+1350*x-54)/exp(1),x, algorithm="giac")

[Out]

1/31250*(50*x + (16975*x - 31358)/(25*x - 3)^2)*e^(-1)

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maple [A]  time = 0.05, size = 25, normalized size = 0.93

method	result	size
risch	\(\frac {{\mathrm e}^{-1} x}{625}+\frac {{\mathrm e}^{-1} \left (\frac {679 x}{781250}-\frac {15679}{9765625}\right )}{x^{2}-\frac {6}{25} x +\frac {9}{625}}\)	\(25\)
default	\(\frac {{\mathrm e}^{-1} \left (\frac {2 x}{625}-\frac {29321}{15625 \left (25 x -3\right )^{2}}+\frac {679}{15625 \left (25 x -3\right )}\right )}{2}\)	\(29\)
gosper	\(\frac {x \left (18 x^{2}+1250 x -291\right ) {\mathrm e}^{-1}}{11250 x^{2}-2700 x +162}\)	\(30\)
norman	\(\frac {-\frac {97 \,{\mathrm e}^{-1} x}{6}+\frac {625 x^{2} {\mathrm e}^{-1}}{9}+{\mathrm e}^{-1} x^{3}}{\left (25 x -3\right )^{2}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((50*x^3-18*x^2-25*x+97)/(31250*x^3-11250*x^2+1350*x-54)/exp(1),x,method=_RETURNVERBOSE)

[Out]

1/625*exp(-1)*x+exp(-1)*(679/781250*x-15679/9765625)/(x^2-6/25*x+9/625)

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maxima [A]  time = 0.47, size = 26, normalized size = 0.96 \begin {gather*} \frac {1}{31250} \, {\left (50 \, x + \frac {16975 \, x - 31358}{625 \, x^{2} - 150 \, x + 9}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x^3-18*x^2-25*x+97)/(31250*x^3-11250*x^2+1350*x-54)/exp(1),x, algorithm="maxima")

[Out]

1/31250*(50*x + (16975*x - 31358)/(625*x^2 - 150*x + 9))*e^(-1)

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mupad [B]  time = 1.29, size = 25, normalized size = 0.93 \begin {gather*} \frac {x\,{\mathrm {e}}^{-1}}{625}-\frac {\frac {15679\,{\mathrm {e}}^{-1}}{15625}-\frac {679\,x\,{\mathrm {e}}^{-1}}{1250}}{{\left (25\,x-3\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-1)*(25*x + 18*x^2 - 50*x^3 - 97))/(1350*x - 11250*x^2 + 31250*x^3 - 54),x)

[Out]

(x*exp(-1))/625 - ((15679*exp(-1))/15625 - (679*x*exp(-1))/1250)/(25*x - 3)^2

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sympy [A]  time = 0.18, size = 31, normalized size = 1.15 \begin {gather*} \frac {x}{625 e} + \frac {16975 x - 31358}{19531250 e x^{2} - 4687500 e x + 281250 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x**3-18*x**2-25*x+97)/(31250*x**3-11250*x**2+1350*x-54)/exp(1),x)

[Out]

x*exp(-1)/625 + (16975*x - 31358)/(19531250*E*x**2 - 4687500*E*x + 281250*E)

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