3.95.57 \(\int \frac {e^5 (-2100+2370 x+160 x^2)}{33075 x^3-49770 x^4+16203 x^5+1896 x^6+48 x^7} \, dx\)

Optimal. Leaf size=22 \[ -\frac {10 e^5}{3 x^2 (21+x) (-5+4 x)} \]

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Rubi [B]  time = 0.06, antiderivative size = 47, normalized size of antiderivative = 2.14, number of steps used = 3, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {12, 2074} \begin {gather*} \frac {2 e^5}{63 x^2}+\frac {158 e^5}{6615 x}+\frac {10 e^5}{117747 (x+21)}+\frac {128 e^5}{1335 (5-4 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^5*(-2100 + 2370*x + 160*x^2))/(33075*x^3 - 49770*x^4 + 16203*x^5 + 1896*x^6 + 48*x^7),x]

[Out]

(128*E^5)/(1335*(5 - 4*x)) + (2*E^5)/(63*x^2) + (158*E^5)/(6615*x) + (10*E^5)/(117747*(21 + x))

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} &=e^5 \int \frac {-2100+2370 x+160 x^2}{33075 x^3-49770 x^4+16203 x^5+1896 x^6+48 x^7} \, dx\\ &=e^5 \int \left (-\frac {4}{63 x^3}-\frac {158}{6615 x^2}-\frac {10}{117747 (21+x)^2}+\frac {512}{1335 (-5+4 x)^2}\right ) \, dx\\ &=\frac {128 e^5}{1335 (5-4 x)}+\frac {2 e^5}{63 x^2}+\frac {158 e^5}{6615 x}+\frac {10 e^5}{117747 (21+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.14 \begin {gather*} \frac {10 e^5}{3 \left (105 x^2-79 x^3-4 x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^5*(-2100 + 2370*x + 160*x^2))/(33075*x^3 - 49770*x^4 + 16203*x^5 + 1896*x^6 + 48*x^7),x]

[Out]

(10*E^5)/(3*(105*x^2 - 79*x^3 - 4*x^4))

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fricas [A]  time = 0.59, size = 22, normalized size = 1.00 \begin {gather*} -\frac {10 \, e^{5}}{3 \, {\left (4 \, x^{4} + 79 \, x^{3} - 105 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((160*x^2+2370*x-2100)*exp(5)/(48*x^7+1896*x^6+16203*x^5-49770*x^4+33075*x^3),x, algorithm="fricas")

[Out]

-10/3*e^5/(4*x^4 + 79*x^3 - 105*x^2)

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giac [A]  time = 0.18, size = 33, normalized size = 1.50 \begin {gather*} -\frac {2}{6615} \, {\left (\frac {316 \, x + 6661}{4 \, x^{2} + 79 \, x - 105} - \frac {79 \, x + 105}{x^{2}}\right )} e^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((160*x^2+2370*x-2100)*exp(5)/(48*x^7+1896*x^6+16203*x^5-49770*x^4+33075*x^3),x, algorithm="giac")

[Out]

-2/6615*((316*x + 6661)/(4*x^2 + 79*x - 105) - (79*x + 105)/x^2)*e^5

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maple [A]  time = 0.04, size = 20, normalized size = 0.91

method	result	size
gosper	\(-\frac {10 \,{\mathrm e}^{5}}{3 x^{2} \left (4 x^{2}+79 x -105\right )}\)	\(20\)
norman	\(-\frac {10 \,{\mathrm e}^{5}}{3 x^{2} \left (4 x^{2}+79 x -105\right )}\)	\(20\)
risch	\(-\frac {10 \,{\mathrm e}^{5}}{3 x^{2} \left (4 x^{2}+79 x -105\right )}\)	\(20\)
default	\(\frac {10 \,{\mathrm e}^{5} \left (\frac {1}{105 x^{2}}+\frac {79}{11025 x}+\frac {1}{39249 x +824229}-\frac {64}{2225 \left (4 x -5\right )}\right )}{3}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((160*x^2+2370*x-2100)*exp(5)/(48*x^7+1896*x^6+16203*x^5-49770*x^4+33075*x^3),x,method=_RETURNVERBOSE)

[Out]

-10/3/x^2*exp(5)/(4*x^2+79*x-105)

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maxima [A]  time = 0.35, size = 22, normalized size = 1.00 \begin {gather*} -\frac {10 \, e^{5}}{3 \, {\left (4 \, x^{4} + 79 \, x^{3} - 105 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((160*x^2+2370*x-2100)*exp(5)/(48*x^7+1896*x^6+16203*x^5-49770*x^4+33075*x^3),x, algorithm="maxima")

[Out]

-10/3*e^5/(4*x^4 + 79*x^3 - 105*x^2)

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mupad [B]  time = 8.32, size = 22, normalized size = 1.00 \begin {gather*} -\frac {10\,{\mathrm {e}}^5}{12\,x^4+237\,x^3-315\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(5)*(2370*x + 160*x^2 - 2100))/(33075*x^3 - 49770*x^4 + 16203*x^5 + 1896*x^6 + 48*x^7),x)

[Out]

-(10*exp(5))/(237*x^3 - 315*x^2 + 12*x^4)

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sympy [A]  time = 0.26, size = 20, normalized size = 0.91 \begin {gather*} - \frac {10 e^{5}}{12 x^{4} + 237 x^{3} - 315 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((160*x**2+2370*x-2100)*exp(5)/(48*x**7+1896*x**6+16203*x**5-49770*x**4+33075*x**3),x)

[Out]

-10*exp(5)/(12*x**4 + 237*x**3 - 315*x**2)

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