3.15.92 \(\int \frac {e^3 (160-40 x^4)+e^6 (-48 x+13 x^3+4 x^5+e^2 (-32+8 x^4))}{25 x^3+e^3 (-10 e^2 x^3-10 x^4)+e^6 (e^4 x^3+2 e^2 x^4+x^5)} \, dx\)

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

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Rubi [B]  time = 0.14, antiderivative size = 109, normalized size of antiderivative = 4.19, number of steps used = 2, number of rules used = 1, integrand size = 91, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {2074} \begin {gather*} -\frac {16 e^3}{\left (5-e^5\right ) x^2}+4 x-\frac {2500-2000 e^5-325 e^6+600 e^{10}+130 e^{11}+16 e^{12}-80 e^{15}-13 e^{16}+4 e^{20}}{e^3 \left (5-e^5\right )^2 \left (-e^3 x-e^5+5\right )}-\frac {16 e^6}{\left (5-e^5\right )^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^3*(160 - 40*x^4) + E^6*(-48*x + 13*x^3 + 4*x^5 + E^2*(-32 + 8*x^4)))/(25*x^3 + E^3*(-10*E^2*x^3 - 10*x^
4) + E^6*(E^4*x^3 + 2*E^2*x^4 + x^5)),x]

[Out]

(-16*E^3)/((5 - E^5)*x^2) - (16*E^6)/((5 - E^5)^2*x) + 4*x - (2500 - 2000*E^5 - 325*E^6 + 600*E^10 + 130*E^11
+ 16*E^12 - 80*E^15 - 13*E^16 + 4*E^20)/(E^3*(5 - E^5)^2*(5 - E^5 - E^3*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 (4-\frac {32 e^3}{\left (-5+e^5\right ) x^3}+\frac {16 e^6}{\left (-5+e^5\right )^2 x^2}+\frac {-2500+2000 e^5+325 e^6-600 e^{10}-130 e^{11}-16 e^{12}+80 e^{15}+13 e^{16}-4 e^{20}}{\left (-5+e^5\right )^2 \left (-5+e^5+e^3 x\right )^2}\right ) \, dx\\ &=-\frac {16 e^3}{\left (5-e^5\right ) x^2}-\frac {16 e^6}{\left (5-e^5\right )^2 x}+4 x-\frac {2500-2000 e^5-325 e^6+600 e^{10}+130 e^{11}+16 e^{12}-80 e^{15}-13 e^{16}+4 e^{20}}{e^3 \left (5-e^5\right )^2 \left (5-e^5-e^3 x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.04, size = 73, normalized size = 2.81 \begin {gather*} \frac {100 x^2-40 e^5 x^2+4 e^{10} x^2-20 e^3 x^3+4 e^8 x^3+e^6 \left (16-13 x^2+4 x^4\right )}{e^3 x^2 \left (-5+e^5+e^3 x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^3*(160 - 40*x^4) + E^6*(-48*x + 13*x^3 + 4*x^5 + E^2*(-32 + 8*x^4)))/(25*x^3 + E^3*(-10*E^2*x^3 -
10*x^4) + E^6*(E^4*x^3 + 2*E^2*x^4 + x^5)),x]

[Out]

(100*x^2 - 40*E^5*x^2 + 4*E^10*x^2 - 20*E^3*x^3 + 4*E^8*x^3 + E^6*(16 - 13*x^2 + 4*x^4))/(E^3*x^2*(-5 + E^5 +
E^3*x))

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fricas [B]  time = 1.58, size = 72, normalized size = 2.77 \begin {gather*} \frac {4 \, x^{3} e^{8} - 20 \, x^{3} e^{3} + 4 \, x^{2} e^{10} - 40 \, x^{2} e^{5} + 100 \, x^{2} + {\left (4 \, x^{4} - 13 \, x^{2} + 16\right )} e^{6}}{x^{3} e^{6} + x^{2} e^{8} - 5 \, x^{2} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^4-32)*exp(2)+4*x^5+13*x^3-48*x)*exp(3)^2+(-40*x^4+160)*exp(3))/((x^3*exp(2)^2+2*x^4*exp(2)+x^
5)*exp(3)^2+(-10*x^3*exp(2)-10*x^4)*exp(3)+25*x^3),x, algorithm="fricas")

[Out]

(4*x^3*e^8 - 20*x^3*e^3 + 4*x^2*e^10 - 40*x^2*e^5 + 100*x^2 + (4*x^4 - 13*x^2 + 16)*e^6)/(x^3*e^6 + x^2*e^8 -
5*x^2*e^3)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^4-32)*exp(2)+4*x^5+13*x^3-48*x)*exp(3)^2+(-40*x^4+160)*exp(3))/((x^3*exp(2)^2+2*x^4*exp(2)+x^
5)*exp(3)^2+(-10*x^3*exp(2)-10*x^4)*exp(3)+25*x^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 4*sageVARx*exp(6)/exp(6)+(-1600*exp(2)^3
*exp(6)*exp(3)^2+320*exp(2)^2*exp(6)^2*exp(3)*exp(4)+8000*exp(2)^2*exp(6)*exp(3)+8000*exp(2)^2*exp(3)^3-16*exp
(2)*exp(6)^3*exp(4)^2

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maple [A]  time = 9.06, size = 32, normalized size = 1.23

method	result	size
gosper	\(\frac {\left (4 x^{4}-13 x^{2}+16\right ) {\mathrm e}^{3}}{x^{2} \left ({\mathrm e}^{2} {\mathrm e}^{3}+x \,{\mathrm e}^{3}-5\right )}\)	\(32\)
norman	\(\frac {-13 x^{2} {\mathrm e}^{3}+4 x^{4} {\mathrm e}^{3}+16 \,{\mathrm e}^{3}}{x^{2} \left ({\mathrm e}^{2} {\mathrm e}^{3}+x \,{\mathrm e}^{3}-5\right )}\)	\(37\)
risch	\(4 x +\frac {\left (4 \,{\mathrm e}^{10}-13 \,{\mathrm e}^{6}-40 \,{\mathrm e}^{5}+100\right ) {\mathrm e}^{-3} x^{2}+16 \,{\mathrm e}^{3}}{x^{2} \left ({\mathrm e}^{5}+x \,{\mathrm e}^{3}-5\right )}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((8*x^4-32)*exp(2)+4*x^5+13*x^3-48*x)*exp(3)^2+(-40*x^4+160)*exp(3))/((x^3*exp(2)^2+2*x^4*exp(2)+x^5)*exp
(3)^2+(-10*x^3*exp(2)-10*x^4)*exp(3)+25*x^3),x,method=_RETURNVERBOSE)

[Out]

1/x^2*(4*x^4-13*x^2+16)*exp(3)/(exp(2)*exp(3)+x*exp(3)-5)

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maxima [A]  time = 0.48, size = 48, normalized size = 1.85 \begin {gather*} 4 \, x + \frac {x^{2} {\left (4 \, e^{10} - 13 \, e^{6} - 40 \, e^{5} + 100\right )} + 16 \, e^{6}}{x^{3} e^{6} + x^{2} {\left (e^{8} - 5 \, e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^4-32)*exp(2)+4*x^5+13*x^3-48*x)*exp(3)^2+(-40*x^4+160)*exp(3))/((x^3*exp(2)^2+2*x^4*exp(2)+x^
5)*exp(3)^2+(-10*x^3*exp(2)-10*x^4)*exp(3)+25*x^3),x, algorithm="maxima")

[Out]

4*x + (x^2*(4*e^10 - 13*e^6 - 40*e^5 + 100) + 16*e^6)/(x^3*e^6 + x^2*(e^8 - 5*e^3))

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mupad [B]  time = 0.22, size = 48, normalized size = 1.85 \begin {gather*} 4\,x+\frac {16\,{\mathrm {e}}^3-x^2\,{\mathrm {e}}^{-3}\,\left (40\,{\mathrm {e}}^5+13\,{\mathrm {e}}^6-4\,{\mathrm {e}}^{10}-100\right )}{{\mathrm {e}}^3\,x^3+\left ({\mathrm {e}}^5-5\right )\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(3)*(40*x^4 - 160) - exp(6)*(exp(2)*(8*x^4 - 32) - 48*x + 13*x^3 + 4*x^5))/(exp(6)*(2*x^4*exp(2) + x^
3*exp(4) + x^5) + 25*x^3 - exp(3)*(10*x^3*exp(2) + 10*x^4)),x)

[Out]

4*x + (16*exp(3) - x^2*exp(-3)*(40*exp(5) + 13*exp(6) - 4*exp(10) - 100))/(x^3*exp(3) + x^2*(exp(5) - 5))

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sympy [B]  time = 5.84, size = 46, normalized size = 1.77 \begin {gather*} 4 x + \frac {x^{2} \left (- 40 e^{5} - 13 e^{6} + 100 + 4 e^{10}\right ) + 16 e^{6}}{x^{3} e^{6} + x^{2} \left (- 5 e^{3} + e^{8}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x**4-32)*exp(2)+4*x**5+13*x**3-48*x)*exp(3)**2+(-40*x**4+160)*exp(3))/((x**3*exp(2)**2+2*x**4*e
xp(2)+x**5)*exp(3)**2+(-10*x**3*exp(2)-10*x**4)*exp(3)+25*x**3),x)

[Out]

4*x + (x**2*(-40*exp(5) - 13*exp(6) + 100 + 4*exp(10)) + 16*exp(6))/(x**3*exp(6) + x**2*(-5*exp(3) + exp(8)))

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