3.97.5 \(\int \frac {-9 x^3-18 x^4-15 x^5-4 x^6+e^4 (27 x^2+36 x^3+25 x^4+6 x^5)}{e^{12}-3 e^8 x+3 e^4 x^2-x^3} \, dx\)

Optimal. Leaf size=24 \[ \frac {x^3 (3+x) \left (3+2 x+x^2\right )}{\left (-e^4+x\right )^2} \]

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Rubi [B]  time = 0.16, antiderivative size = 109, normalized size of antiderivative = 4.54, number of steps used = 2, number of rules used = 1, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {2074} \begin {gather*} x^4+\left (5+2 e^4\right ) x^3+\left (9+10 e^4+3 e^8\right ) x^2+\left (9+18 e^4+15 e^8+4 e^{12}\right ) x-\frac {e^8 \left (27+36 e^4+25 e^8+6 e^{12}\right )}{e^4-x}+\frac {e^{12} \left (9+9 e^4+5 e^8+e^{12}\right )}{\left (e^4-x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9*x^3 - 18*x^4 - 15*x^5 - 4*x^6 + E^4*(27*x^2 + 36*x^3 + 25*x^4 + 6*x^5))/(E^12 - 3*E^8*x + 3*E^4*x^2 -
x^3),x]

[Out]

(E^12*(9 + 9*E^4 + 5*E^8 + E^12))/(E^4 - x)^2 - (E^8*(27 + 36*E^4 + 25*E^8 + 6*E^12))/(E^4 - x) + (9 + 18*E^4
+ 15*E^8 + 4*E^12)*x + (9 + 10*E^4 + 3*E^8)*x^2 + (5 + 2*E^4)*x^3 + x^4

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 (9 \left (1+\frac {1}{9} e^4 \left (18+15 e^4+4 e^8\right )\right )+\frac {2 e^{12} \left (9+9 e^4+5 e^8+e^{12}\right )}{\left (e^4-x\right )^3}-\frac {e^8 \left (27+36 e^4+25 e^8+6 e^{12}\right )}{\left (e^4-x\right )^2}+2 \left (9+10 e^4+3 e^8\right ) x+3 \left (5+2 e^4\right ) x^2+4 x^3\right ) \, dx\\ &=\frac {e^{12} \left (9+9 e^4+5 e^8+e^{12}\right )}{\left (e^4-x\right )^2}-\frac {e^8 \left (27+36 e^4+25 e^8+6 e^{12}\right )}{e^4-x}+\left (9+18 e^4+15 e^8+4 e^{12}\right ) x+\left (9+10 e^4+3 e^8\right ) x^2+\left (5+2 e^4\right ) x^3+x^4\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.04, size = 88, normalized size = 3.67 \begin {gather*} \frac {-15 e^{24}-54 e^8 (-1+x) x-27 e^4 x^2+10 e^{20} (-5+3 x)+e^{12} \left (-27+108 x-50 x^2\right )+e^{16} \left (-54+100 x-15 x^2\right )+x^3 \left (9+9 x+5 x^2+x^3\right )}{\left (e^4-x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9*x^3 - 18*x^4 - 15*x^5 - 4*x^6 + E^4*(27*x^2 + 36*x^3 + 25*x^4 + 6*x^5))/(E^12 - 3*E^8*x + 3*E^4*
x^2 - x^3),x]

[Out]

(-15*E^24 - 54*E^8*(-1 + x)*x - 27*E^4*x^2 + 10*E^20*(-5 + 3*x) + E^12*(-27 + 108*x - 50*x^2) + E^16*(-54 + 10
0*x - 15*x^2) + x^3*(9 + 9*x + 5*x^2 + x^3))/(E^4 - x)^2

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fricas [B]  time = 0.54, size = 92, normalized size = 3.83 \begin {gather*} \frac {x^{6} + 5 \, x^{5} + 9 \, x^{4} + 9 \, x^{3} - 18 \, x^{2} e^{4} + 10 \, {\left (x - 2\right )} e^{20} - {\left (5 \, x^{2} - 40 \, x + 27\right )} e^{16} - 2 \, {\left (10 \, x^{2} - 27 \, x + 9\right )} e^{12} - 9 \, {\left (3 \, x^{2} - 4 \, x\right )} e^{8} - 5 \, e^{24}}{x^{2} - 2 \, x e^{4} + e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^5+25*x^4+36*x^3+27*x^2)*exp(4)-4*x^6-15*x^5-18*x^4-9*x^3)/(exp(4)^3-3*x*exp(4)^2+3*x^2*exp(4)-
x^3),x, algorithm="fricas")

[Out]

(x^6 + 5*x^5 + 9*x^4 + 9*x^3 - 18*x^2*e^4 + 10*(x - 2)*e^20 - (5*x^2 - 40*x + 27)*e^16 - 2*(10*x^2 - 27*x + 9)
*e^12 - 9*(3*x^2 - 4*x)*e^8 - 5*e^24)/(x^2 - 2*x*e^4 + e^8)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 \, x^{6} + 15 \, x^{5} + 18 \, x^{4} + 9 \, x^{3} - {\left (6 \, x^{5} + 25 \, x^{4} + 36 \, x^{3} + 27 \, x^{2}\right )} e^{4}}{x^{3} - 3 \, x^{2} e^{4} + 3 \, x e^{8} - e^{12}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^5+25*x^4+36*x^3+27*x^2)*exp(4)-4*x^6-15*x^5-18*x^4-9*x^3)/(exp(4)^3-3*x*exp(4)^2+3*x^2*exp(4)-
x^3),x, algorithm="giac")

[Out]

integrate((4*x^6 + 15*x^5 + 18*x^4 + 9*x^3 - (6*x^5 + 25*x^4 + 36*x^3 + 27*x^2)*e^4)/(x^3 - 3*x^2*e^4 + 3*x*e^
8 - e^12), x)

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maple [A]  time = 0.08, size = 29, normalized size = 1.21

method	result	size
norman	\(\frac {x^{6}+5 x^{5}+9 x^{4}+9 x^{3}}{\left ({\mathrm e}^{4}-x \right )^{2}}\)	\(29\)
gosper	\(\frac {x^{3} \left (x^{3}+5 x^{2}+9 x +9\right )}{{\mathrm e}^{8}-2 x \,{\mathrm e}^{4}+x^{2}}\)	\(33\)
risch	\(4 x \,{\mathrm e}^{12}+3 x^{2} {\mathrm e}^{8}+2 x^{3} {\mathrm e}^{4}+x^{4}+15 x \,{\mathrm e}^{8}+10 x^{2} {\mathrm e}^{4}+5 x^{3}+18 x \,{\mathrm e}^{4}+9 x^{2}+9 x +\frac {\left (6 \,{\mathrm e}^{20}+25 \,{\mathrm e}^{16}+36 \,{\mathrm e}^{12}+27 \,{\mathrm e}^{8}\right ) x -5 \,{\mathrm e}^{24}-20 \,{\mathrm e}^{20}-27 \,{\mathrm e}^{16}-18 \,{\mathrm e}^{12}}{{\mathrm e}^{8}-2 x \,{\mathrm e}^{4}+x^{2}}\)	\(104\)
default	\(58 x \,{\mathrm e}^{12}+3 x^{2} {\mathrm e}^{8}+2 x^{3} {\mathrm e}^{4}+x^{4}+15 x \,{\mathrm e}^{8}-54 \,{\mathrm e}^{4} {\mathrm e}^{8} x +10 x^{2} {\mathrm e}^{4}+5 x^{3}+18 x \,{\mathrm e}^{4}+9 x^{2}+9 x -\frac {\left (\munderset {\textit {\_R} =\RootOf \left (-{\mathrm e}^{12}+3 \textit {\_Z} \,{\mathrm e}^{8}-3 \textit {\_Z}^{2} {\mathrm e}^{4}+\textit {\_Z}^{3}\right )}{\sum }\frac {\left (-18 \,{\mathrm e}^{12} {\mathrm e}^{4}+6 \textit {\_R} \,{\mathrm e}^{20}+25 \textit {\_R} \,{\mathrm e}^{16}-15 \,{\mathrm e}^{8} {\mathrm e}^{12}+27 \textit {\_R} \,{\mathrm e}^{8}-4 \,{\mathrm e}^{24}+36 \textit {\_R} \,{\mathrm e}^{12}-9 \,{\mathrm e}^{12}\right ) \ln \left (x -\textit {\_R} \right )}{{\mathrm e}^{8}-2 \textit {\_R} \,{\mathrm e}^{4}+\textit {\_R}^{2}}\right )}{3}\)	\(173\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x^5+25*x^4+36*x^3+27*x^2)*exp(4)-4*x^6-15*x^5-18*x^4-9*x^3)/(exp(4)^3-3*x*exp(4)^2+3*x^2*exp(4)-x^3),x
,method=_RETURNVERBOSE)

[Out]

(x^6+5*x^5+9*x^4+9*x^3)/(exp(4)-x)^2

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maxima [B]  time = 0.36, size = 94, normalized size = 3.92 \begin {gather*} x^{4} + x^{3} {\left (2 \, e^{4} + 5\right )} + x^{2} {\left (3 \, e^{8} + 10 \, e^{4} + 9\right )} + x {\left (4 \, e^{12} + 15 \, e^{8} + 18 \, e^{4} + 9\right )} + \frac {x {\left (6 \, e^{20} + 25 \, e^{16} + 36 \, e^{12} + 27 \, e^{8}\right )} - 5 \, e^{24} - 20 \, e^{20} - 27 \, e^{16} - 18 \, e^{12}}{x^{2} - 2 \, x e^{4} + e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^5+25*x^4+36*x^3+27*x^2)*exp(4)-4*x^6-15*x^5-18*x^4-9*x^3)/(exp(4)^3-3*x*exp(4)^2+3*x^2*exp(4)-
x^3),x, algorithm="maxima")

[Out]

x^4 + x^3*(2*e^4 + 5) + x^2*(3*e^8 + 10*e^4 + 9) + x*(4*e^12 + 15*e^8 + 18*e^4 + 9) + (x*(6*e^20 + 25*e^16 + 3
6*e^12 + 27*e^8) - 5*e^24 - 20*e^20 - 27*e^16 - 18*e^12)/(x^2 - 2*x*e^4 + e^8)

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mupad [B]  time = 5.56, size = 138, normalized size = 5.75 \begin {gather*} x^3\,\left (2\,{\mathrm {e}}^4+5\right )-x\,\left (36\,{\mathrm {e}}^4-4\,{\mathrm {e}}^{12}+3\,{\mathrm {e}}^4\,\left (25\,{\mathrm {e}}^4+12\,{\mathrm {e}}^8-3\,{\mathrm {e}}^4\,\left (6\,{\mathrm {e}}^4+15\right )-18\right )+3\,{\mathrm {e}}^8\,\left (6\,{\mathrm {e}}^4+15\right )-9\right )-\frac {18\,{\mathrm {e}}^{12}+27\,{\mathrm {e}}^{16}+20\,{\mathrm {e}}^{20}+5\,{\mathrm {e}}^{24}-x\,\left (27\,{\mathrm {e}}^8+36\,{\mathrm {e}}^{12}+25\,{\mathrm {e}}^{16}+6\,{\mathrm {e}}^{20}\right )}{x^2-2\,{\mathrm {e}}^4\,x+{\mathrm {e}}^8}-x^2\,\left (\frac {25\,{\mathrm {e}}^4}{2}+6\,{\mathrm {e}}^8-\frac {3\,{\mathrm {e}}^4\,\left (6\,{\mathrm {e}}^4+15\right )}{2}-9\right )+x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(9*x^3 + 18*x^4 + 15*x^5 + 4*x^6 - exp(4)*(27*x^2 + 36*x^3 + 25*x^4 + 6*x^5))/(exp(12) - 3*x*exp(8) + 3*x
^2*exp(4) - x^3),x)

[Out]

x^3*(2*exp(4) + 5) - x*(36*exp(4) - 4*exp(12) + 3*exp(4)*(25*exp(4) + 12*exp(8) - 3*exp(4)*(6*exp(4) + 15) - 1
8) + 3*exp(8)*(6*exp(4) + 15) - 9) - (18*exp(12) + 27*exp(16) + 20*exp(20) + 5*exp(24) - x*(27*exp(8) + 36*exp
(12) + 25*exp(16) + 6*exp(20)))/(exp(8) - 2*x*exp(4) + x^2) - x^2*((25*exp(4))/2 + 6*exp(8) - (3*exp(4)*(6*exp
(4) + 15))/2 - 9) + x^4

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sympy [B]  time = 0.53, size = 102, normalized size = 4.25 \begin {gather*} x^{4} + x^{3} \left (5 + 2 e^{4}\right ) + x^{2} \left (9 + 10 e^{4} + 3 e^{8}\right ) + x \left (9 + 18 e^{4} + 15 e^{8} + 4 e^{12}\right ) + \frac {x \left (27 e^{8} + 36 e^{12} + 25 e^{16} + 6 e^{20}\right ) - 5 e^{24} - 20 e^{20} - 27 e^{16} - 18 e^{12}}{x^{2} - 2 x e^{4} + e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x**5+25*x**4+36*x**3+27*x**2)*exp(4)-4*x**6-15*x**5-18*x**4-9*x**3)/(exp(4)**3-3*x*exp(4)**2+3*x
**2*exp(4)-x**3),x)

[Out]

x**4 + x**3*(5 + 2*exp(4)) + x**2*(9 + 10*exp(4) + 3*exp(8)) + x*(9 + 18*exp(4) + 15*exp(8) + 4*exp(12)) + (x*
(27*exp(8) + 36*exp(12) + 25*exp(16) + 6*exp(20)) - 5*exp(24) - 20*exp(20) - 27*exp(16) - 18*exp(12))/(x**2 -
2*x*exp(4) + exp(8))

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