3.36.18 \(\int \frac {-162 x^4+50 e x^4-246 x^5-4 x^6+e^9 (2 x-50 e x+306 x^2+4 x^3)+e^6 (54 x^2+150 e x^2-838 x^3-12 x^4)+e^3 (306 x^3-150 e x^3+778 x^4+12 x^5)}{25 e^9-75 e^6 x+75 e^3 x^2-25 x^3} \, dx\)

Optimal. Leaf size=33 \[ x^2 \left (-e+4 x+\left (\frac {2 x}{e^3-x}+\frac {1+x}{5}\right )^2\right ) \]

________________________________________________________________________________________

Rubi [B]  time = 0.14, antiderivative size = 84, normalized size of antiderivative = 2.55, number of steps used = 3, number of rules used = 2, integrand size = 124, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {6, 2074} \begin {gather*} \frac {x^4}{25}+\frac {82 x^3}{25}+\frac {1}{25} \left (81-25 e-20 e^3\right ) x^2+\frac {4}{5} e^3 \left (9-e^3\right ) x-\frac {4 e^9 \left (19-e^3\right )}{5 \left (e^3-x\right )}+\frac {4 e^{12}}{\left (e^3-x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-162*x^4 + 50*E*x^4 - 246*x^5 - 4*x^6 + E^9*(2*x - 50*E*x + 306*x^2 + 4*x^3) + E^6*(54*x^2 + 150*E*x^2 -
838*x^3 - 12*x^4) + E^3*(306*x^3 - 150*E*x^3 + 778*x^4 + 12*x^5))/(25*E^9 - 75*E^6*x + 75*E^3*x^2 - 25*x^3),x]

[Out]

(4*E^12)/(E^3 - x)^2 - (4*E^9*(19 - E^3))/(5*(E^3 - x)) + (4*E^3*(9 - E^3)*x)/5 + ((81 - 25*E - 20*E^3)*x^2)/2
5 + (82*x^3)/25 + x^4/25

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 \frac {(-162+50 e) x^4-246 x^5-4 x^6+e^9 \left (2 x-50 e x+306 x^2+4 x^3\right )+e^6 \left (54 x^2+150 e x^2-838 x^3-12 x^4\right )+e^3 \left (306 x^3-150 e x^3+778 x^4+12 x^5\right )}{25 e^9-75 e^6 x+75 e^3 x^2-25 x^3} \, dx\\ &=\int \left (-\frac {4}{5} e^3 \left (-9+e^3\right )+\frac {8 e^{12}}{\left (e^3-x\right )^3}+\frac {4 e^9 \left (-19+e^3\right )}{5 \left (e^3-x\right )^2}-\frac {2}{25} \left (-81+25 e+20 e^3\right ) x+\frac {246 x^2}{25}+\frac {4 x^3}{25}\right ) \, dx\\ &=\frac {4 e^{12}}{\left (e^3-x\right )^2}-\frac {4 e^9 \left (19-e^3\right )}{5 \left (e^3-x\right )}+\frac {4}{5} e^3 \left (9-e^3\right ) x+\frac {1}{25} \left (81-25 e-20 e^3\right ) x^2+\frac {82 x^3}{25}+\frac {x^4}{25}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.08, size = 94, normalized size = 2.85 \begin {gather*} \frac {1}{25} \left (25 e^7-e^{12} \left (1-\frac {100}{\left (e^3-x\right )^2}-\frac {20}{e^3-x}\right )-e^9 \left (42+\frac {380}{e^3-x}\right )-20 e^3 (-9+x) x-25 e x^2-e^6 (261+20 x)+x^2 \left (81+82 x+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-162*x^4 + 50*E*x^4 - 246*x^5 - 4*x^6 + E^9*(2*x - 50*E*x + 306*x^2 + 4*x^3) + E^6*(54*x^2 + 150*E*
x^2 - 838*x^3 - 12*x^4) + E^3*(306*x^3 - 150*E*x^3 + 778*x^4 + 12*x^5))/(25*E^9 - 75*E^6*x + 75*E^3*x^2 - 25*x
^3),x]

[Out]

(25*E^7 - E^12*(1 - 100/(E^3 - x)^2 - 20/(E^3 - x)) - E^9*(42 + 380/(E^3 - x)) - 20*E^3*(-9 + x)*x - 25*E*x^2
- E^6*(261 + 20*x) + x^2*(81 + 82*x + x^2))/25

________________________________________________________________________________________

fricas [B]  time = 0.97, size = 107, normalized size = 3.24 \begin {gather*} \frac {x^{6} + 82 \, x^{5} - 25 \, x^{4} e + 81 \, x^{4} + 50 \, x^{3} e^{4} - 25 \, x^{2} e^{7} - 40 \, {\left (x + 7\right )} e^{12} + 20 \, {\left (x^{2} + 28 \, x\right )} e^{9} + {\left (x^{4} + 102 \, x^{3} - 279 \, x^{2}\right )} e^{6} - 2 \, {\left (x^{5} + 92 \, x^{4} - 9 \, x^{3}\right )} e^{3} + 20 \, e^{15}}{25 \, {\left (x^{2} - 2 \, x e^{3} + e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*x*exp(1)+4*x^3+306*x^2+2*x)*exp(3)^3+(150*x^2*exp(1)-12*x^4-838*x^3+54*x^2)*exp(3)^2+(-150*x^3
*exp(1)+12*x^5+778*x^4+306*x^3)*exp(3)+50*x^4*exp(1)-4*x^6-246*x^5-162*x^4)/(25*exp(3)^3-75*x*exp(3)^2+75*x^2*
exp(3)-25*x^3),x, algorithm="fricas")

[Out]

1/25*(x^6 + 82*x^5 - 25*x^4*e + 81*x^4 + 50*x^3*e^4 - 25*x^2*e^7 - 40*(x + 7)*e^12 + 20*(x^2 + 28*x)*e^9 + (x^
4 + 102*x^3 - 279*x^2)*e^6 - 2*(x^5 + 92*x^4 - 9*x^3)*e^3 + 20*e^15)/(x^2 - 2*x*e^3 + e^6)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (2 \, x^{6} + 123 \, x^{5} - 25 \, x^{4} e + 81 \, x^{4} - {\left (2 \, x^{3} + 153 \, x^{2} - 25 \, x e + x\right )} e^{9} + {\left (6 \, x^{4} + 419 \, x^{3} - 75 \, x^{2} e - 27 \, x^{2}\right )} e^{6} - {\left (6 \, x^{5} + 389 \, x^{4} - 75 \, x^{3} e + 153 \, x^{3}\right )} e^{3}\right )}}{25 \, {\left (x^{3} - 3 \, x^{2} e^{3} + 3 \, x e^{6} - e^{9}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*x*exp(1)+4*x^3+306*x^2+2*x)*exp(3)^3+(150*x^2*exp(1)-12*x^4-838*x^3+54*x^2)*exp(3)^2+(-150*x^3
*exp(1)+12*x^5+778*x^4+306*x^3)*exp(3)+50*x^4*exp(1)-4*x^6-246*x^5-162*x^4)/(25*exp(3)^3-75*x*exp(3)^2+75*x^2*
exp(3)-25*x^3),x, algorithm="giac")

[Out]

integrate(2/25*(2*x^6 + 123*x^5 - 25*x^4*e + 81*x^4 - (2*x^3 + 153*x^2 - 25*x*e + x)*e^9 + (6*x^4 + 419*x^3 -
75*x^2*e - 27*x^2)*e^6 - (6*x^5 + 389*x^4 - 75*x^3*e + 153*x^3)*e^3)/(x^3 - 3*x^2*e^3 + 3*x*e^6 - e^9), x)

________________________________________________________________________________________

maple [B]  time = 0.09, size = 76, normalized size = 2.30

method	result	size
risch	\(-\frac {4 x \,{\mathrm e}^{6}}{5}-\frac {4 x^{2} {\mathrm e}^{3}}{5}+\frac {36 x \,{\mathrm e}^{3}}{5}+\frac {x^{4}}{25}-x^{2} {\mathrm e}+\frac {82 x^{3}}{25}+\frac {81 x^{2}}{25}+\frac {\frac {\left (-20 \,{\mathrm e}^{12}+380 \,{\mathrm e}^{9}\right ) x}{25}+\frac {4 \,{\mathrm e}^{15}}{5}-\frac {56 \,{\mathrm e}^{12}}{5}}{{\mathrm e}^{6}-2 x \,{\mathrm e}^{3}+x^{2}}\)	\(76\)
norman	\(\frac {\left (\frac {82}{25}-\frac {2 \,{\mathrm e}^{3}}{25}\right ) x^{5}+\left (2 \,{\mathrm e} \,{\mathrm e}^{3}+\frac {102 \,{\mathrm e}^{6}}{25}+\frac {18 \,{\mathrm e}^{3}}{25}\right ) x^{3}+\left (-2 \,{\mathrm e} \,{\mathrm e}^{9}+\frac {2 \,{\mathrm e}^{9}}{25}\right ) x +\left (\frac {{\mathrm e}^{6}}{25}-{\mathrm e}-\frac {184 \,{\mathrm e}^{3}}{25}+\frac {81}{25}\right ) x^{4}+\frac {x^{6}}{25}+\frac {{\mathrm e}^{12} \left (25 \,{\mathrm e}-1\right )}{25}}{\left (-x +{\mathrm e}^{3}\right )^{2}}\)	\(96\)
gosper	\(\frac {x^{4} {\mathrm e}^{6}-2 x^{5} {\mathrm e}^{3}+x^{6}+25 \,{\mathrm e} \,{\mathrm e}^{12}-50 \,{\mathrm e} \,{\mathrm e}^{9} x +50 \,{\mathrm e} \,{\mathrm e}^{3} x^{3}-25 x^{4} {\mathrm e}+102 x^{3} {\mathrm e}^{6}-184 x^{4} {\mathrm e}^{3}+82 x^{5}-{\mathrm e}^{12}+2 x \,{\mathrm e}^{9}+18 x^{3} {\mathrm e}^{3}+81 x^{4}}{25 \,{\mathrm e}^{6}-50 x \,{\mathrm e}^{3}+25 x^{2}}\)	\(116\)
default	\(\frac {x^{4}}{25}-6 x \,{\mathrm e} \,{\mathrm e}^{3}-x^{2} {\mathrm e}-\frac {4 x \,{\mathrm e}^{6}}{5}-\frac {4 x^{2} {\mathrm e}^{3}}{5}+\frac {82 x^{3}}{25}+6 x \,{\mathrm e}^{4}+\frac {36 x \,{\mathrm e}^{3}}{5}+\frac {81 x^{2}}{25}-\frac {4 \left (\munderset {\textit {\_R} =\RootOf \left (-3 \textit {\_Z}^{2} {\mathrm e}^{3}+\textit {\_Z}^{3}+3 \textit {\_Z} \,{\mathrm e}^{6}-{\mathrm e}^{9}\right )}{\sum }\frac {\left (-9 \,{\mathrm e}^{9} {\mathrm e}^{3}-\textit {\_R} \,{\mathrm e}^{12}+{\mathrm e}^{9} {\mathrm e}^{6}+19 \textit {\_R} \,{\mathrm e}^{9}\right ) \ln \left (x -\textit {\_R} \right )}{-2 \textit {\_R} \,{\mathrm e}^{3}+\textit {\_R}^{2}+{\mathrm e}^{6}}\right )}{15}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-50*x*exp(1)+4*x^3+306*x^2+2*x)*exp(3)^3+(150*x^2*exp(1)-12*x^4-838*x^3+54*x^2)*exp(3)^2+(-150*x^3*exp(1
)+12*x^5+778*x^4+306*x^3)*exp(3)+50*x^4*exp(1)-4*x^6-246*x^5-162*x^4)/(25*exp(3)^3-75*x*exp(3)^2+75*x^2*exp(3)
-25*x^3),x,method=_RETURNVERBOSE)

[Out]

-4/5*x*exp(6)-4/5*x^2*exp(3)+36/5*x*exp(3)+1/25*x^4-x^2*exp(1)+82/25*x^3+81/25*x^2+(1/25*(-20*exp(12)+380*exp(
9))*x+4/5*exp(15)-56/5*exp(12))/(exp(6)-2*x*exp(3)+x^2)

________________________________________________________________________________________

maxima [B]  time = 0.51, size = 69, normalized size = 2.09 \begin {gather*} \frac {1}{25} \, x^{4} + \frac {82}{25} \, x^{3} - \frac {1}{25} \, x^{2} {\left (20 \, e^{3} + 25 \, e - 81\right )} - \frac {4}{5} \, x {\left (e^{6} - 9 \, e^{3}\right )} - \frac {4 \, {\left (x {\left (e^{12} - 19 \, e^{9}\right )} - e^{15} + 14 \, e^{12}\right )}}{5 \, {\left (x^{2} - 2 \, x e^{3} + e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*x*exp(1)+4*x^3+306*x^2+2*x)*exp(3)^3+(150*x^2*exp(1)-12*x^4-838*x^3+54*x^2)*exp(3)^2+(-150*x^3
*exp(1)+12*x^5+778*x^4+306*x^3)*exp(3)+50*x^4*exp(1)-4*x^6-246*x^5-162*x^4)/(25*exp(3)^3-75*x*exp(3)^2+75*x^2*
exp(3)-25*x^3),x, algorithm="maxima")

[Out]

1/25*x^4 + 82/25*x^3 - 1/25*x^2*(20*e^3 + 25*e - 81) - 4/5*x*(e^6 - 9*e^3) - 4/5*(x*(e^12 - 19*e^9) - e^15 + 1
4*e^12)/(x^2 - 2*x*e^3 + e^6)

________________________________________________________________________________________

mupad [B]  time = 0.23, size = 106, normalized size = 3.21 \begin {gather*} \frac {4\,{\mathrm {e}}^{15}-56\,{\mathrm {e}}^{12}+x\,\left (76\,{\mathrm {e}}^9-4\,{\mathrm {e}}^{12}\right )}{5\,x^2-10\,{\mathrm {e}}^3\,x+5\,{\mathrm {e}}^6}-x^2\,\left (\mathrm {e}+\frac {4\,{\mathrm {e}}^3}{5}-\frac {81}{25}\right )+\frac {82\,x^3}{25}+\frac {x^4}{25}-x\,\left (\frac {738\,{\mathrm {e}}^6}{25}-\frac {4\,{\mathrm {e}}^9}{25}-\frac {2\,{\mathrm {e}}^3\,\left (75\,\mathrm {e}+419\,{\mathrm {e}}^3-2\,{\mathrm {e}}^6-153\right )}{25}+3\,{\mathrm {e}}^3\,\left (2\,\mathrm {e}+\frac {8\,{\mathrm {e}}^3}{5}-\frac {162}{25}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(6)*(150*x^2*exp(1) + 54*x^2 - 838*x^3 - 12*x^4) + exp(3)*(306*x^3 - 150*x^3*exp(1) + 778*x^4 + 12*x^5
) + 50*x^4*exp(1) + exp(9)*(2*x - 50*x*exp(1) + 306*x^2 + 4*x^3) - 162*x^4 - 246*x^5 - 4*x^6)/(25*exp(9) - 75*
x*exp(6) + 75*x^2*exp(3) - 25*x^3),x)

[Out]

(4*exp(15) - 56*exp(12) + x*(76*exp(9) - 4*exp(12)))/(5*exp(6) - 10*x*exp(3) + 5*x^2) - x^2*(exp(1) + (4*exp(3
))/5 - 81/25) + (82*x^3)/25 + x^4/25 - x*((738*exp(6))/25 - (4*exp(9))/25 - (2*exp(3)*(75*exp(1) + 419*exp(3)
- 2*exp(6) - 153))/25 + 3*exp(3)*(2*exp(1) + (8*exp(3))/5 - 162/25))

________________________________________________________________________________________

sympy [B]  time = 0.48, size = 82, normalized size = 2.48 \begin {gather*} \frac {x^{4}}{25} + \frac {82 x^{3}}{25} + x^{2} \left (- \frac {4 e^{3}}{5} - e + \frac {81}{25}\right ) + x \left (- \frac {4 e^{6}}{5} + \frac {36 e^{3}}{5}\right ) + \frac {x \left (- 4 e^{12} + 76 e^{9}\right ) - 56 e^{12} + 4 e^{15}}{5 x^{2} - 10 x e^{3} + 5 e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*x*exp(1)+4*x**3+306*x**2+2*x)*exp(3)**3+(150*x**2*exp(1)-12*x**4-838*x**3+54*x**2)*exp(3)**2+(
-150*x**3*exp(1)+12*x**5+778*x**4+306*x**3)*exp(3)+50*x**4*exp(1)-4*x**6-246*x**5-162*x**4)/(25*exp(3)**3-75*x
*exp(3)**2+75*x**2*exp(3)-25*x**3),x)

[Out]

x**4/25 + 82*x**3/25 + x**2*(-4*exp(3)/5 - E + 81/25) + x*(-4*exp(6)/5 + 36*exp(3)/5) + (x*(-4*exp(12) + 76*ex
p(9)) - 56*exp(12) + 4*exp(15))/(5*x**2 - 10*x*exp(3) + 5*exp(6))

________________________________________________________________________________________