\(\int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 (9 x^2+6 x^3+x^4)+e (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6)} \, dx\) [6521]

Optimal result

Integrand size = 107, antiderivative size = 25 \[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx=-\frac {2}{x^2 (3+x) \left (\frac {25-e}{x}+2 x\right )} \]

[Out]

-2/x^2/(2*x+(25-exp(1))/x)/(3+x)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.

Time = 0.16 (sec) , antiderivative size = 143, normalized size of antiderivative = 5.72, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {2099, 653, 210} \[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx=-\frac {12 \sqrt {\frac {2}{25-e}} \arctan \left (\sqrt {\frac {2}{25-e}} x\right )}{1075-68 e+e^2}+\frac {12 \sqrt {2} \arctan \left (\sqrt {\frac {2}{25-e}} x\right )}{(25-e)^{3/2} (43-e)}+\frac {4 (6 x-e+25)}{\left (1075-68 e+e^2\right ) \left (2 x^2-e+25\right )}-\frac {2}{3 (25-e) x}+\frac {2}{3 (43-e) (x+3)} \]

[In]

Int[(150 + E*(-6 - 4*x) + 100*x + 36*x^2 + 16*x^3)/(5625*x^2 + 3750*x^3 + 1525*x^4 + 600*x^5 + 136*x^6 + 24*x^
7 + 4*x^8 + E^2*(9*x^2 + 6*x^3 + x^4) + E*(-450*x^2 - 300*x^3 - 86*x^4 - 24*x^5 - 4*x^6)),x]

[Out]

-2/(3*(25 - E)*x) + 2/(3*(43 - E)*(3 + x)) + (4*(25 - E + 6*x))/((1075 - 68*E + E^2)*(25 - E + 2*x^2)) + (12*S
qrt[2]*ArcTan[Sqrt[2/(25 - E)]*x])/((25 - E)^(3/2)*(43 - E)) - (12*Sqrt[2/(25 - E)]*ArcTan[Sqrt[2/(25 - E)]*x]
)/(1075 - 68*E + E^2)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 653

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)/(2*a*c*(p + 1)))*(a + c*x
^2)^(p + 1), x] + Dist[d*((2*p + 3)/(2*a*(p + 1))), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 2099

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{align*} \text {integral}& = \int \left (-\frac {2}{3 (-25+e) x^2}+\frac {2}{3 (-43+e) (3+x)^2}+\frac {16 (-3+x)}{(-43+e) \left (-25+e-2 x^2\right )^2}+\frac {24}{(-43+e) (-25+e) \left (-25+e-2 x^2\right )}\right ) \, dx \\ & = -\frac {2}{3 (25-e) x}+\frac {2}{3 (43-e) (3+x)}-\frac {16 \int \frac {-3+x}{\left (-25+e-2 x^2\right )^2} \, dx}{43-e}+\frac {24 \int \frac {1}{-25+e-2 x^2} \, dx}{1075-68 e+e^2} \\ & = -\frac {2}{3 (25-e) x}+\frac {2}{3 (43-e) (3+x)}+\frac {4 (25-e+6 x)}{\left (1075-68 e+e^2\right ) \left (25-e+2 x^2\right )}-\frac {12 \sqrt {\frac {2}{25-e}} \tan ^{-1}\left (\sqrt {\frac {2}{25-e}} x\right )}{1075-68 e+e^2}-\frac {24 \int \frac {1}{-25+e-2 x^2} \, dx}{1075-68 e+e^2} \\ & = -\frac {2}{3 (25-e) x}+\frac {2}{3 (43-e) (3+x)}+\frac {4 (25-e+6 x)}{\left (1075-68 e+e^2\right ) \left (25-e+2 x^2\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx=-\frac {2}{x (3+x) \left (25-e+2 x^2\right )} \]

[In]

Integrate[(150 + E*(-6 - 4*x) + 100*x + 36*x^2 + 16*x^3)/(5625*x^2 + 3750*x^3 + 1525*x^4 + 600*x^5 + 136*x^6 +
24*x^7 + 4*x^8 + E^2*(9*x^2 + 6*x^3 + x^4) + E*(-450*x^2 - 300*x^3 - 86*x^4 - 24*x^5 - 4*x^6)),x]

[Out]

-2/(x*(3 + x)*(25 - E + 2*x^2))

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88

method	result	size
norman	\(\frac {2}{x \left (3+x \right ) \left (-2 x^{2}+{\mathrm e}-25\right )}\)	\(22\)
gosper	\(\frac {2}{x \left (-2 x^{3}+x \,{\mathrm e}-6 x^{2}+3 \,{\mathrm e}-25 x -75\right )}\)	\(31\)
risch	\(\frac {2}{x \left (-2 x^{3}+x \,{\mathrm e}-6 x^{2}+3 \,{\mathrm e}-25 x -75\right )}\)	\(31\)
parallelrisch	\(\frac {2}{x \left (-2 x^{3}+x \,{\mathrm e}-6 x^{2}+3 \,{\mathrm e}-25 x -75\right )}\)	\(31\)
default	\(-\frac {2 \left (25-{\mathrm e}\right )}{\left (3 \,{\mathrm e}^{2}-150 \,{\mathrm e}+1875\right ) x}+\frac {\frac {2 \left (-\frac {6 \left (8772 \,{\mathrm e} \,{\mathrm e}^{2}-874964 \,{\mathrm e}^{2}+{\mathrm e} \,{\mathrm e}^{4}-154 \,{\mathrm e} \,{\mathrm e}^{3}+{\mathrm e}^{5}[phone] \,{\mathrm e}-204 \,{\mathrm e}^{4}+16472 \,{\mathrm e}^{3}-99383750\right ) x}{{\mathrm e}-25}+2 \,{\mathrm e}^{5}-874964 \,{\mathrm e}^{2}[phone] \,{\mathrm e}-358 \,{\mathrm e}^{4}+25244 \,{\mathrm e}^{3}-99383750\right )}{-2 x^{2}+{\mathrm e}-25}+\frac {12 \left (-8772 \,{\mathrm e} \,{\mathrm e}^{2}-{\mathrm e} \,{\mathrm e}^{4}+154 \,{\mathrm e} \,{\mathrm e}^{3}+{\mathrm e}^{5}-154 \,{\mathrm e}^{4}+8772 \,{\mathrm e}^{3}\right ) \arctan \left (\frac {2 x}{\sqrt {-2 \,{\mathrm e}+50}}\right )}{\left ({\mathrm e}-25\right ) \sqrt {-2 \,{\mathrm e}+50}}}{\left (1849-86 \,{\mathrm e}+{\mathrm e}^{2}\right )^{2} \left ({\mathrm e}^{2}-50 \,{\mathrm e}+625\right )}-\frac {2 \left ({\mathrm e}^{3}-129 \,{\mathrm e}^{2}+5547 \,{\mathrm e}-79507\right )}{3 \left (1849-86 \,{\mathrm e}+{\mathrm e}^{2}\right )^{2} \left (3+x \right )}\)	\(231\)

[In]

int(((-4*x-6)*exp(1)+16*x^3+36*x^2+100*x+150)/((x^4+6*x^3+9*x^2)*exp(1)^2+(-4*x^6-24*x^5-86*x^4-300*x^3-450*x^
2)*exp(1)+4*x^8+24*x^7+136*x^6+600*x^5+1525*x^4+3750*x^3+5625*x^2),x,method=_RETURNVERBOSE)

[Out]

2/x/(3+x)/(-2*x^2+exp(1)-25)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx=-\frac {2}{2 \, x^{4} + 6 \, x^{3} + 25 \, x^{2} - {\left (x^{2} + 3 \, x\right )} e + 75 \, x} \]

[In]

integrate(((-4*x-6)*exp(1)+16*x^3+36*x^2+100*x+150)/((x^4+6*x^3+9*x^2)*exp(1)^2+(-4*x^6-24*x^5-86*x^4-300*x^3-
450*x^2)*exp(1)+4*x^8+24*x^7+136*x^6+600*x^5+1525*x^4+3750*x^3+5625*x^2),x, algorithm="fricas")

[Out]

-2/(2*x^4 + 6*x^3 + 25*x^2 - (x^2 + 3*x)*e + 75*x)

Sympy [A] (verification not implemented)

Time = 1.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx=- \frac {2}{2 x^{4} + 6 x^{3} + x^{2} \cdot \left (25 - e\right ) + x \left (75 - 3 e\right )} \]

[In]

integrate(((-4*x-6)*exp(1)+16*x**3+36*x**2+100*x+150)/((x**4+6*x**3+9*x**2)*exp(1)**2+(-4*x**6-24*x**5-86*x**4
-300*x**3-450*x**2)*exp(1)+4*x**8+24*x**7+136*x**6+600*x**5+1525*x**4+3750*x**3+5625*x**2),x)

[Out]

-2/(2*x**4 + 6*x**3 + x**2*(25 - E) + x*(75 - 3*E))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx=-\frac {2}{2 \, x^{4} + 6 \, x^{3} - x^{2} {\left (e - 25\right )} - 3 \, x {\left (e - 25\right )}} \]

[In]

integrate(((-4*x-6)*exp(1)+16*x^3+36*x^2+100*x+150)/((x^4+6*x^3+9*x^2)*exp(1)^2+(-4*x^6-24*x^5-86*x^4-300*x^3-
450*x^2)*exp(1)+4*x^8+24*x^7+136*x^6+600*x^5+1525*x^4+3750*x^3+5625*x^2),x, algorithm="maxima")

[Out]

-2/(2*x^4 + 6*x^3 - x^2*(e - 25) - 3*x*(e - 25))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx=-\frac {2}{2 \, x^{4} + 6 \, x^{3} + 25 \, x^{2} - {\left (x^{2} + 3 \, x\right )} e + 75 \, x} \]

[In]

integrate(((-4*x-6)*exp(1)+16*x^3+36*x^2+100*x+150)/((x^4+6*x^3+9*x^2)*exp(1)^2+(-4*x^6-24*x^5-86*x^4-300*x^3-
450*x^2)*exp(1)+4*x^8+24*x^7+136*x^6+600*x^5+1525*x^4+3750*x^3+5625*x^2),x, algorithm="giac")

[Out]

-2/(2*x^4 + 6*x^3 + 25*x^2 - (x^2 + 3*x)*e + 75*x)

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx=-\frac {2}{2\,x^4+6\,x^3+\left (25-\mathrm {e}\right )\,x^2+\left (75-3\,\mathrm {e}\right )\,x} \]

[In]

int((100*x + 36*x^2 + 16*x^3 - exp(1)*(4*x + 6) + 150)/(exp(2)*(9*x^2 + 6*x^3 + x^4) - exp(1)*(450*x^2 + 300*x
^3 + 86*x^4 + 24*x^5 + 4*x^6) + 5625*x^2 + 3750*x^3 + 1525*x^4 + 600*x^5 + 136*x^6 + 24*x^7 + 4*x^8),x)

[Out]

-2/(6*x^3 + 2*x^4 - x*(3*exp(1) - 75) - x^2*(exp(1) - 25))