\(\int \frac {270+90 e^{13/2}-90 x^2}{9+e^{13}+12 x-4 e^{39/4} x+10 x^2+4 x^3+x^4+e^{13/2} (6+4 x+6 x^2)+e^{13/4} (-12 x-8 x^2-4 x^3)} \, dx\) [7917]

Optimal result

Integrand size = 82, antiderivative size = 23 \[ \int \frac {270+90 e^{13/2}-90 x^2}{9+e^{13}+12 x-4 e^{39/4} x+10 x^2+4 x^3+x^4+e^{13/2} \left (6+4 x+6 x^2\right )+e^{13/4} \left (-12 x-8 x^2-4 x^3\right )} \, dx=1+\frac {90 x}{3+2 x+\left (-e^{13/4}+x\right )^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {6, 1694, 12, 1828, 8} \[ \int \frac {270+90 e^{13/2}-90 x^2}{9+e^{13}+12 x-4 e^{39/4} x+10 x^2+4 x^3+x^4+e^{13/2} \left (6+4 x+6 x^2\right )+e^{13/4} \left (-12 x-8 x^2-4 x^3\right )} \, dx=\frac {90 x}{x^2+2 \left (1-e^{13/4}\right ) x+e^{13/2}+3} \]

[In]

Int[(270 + 90*E^(13/2) - 90*x^2)/(9 + E^13 + 12*x - 4*E^(39/4)*x + 10*x^2 + 4*x^3 + x^4 + E^(13/2)*(6 + 4*x +
6*x^2) + E^(13/4)*(-12*x - 8*x^2 - 4*x^3)),x]

[Out]

(90*x)/(3 + E^(13/2) + 2*(1 - E^(13/4))*x + x^2)

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 1694

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -d/(4*e) + x)*(a + d^4/(256*e^3)
- b*(d/(8*e)) + (c - 3*(d^2/(8*e)))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0]
&& NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rule 1828

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*
g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \int \frac {270+90 e^{13/2}-90 x^2}{9+e^{13}+\left (12-4 e^{39/4}\right ) x+10 x^2+4 x^3+x^4+e^{13/2} \left (6+4 x+6 x^2\right )+e^{13/4} \left (-12 x-8 x^2-4 x^3\right )} \, dx \\ & = \text {Subst}\left (\int \frac {90 \left (2 \left (1+e^{13/4}\right )+2 \left (1-e^{13/4}\right ) x-x^2\right )}{\left (2+2 e^{13/4}+x^2\right )^2} \, dx,x,\frac {1}{4} \left (4-4 e^{13/4}\right )+x\right ) \\ & = 90 \text {Subst}\left (\int \frac {2 \left (1+e^{13/4}\right )+2 \left (1-e^{13/4}\right ) x-x^2}{\left (2+2 e^{13/4}+x^2\right )^2} \, dx,x,\frac {1}{4} \left (4-4 e^{13/4}\right )+x\right ) \\ & = \frac {90 x}{3+e^{13/2}+2 \left (1-e^{13/4}\right ) x+x^2}-\frac {45 \text {Subst}\left (\int 0 \, dx,x,\frac {1}{4} \left (4-4 e^{13/4}\right )+x\right )}{2 \left (1+e^{13/4}\right )} \\ & = \frac {90 x}{3+e^{13/2}+2 \left (1-e^{13/4}\right ) x+x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {270+90 e^{13/2}-90 x^2}{9+e^{13}+12 x-4 e^{39/4} x+10 x^2+4 x^3+x^4+e^{13/2} \left (6+4 x+6 x^2\right )+e^{13/4} \left (-12 x-8 x^2-4 x^3\right )} \, dx=\frac {90 x}{3+e^{13/2}+2 x-2 e^{13/4} x+x^2} \]

[In]

Integrate[(270 + 90*E^(13/2) - 90*x^2)/(9 + E^13 + 12*x - 4*E^(39/4)*x + 10*x^2 + 4*x^3 + x^4 + E^(13/2)*(6 +
4*x + 6*x^2) + E^(13/4)*(-12*x - 8*x^2 - 4*x^3)),x]

[Out]

(90*x)/(3 + E^(13/2) + 2*x - 2*E^(13/4)*x + x^2)

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

method	result	size
risch	\(\frac {90 x}{{\mathrm e}^{\frac {13}{2}}-2 \,{\mathrm e}^{\frac {13}{4}} x +x^{2}+2 x +3}\)	\(21\)
gosper	\(\frac {90 x}{{\mathrm e}^{\frac {13}{2}}-2 \,{\mathrm e}^{\frac {13}{4}} x +x^{2}+2 x +3}\)	\(23\)
norman	\(\frac {90 x}{{\mathrm e}^{\frac {13}{2}}-2 \,{\mathrm e}^{\frac {13}{4}} x +x^{2}+2 x +3}\)	\(23\)
parallelrisch	\(\frac {90 x}{{\mathrm e}^{\frac {13}{2}}-2 \,{\mathrm e}^{\frac {13}{4}} x +x^{2}+2 x +3}\)	\(23\)
default	\(\frac {45 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\left (-4 \,{\mathrm e}^{\frac {13}{4}}+4\right ) \textit {\_Z}^{3}+\left (6 \,{\mathrm e}^{\frac {13}{2}}-8 \,{\mathrm e}^{\frac {13}{4}}+10\right ) \textit {\_Z}^{2}+\left (-4 \,{\mathrm e}^{\frac {39}{4}}+4 \,{\mathrm e}^{\frac {13}{2}}-12 \,{\mathrm e}^{\frac {13}{4}}+12\right ) \textit {\_Z} +{\mathrm e}^{13}+9+6 \,{\mathrm e}^{\frac {13}{2}}\right )}{\sum }\frac {\left ({\mathrm e}^{\frac {13}{2}}-\textit {\_R}^{2}+3\right ) \ln \left (x -\textit {\_R} \right )}{3-{\mathrm e}^{\frac {39}{4}}+3 \,{\mathrm e}^{\frac {13}{2}} \textit {\_R} -3 \,{\mathrm e}^{\frac {13}{4}} \textit {\_R}^{2}+\textit {\_R}^{3}+{\mathrm e}^{\frac {13}{2}}-4 \,{\mathrm e}^{\frac {13}{4}} \textit {\_R} +3 \textit {\_R}^{2}-3 \,{\mathrm e}^{\frac {13}{4}}+5 \textit {\_R}}\right )}{2}\)	\(116\)

[In]

int((90*exp(13/4)^2-90*x^2+270)/(exp(13/4)^4-4*x*exp(13/4)^3+(6*x^2+4*x+6)*exp(13/4)^2+(-4*x^3-8*x^2-12*x)*exp
(13/4)+x^4+4*x^3+10*x^2+12*x+9),x,method=_RETURNVERBOSE)

[Out]

90*x/(exp(13/2)-2*exp(13/4)*x+x^2+2*x+3)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {270+90 e^{13/2}-90 x^2}{9+e^{13}+12 x-4 e^{39/4} x+10 x^2+4 x^3+x^4+e^{13/2} \left (6+4 x+6 x^2\right )+e^{13/4} \left (-12 x-8 x^2-4 x^3\right )} \, dx=\frac {90 \, x}{x^{2} - 2 \, x e^{\frac {13}{4}} + 2 \, x + e^{\frac {13}{2}} + 3} \]

[In]

integrate((90*exp(13/4)^2-90*x^2+270)/(exp(13/4)^4-4*x*exp(13/4)^3+(6*x^2+4*x+6)*exp(13/4)^2+(-4*x^3-8*x^2-12*
x)*exp(13/4)+x^4+4*x^3+10*x^2+12*x+9),x, algorithm="fricas")

[Out]

90*x/(x^2 - 2*x*e^(13/4) + 2*x + e^(13/2) + 3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (17) = 34\).

Time = 0.70 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.22 \[ \int \frac {270+90 e^{13/2}-90 x^2}{9+e^{13}+12 x-4 e^{39/4} x+10 x^2+4 x^3+x^4+e^{13/2} \left (6+4 x+6 x^2\right )+e^{13/4} \left (-12 x-8 x^2-4 x^3\right )} \, dx=- \frac {x \left (- 90 e^{\frac {13}{4}} - 90\right )}{x^{2} \cdot \left (1 + e^{\frac {13}{4}}\right ) + x \left (2 - 2 e^{\frac {13}{2}}\right ) + 3 + 3 e^{\frac {13}{4}} + e^{\frac {13}{2}} + e^{\frac {39}{4}}} \]

[In]

integrate((90*exp(13/4)**2-90*x**2+270)/(exp(13/4)**4-4*x*exp(13/4)**3+(6*x**2+4*x+6)*exp(13/4)**2+(-4*x**3-8*
x**2-12*x)*exp(13/4)+x**4+4*x**3+10*x**2+12*x+9),x)

[Out]

-x*(-90*exp(13/4) - 90)/(x**2*(1 + exp(13/4)) + x*(2 - 2*exp(13/2)) + 3 + 3*exp(13/4) + exp(13/2) + exp(39/4))

Maxima [F]

\[ \int \frac {270+90 e^{13/2}-90 x^2}{9+e^{13}+12 x-4 e^{39/4} x+10 x^2+4 x^3+x^4+e^{13/2} \left (6+4 x+6 x^2\right )+e^{13/4} \left (-12 x-8 x^2-4 x^3\right )} \, dx=\int { -\frac {90 \, {\left (x^{2} - e^{\frac {13}{2}} - 3\right )}}{x^{4} + 4 \, x^{3} + 10 \, x^{2} - 4 \, x e^{\frac {39}{4}} + 2 \, {\left (3 \, x^{2} + 2 \, x + 3\right )} e^{\frac {13}{2}} - 4 \, {\left (x^{3} + 2 \, x^{2} + 3 \, x\right )} e^{\frac {13}{4}} + 12 \, x + e^{13} + 9} \,d x } \]

[In]

integrate((90*exp(13/4)^2-90*x^2+270)/(exp(13/4)^4-4*x*exp(13/4)^3+(6*x^2+4*x+6)*exp(13/4)^2+(-4*x^3-8*x^2-12*
x)*exp(13/4)+x^4+4*x^3+10*x^2+12*x+9),x, algorithm="maxima")

[Out]

-90*integrate((x^2 - e^(13/2) - 3)/(x^4 + 4*x^3 + 10*x^2 - 4*x*e^(39/4) + 2*(3*x^2 + 2*x + 3)*e^(13/2) - 4*(x^
3 + 2*x^2 + 3*x)*e^(13/4) + 12*x + e^13 + 9), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {270+90 e^{13/2}-90 x^2}{9+e^{13}+12 x-4 e^{39/4} x+10 x^2+4 x^3+x^4+e^{13/2} \left (6+4 x+6 x^2\right )+e^{13/4} \left (-12 x-8 x^2-4 x^3\right )} \, dx=\frac {90 \, x}{x^{2} - 2 \, x e^{\frac {13}{4}} + 2 \, x + e^{\frac {13}{2}} + 3} \]

[In]

integrate((90*exp(13/4)^2-90*x^2+270)/(exp(13/4)^4-4*x*exp(13/4)^3+(6*x^2+4*x+6)*exp(13/4)^2+(-4*x^3-8*x^2-12*
x)*exp(13/4)+x^4+4*x^3+10*x^2+12*x+9),x, algorithm="giac")

[Out]

90*x/(x^2 - 2*x*e^(13/4) + 2*x + e^(13/2) + 3)

Mupad [B] (verification not implemented)

Time = 12.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {270+90 e^{13/2}-90 x^2}{9+e^{13}+12 x-4 e^{39/4} x+10 x^2+4 x^3+x^4+e^{13/2} \left (6+4 x+6 x^2\right )+e^{13/4} \left (-12 x-8 x^2-4 x^3\right )} \, dx=\frac {90\,x}{x^2+\left (2-2\,{\mathrm {e}}^{13/4}\right )\,x+{\mathrm {e}}^{13/2}+3} \]

[In]

int((90*exp(13/2) - 90*x^2 + 270)/(12*x + exp(13) + exp(13/2)*(4*x + 6*x^2 + 6) - 4*x*exp(39/4) - exp(13/4)*(1
2*x + 8*x^2 + 4*x^3) + 10*x^2 + 4*x^3 + x^4 + 9),x)

[Out]

(90*x)/(exp(13/2) + x^2 - x*(2*exp(13/4) - 2) + 3)