\(\int \frac {432 x+216 e^3 x^2+e^6 (-18900+36 x^3)+e^9 (3150 x+2 x^4)}{216+108 e^3 x+18 e^6 x^2+e^9 x^3} \, dx\) [3827]

Optimal result

Integrand size = 62, antiderivative size = 15 \[ \int \frac {432 x+216 e^3 x^2+e^6 \left (-18900+36 x^3\right )+e^9 \left (3150 x+2 x^4\right )}{216+108 e^3 x+18 e^6 x^2+e^9 x^3} \, dx=x \left (x-\frac {3150}{\left (\frac {6}{e^3}+x\right )^2}\right ) \]

[Out]

(x-3150/(x+6/exp(3))^2)*x

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(32\) vs. \(2(15)=30\).

Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.13, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {2099} \[ \int \frac {432 x+216 e^3 x^2+e^6 \left (-18900+36 x^3\right )+e^9 \left (3150 x+2 x^4\right )}{216+108 e^3 x+18 e^6 x^2+e^9 x^3} \, dx=x^2-\frac {3150 e^3}{e^3 x+6}+\frac {18900 e^3}{\left (e^3 x+6\right )^2} \]

[In]

Int[(432*x + 216*E^3*x^2 + E^6*(-18900 + 36*x^3) + E^9*(3150*x + 2*x^4))/(216 + 108*E^3*x + 18*E^6*x^2 + E^9*x
^3),x]

[Out]

x^2 + (18900*E^3)/(6 + E^3*x)^2 - (3150*E^3)/(6 + E^3*x)

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 (2 x-\frac {37800 e^6}{\left (6+e^3 x\right )^3}+\frac {3150 e^6}{\left (6+e^3 x\right )^2}\right ) \, dx \\ & = x^2+\frac {18900 e^3}{\left (6+e^3 x\right )^2}-\frac {3150 e^3}{6+e^3 x} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(32\) vs. \(2(15)=30\).

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.13 \[ \int \frac {432 x+216 e^3 x^2+e^6 \left (-18900+36 x^3\right )+e^9 \left (3150 x+2 x^4\right )}{216+108 e^3 x+18 e^6 x^2+e^9 x^3} \, dx=x^2+\frac {18900 e^3}{\left (6+e^3 x\right )^2}-\frac {3150 e^3}{6+e^3 x} \]

[In]

Integrate[(432*x + 216*E^3*x^2 + E^6*(-18900 + 36*x^3) + E^9*(3150*x + 2*x^4))/(216 + 108*E^3*x + 18*E^6*x^2 +
E^9*x^3),x]

[Out]

x^2 + (18900*E^3)/(6 + E^3*x)^2 - (3150*E^3)/(6 + E^3*x)

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.67

method	result	size
risch	\(x^{2}-\frac {3150 x \,{\mathrm e}^{6}}{x^{2} {\mathrm e}^{6}+12 x \,{\mathrm e}^{3}+36}\)	\(25\)
norman	\(\frac {{\mathrm e}^{6} x^{4}-3150 x \,{\mathrm e}^{6}+36 x^{2}+12 x^{3} {\mathrm e}^{3}}{\left (x \,{\mathrm e}^{3}+6\right )^{2}}\)	\(38\)
gosper	\(\frac {x \left (x^{3} {\mathrm e}^{6}+12 x^{2} {\mathrm e}^{3}-3150 \,{\mathrm e}^{6}+36 x \right )}{x^{2} {\mathrm e}^{6}+12 x \,{\mathrm e}^{3}+36}\)	\(45\)
parallelrisch	\(\frac {36 \,{\mathrm e}^{6} x^{4}+432 x^{3} {\mathrm e}^{3}-113400 x \,{\mathrm e}^{6}+1296 x^{2}}{36 x^{2} {\mathrm e}^{6}+432 x \,{\mathrm e}^{3}+1296}\)	\(49\)
default	\(x^{2}+1050 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} {\mathrm e}^{9}+18 \textit {\_Z}^{2} {\mathrm e}^{6}+108 \,{\mathrm e}^{3} \textit {\_Z} +216\right )}{\sum }\frac {\left (\textit {\_R} \,{\mathrm e}^{9}-6 \,{\mathrm e}^{6}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2} {\mathrm e}^{9}+12 \textit {\_R} \,{\mathrm e}^{6}+36 \,{\mathrm e}^{3}}\right )\)	\(65\)

[In]

int(((2*x^4+3150*x)*exp(3)^3+(36*x^3-18900)*exp(3)^2+216*x^2*exp(3)+432*x)/(x^3*exp(3)^3+18*x^2*exp(3)^2+108*x
*exp(3)+216),x,method=_RETURNVERBOSE)

[Out]

x^2-3150*x*exp(6)/(x^2*exp(6)+12*x*exp(3)+36)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (14) = 28\).

Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.60 \[ \int \frac {432 x+216 e^3 x^2+e^6 \left (-18900+36 x^3\right )+e^9 \left (3150 x+2 x^4\right )}{216+108 e^3 x+18 e^6 x^2+e^9 x^3} \, dx=\frac {12 \, x^{3} e^{3} + 36 \, x^{2} + {\left (x^{4} - 3150 \, x\right )} e^{6}}{x^{2} e^{6} + 12 \, x e^{3} + 36} \]

[In]

integrate(((2*x^4+3150*x)*exp(3)^3+(36*x^3-18900)*exp(3)^2+216*x^2*exp(3)+432*x)/(x^3*exp(3)^3+18*x^2*exp(3)^2
+108*x*exp(3)+216),x, algorithm="fricas")

[Out]

(12*x^3*e^3 + 36*x^2 + (x^4 - 3150*x)*e^6)/(x^2*e^6 + 12*x*e^3 + 36)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.60 \[ \int \frac {432 x+216 e^3 x^2+e^6 \left (-18900+36 x^3\right )+e^9 \left (3150 x+2 x^4\right )}{216+108 e^3 x+18 e^6 x^2+e^9 x^3} \, dx=x^{2} - \frac {3150 x e^{6}}{x^{2} e^{6} + 12 x e^{3} + 36} \]

[In]

integrate(((2*x**4+3150*x)*exp(3)**3+(36*x**3-18900)*exp(3)**2+216*x**2*exp(3)+432*x)/(x**3*exp(3)**3+18*x**2*
exp(3)**2+108*x*exp(3)+216),x)

[Out]

x**2 - 3150*x*exp(6)/(x**2*exp(6) + 12*x*exp(3) + 36)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.60 \[ \int \frac {432 x+216 e^3 x^2+e^6 \left (-18900+36 x^3\right )+e^9 \left (3150 x+2 x^4\right )}{216+108 e^3 x+18 e^6 x^2+e^9 x^3} \, dx=x^{2} - \frac {3150 \, x e^{6}}{x^{2} e^{6} + 12 \, x e^{3} + 36} \]

[In]

integrate(((2*x^4+3150*x)*exp(3)^3+(36*x^3-18900)*exp(3)^2+216*x^2*exp(3)+432*x)/(x^3*exp(3)^3+18*x^2*exp(3)^2
+108*x*exp(3)+216),x, algorithm="maxima")

[Out]

x^2 - 3150*x*e^6/(x^2*e^6 + 12*x*e^3 + 36)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {432 x+216 e^3 x^2+e^6 \left (-18900+36 x^3\right )+e^9 \left (3150 x+2 x^4\right )}{216+108 e^3 x+18 e^6 x^2+e^9 x^3} \, dx=x^{2} - \frac {3150 \, x e^{6}}{{\left (x e^{3} + 6\right )}^{2}} \]

[In]

integrate(((2*x^4+3150*x)*exp(3)^3+(36*x^3-18900)*exp(3)^2+216*x^2*exp(3)+432*x)/(x^3*exp(3)^3+18*x^2*exp(3)^2
+108*x*exp(3)+216),x, algorithm="giac")

[Out]

x^2 - 3150*x*e^6/(x*e^3 + 6)^2

Mupad [B] (verification not implemented)

Time = 9.34 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {432 x+216 e^3 x^2+e^6 \left (-18900+36 x^3\right )+e^9 \left (3150 x+2 x^4\right )}{216+108 e^3 x+18 e^6 x^2+e^9 x^3} \, dx=x^2-\frac {3150\,x\,{\mathrm {e}}^6}{{\left (x\,{\mathrm {e}}^3+6\right )}^2} \]

[In]

int((432*x + exp(9)*(3150*x + 2*x^4) + exp(6)*(36*x^3 - 18900) + 216*x^2*exp(3))/(108*x*exp(3) + 18*x^2*exp(6)
+ x^3*exp(9) + 216),x)

[Out]

x^2 - (3150*x*exp(6))/(x*exp(3) + 6)^2