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

Optimal. Leaf size=15 \[ x \left (x-\frac {3150}{\left (\frac {6}{e^3}+x\right )^2}\right ) \]

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Rubi [B]  time = 0.06, antiderivative size = 32, normalized size of antiderivative = 2.13, number of steps used = 2, number of rules used = 1, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {2074} \begin {gather*} x^2-\frac {3150 e^3}{e^3 x+6}+\frac {18900 e^3}{\left (e^3 x+6\right )^2} \end {gather*}

Antiderivative was successfully verified.

[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 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 (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 {aligned} \end {gather*}

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Mathematica [B]  time = 0.02, size = 32, normalized size = 2.13 \begin {gather*} x^2+\frac {18900 e^3}{\left (6+e^3 x\right )^2}-\frac {3150 e^3}{6+e^3 x} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [B]  time = 0.59, size = 39, normalized size = 2.60 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (108 \, x^{2} e^{3} + {\left (x^{4} + 1575 \, x\right )} e^{9} + 18 \, {\left (x^{3} - 525\right )} e^{6} + 216 \, x\right )}}{x^{3} e^{9} + 18 \, x^{2} e^{6} + 108 \, x e^{3} + 216}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

integrate(2*(108*x^2*e^3 + (x^4 + 1575*x)*e^9 + 18*(x^3 - 525)*e^6 + 216*x)/(x^3*e^9 + 18*x^2*e^6 + 108*x*e^3
+ 216), x)

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maple [A]  time = 0.05, size = 25, normalized size = 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	\(\frac {x^{4} {\mathrm e}^{6}-3150 x \,{\mathrm e}^{6}+36 x^{2}+12 x^{3} {\mathrm e}^{3}}{\left (x \,{\mathrm e}^{3}+6\right )^{2}}\)	\(38\)
gosper	\(\frac {x \left (x^{3} {\mathrm e}^{6}+12 x^{2} {\mathrm e}^{3}-3150 \,{\mathrm e}^{6}+36 x \right )}{x^{2} {\mathrm e}^{6}+12 x \,{\mathrm e}^{3}+36}\)	\(45\)
default	\(x^{2}+1050 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3} {\mathrm e}^{9}+18 \textit {\_Z}^{2} {\mathrm e}^{6}+108 \textit {\_Z} \,{\mathrm e}^{3}+216\right )}{\sum }\frac {\left (\textit {\_R} \,{\mathrm e}^{9}-6 \,{\mathrm e}^{6}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2} {\mathrm e}^{9}+12 \textit {\_R} \,{\mathrm e}^{6}+36 \,{\mathrm e}^{3}}\right )\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.45, size = 24, normalized size = 1.60 \begin {gather*} x^{2} - \frac {3150 \, x e^{6}}{x^{2} e^{6} + 12 \, x e^{3} + 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 2.25, size = 17, normalized size = 1.13 \begin {gather*} x^2-\frac {3150\,x\,{\mathrm {e}}^6}{{\left (x\,{\mathrm {e}}^3+6\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.20, size = 24, normalized size = 1.60 \begin {gather*} x^{2} - \frac {3150 x e^{6}}{x^{2} e^{6} + 12 x e^{3} + 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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