3.71.48 \(\int \frac {-270 x^2+54 x^4+e^x (-108 x^3+54 x^4)}{25-30 x^2+9 e^{2 x} x^2-60 x^3+9 x^4+36 x^5+36 x^6+e^x (30 x-18 x^3-36 x^4)} \, dx\)

Optimal. Leaf size=28 \[ \frac {3 x}{x+\frac {-e^x-\frac {5}{3 x}+x}{2 x}} \]

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Rubi [F]  time = 1.85, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-270 x^2+54 x^4+e^x \left (-108 x^3+54 x^4\right )}{25-30 x^2+9 e^{2 x} x^2-60 x^3+9 x^4+36 x^5+36 x^6+e^x \left (30 x-18 x^3-36 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-270*x^2 + 54*x^4 + E^x*(-108*x^3 + 54*x^4))/(25 - 30*x^2 + 9*E^(2*x)*x^2 - 60*x^3 + 9*x^4 + 36*x^5 + 36*
x^6 + E^x*(30*x - 18*x^3 - 36*x^4)),x]

[Out]

-90*Defer[Int][x^2/(-5 - 3*E^x*x + 3*x^2 + 6*x^3)^2, x] - 90*Defer[Int][x^3/(-5 - 3*E^x*x + 3*x^2 + 6*x^3)^2,
x] - 54*Defer[Int][x^4/(-5 - 3*E^x*x + 3*x^2 + 6*x^3)^2, x] - 162*Defer[Int][x^5/(-5 - 3*E^x*x + 3*x^2 + 6*x^3
)^2, x] + 108*Defer[Int][x^6/(-5 - 3*E^x*x + 3*x^2 + 6*x^3)^2, x] + 36*Defer[Int][x^2/(-5 - 3*E^x*x + 3*x^2 +
6*x^3), x] - 18*Defer[Int][x^3/(-5 - 3*E^x*x + 3*x^2 + 6*x^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {54 x^2 \left (-5+e^x (-2+x) x+x^2\right )}{\left (5+3 e^x x-3 x^2-6 x^3\right )^2} \, dx\\ &=54 \int \frac {x^2 \left (-5+e^x (-2+x) x+x^2\right )}{\left (5+3 e^x x-3 x^2-6 x^3\right )^2} \, dx\\ &=54 \int \left (-\frac {(-2+x) x^2}{3 \left (-5-3 e^x x+3 x^2+6 x^3\right )}+\frac {x^2 \left (-5-5 x-3 x^2-9 x^3+6 x^4\right )}{3 \left (-5-3 e^x x+3 x^2+6 x^3\right )^2}\right ) \, dx\\ &=-\left (18 \int \frac {(-2+x) x^2}{-5-3 e^x x+3 x^2+6 x^3} \, dx\right )+18 \int \frac {x^2 \left (-5-5 x-3 x^2-9 x^3+6 x^4\right )}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2} \, dx\\ &=18 \int \left (-\frac {5 x^2}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2}-\frac {5 x^3}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2}-\frac {3 x^4}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2}-\frac {9 x^5}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2}+\frac {6 x^6}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2}\right ) \, dx-18 \int \left (-\frac {2 x^2}{-5-3 e^x x+3 x^2+6 x^3}+\frac {x^3}{-5-3 e^x x+3 x^2+6 x^3}\right ) \, dx\\ &=-\left (18 \int \frac {x^3}{-5-3 e^x x+3 x^2+6 x^3} \, dx\right )+36 \int \frac {x^2}{-5-3 e^x x+3 x^2+6 x^3} \, dx-54 \int \frac {x^4}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2} \, dx-90 \int \frac {x^2}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2} \, dx-90 \int \frac {x^3}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2} \, dx+108 \int \frac {x^6}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2} \, dx-162 \int \frac {x^5}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.39, size = 25, normalized size = 0.89 \begin {gather*} -\frac {18 x^3}{5+3 e^x x-3 x^2-6 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-270*x^2 + 54*x^4 + E^x*(-108*x^3 + 54*x^4))/(25 - 30*x^2 + 9*E^(2*x)*x^2 - 60*x^3 + 9*x^4 + 36*x^5
+ 36*x^6 + E^x*(30*x - 18*x^3 - 36*x^4)),x]

[Out]

(-18*x^3)/(5 + 3*E^x*x - 3*x^2 - 6*x^3)

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fricas [A]  time = 0.56, size = 24, normalized size = 0.86 \begin {gather*} \frac {18 \, x^{3}}{6 \, x^{3} + 3 \, x^{2} - 3 \, x e^{x} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((54*x^4-108*x^3)*exp(x)+54*x^4-270*x^2)/(9*exp(x)^2*x^2+(-36*x^4-18*x^3+30*x)*exp(x)+36*x^6+36*x^5+
9*x^4-60*x^3-30*x^2+25),x, algorithm="fricas")

[Out]

18*x^3/(6*x^3 + 3*x^2 - 3*x*e^x - 5)

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giac [A]  time = 0.26, size = 24, normalized size = 0.86 \begin {gather*} \frac {18 \, x^{3}}{6 \, x^{3} + 3 \, x^{2} - 3 \, x e^{x} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((54*x^4-108*x^3)*exp(x)+54*x^4-270*x^2)/(9*exp(x)^2*x^2+(-36*x^4-18*x^3+30*x)*exp(x)+36*x^6+36*x^5+
9*x^4-60*x^3-30*x^2+25),x, algorithm="giac")

[Out]

18*x^3/(6*x^3 + 3*x^2 - 3*x*e^x - 5)

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maple [A]  time = 0.07, size = 25, normalized size = 0.89

method	result	size
risch	\(\frac {18 x^{3}}{6 x^{3}+3 x^{2}-3 \,{\mathrm e}^{x} x -5}\)	\(25\)
norman	\(\frac {-9 x^{2}+9 \,{\mathrm e}^{x} x +15}{6 x^{3}+3 x^{2}-3 \,{\mathrm e}^{x} x -5}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((54*x^4-108*x^3)*exp(x)+54*x^4-270*x^2)/(9*exp(x)^2*x^2+(-36*x^4-18*x^3+30*x)*exp(x)+36*x^6+36*x^5+9*x^4-
60*x^3-30*x^2+25),x,method=_RETURNVERBOSE)

[Out]

18*x^3/(6*x^3+3*x^2-3*exp(x)*x-5)

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maxima [A]  time = 0.41, size = 24, normalized size = 0.86 \begin {gather*} \frac {18 \, x^{3}}{6 \, x^{3} + 3 \, x^{2} - 3 \, x e^{x} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((54*x^4-108*x^3)*exp(x)+54*x^4-270*x^2)/(9*exp(x)^2*x^2+(-36*x^4-18*x^3+30*x)*exp(x)+36*x^6+36*x^5+
9*x^4-60*x^3-30*x^2+25),x, algorithm="maxima")

[Out]

18*x^3/(6*x^3 + 3*x^2 - 3*x*e^x - 5)

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mupad [B]  time = 4.21, size = 24, normalized size = 0.86 \begin {gather*} -\frac {18\,x^3}{3\,x\,{\mathrm {e}}^x-3\,x^2-6\,x^3+5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(108*x^3 - 54*x^4) + 270*x^2 - 54*x^4)/(9*x^2*exp(2*x) - 30*x^2 - 60*x^3 + 9*x^4 + 36*x^5 + 36*x^
6 - exp(x)*(18*x^3 - 30*x + 36*x^4) + 25),x)

[Out]

-(18*x^3)/(3*x*exp(x) - 3*x^2 - 6*x^3 + 5)

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sympy [A]  time = 0.17, size = 24, normalized size = 0.86 \begin {gather*} - \frac {18 x^{3}}{- 6 x^{3} - 3 x^{2} + 3 x e^{x} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((54*x**4-108*x**3)*exp(x)+54*x**4-270*x**2)/(9*exp(x)**2*x**2+(-36*x**4-18*x**3+30*x)*exp(x)+36*x**
6+36*x**5+9*x**4-60*x**3-30*x**2+25),x)

[Out]

-18*x**3/(-6*x**3 - 3*x**2 + 3*x*exp(x) + 5)

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