3.10.77 \(\int \frac {3+2 x^3+e^{x+17 x^2} (-1+x+34 x^2)}{9+e^{2 x+34 x^2}-18 x+9 x^2-6 x^3+6 x^4+x^6+e^{x+17 x^2} (-6+6 x+2 x^3)} \, dx\)

Optimal. Leaf size=28 \[ \frac {x}{3-x \left (3+\frac {e^{x+17 x^2}+x^3}{x}\right )} \]

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Rubi [F]  time = 1.49, 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 {3+2 x^3+e^{x+17 x^2} \left (-1+x+34 x^2\right )}{9+e^{2 x+34 x^2}-18 x+9 x^2-6 x^3+6 x^4+x^6+e^{x+17 x^2} \left (-6+6 x+2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3 + 2*x^3 + E^(x + 17*x^2)*(-1 + x + 34*x^2))/(9 + E^(2*x + 34*x^2) - 18*x + 9*x^2 - 6*x^3 + 6*x^4 + x^6
+ E^(x + 17*x^2)*(-6 + 6*x + 2*x^3)),x]

[Out]

6*Defer[Int][x/(-3 + E^(x + 17*x^2) + 3*x + x^3)^2, x] + 99*Defer[Int][x^2/(-3 + E^(x + 17*x^2) + 3*x + x^3)^2
, x] - 99*Defer[Int][x^3/(-3 + E^(x + 17*x^2) + 3*x + x^3)^2, x] - Defer[Int][x^4/(-3 + E^(x + 17*x^2) + 3*x +
x^3)^2, x] - 34*Defer[Int][x^5/(-3 + E^(x + 17*x^2) + 3*x + x^3)^2, x] - Defer[Int][(-3 + E^(x + 17*x^2) + 3*
x + x^3)^(-1), x] + Defer[Int][x/(-3 + E^(x + 17*x^2) + 3*x + x^3), x] + 34*Defer[Int][x^2/(-3 + E^(x + 17*x^2
) + 3*x + x^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3+2 x^3+e^{x+17 x^2} \left (-1+x+34 x^2\right )}{\left (3-e^{x+17 x^2}-3 x-x^3\right )^2} \, dx\\ &=\int \left (\frac {-1+x+34 x^2}{-3+e^{x+17 x^2}+3 x+x^3}-\frac {x \left (-6-99 x+99 x^2+x^3+34 x^4\right )}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2}\right ) \, dx\\ &=\int \frac {-1+x+34 x^2}{-3+e^{x+17 x^2}+3 x+x^3} \, dx-\int \frac {x \left (-6-99 x+99 x^2+x^3+34 x^4\right )}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2} \, dx\\ &=-\int \left (-\frac {6 x}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2}-\frac {99 x^2}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2}+\frac {99 x^3}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2}+\frac {x^4}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2}+\frac {34 x^5}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2}\right ) \, dx+\int \left (-\frac {1}{-3+e^{x+17 x^2}+3 x+x^3}+\frac {x}{-3+e^{x+17 x^2}+3 x+x^3}+\frac {34 x^2}{-3+e^{x+17 x^2}+3 x+x^3}\right ) \, dx\\ &=6 \int \frac {x}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2} \, dx-34 \int \frac {x^5}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2} \, dx+34 \int \frac {x^2}{-3+e^{x+17 x^2}+3 x+x^3} \, dx+99 \int \frac {x^2}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2} \, dx-99 \int \frac {x^3}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2} \, dx-\int \frac {x^4}{\left (-3+e^{x+17 x^2}+3 x+x^3\right )^2} \, dx-\int \frac {1}{-3+e^{x+17 x^2}+3 x+x^3} \, dx+\int \frac {x}{-3+e^{x+17 x^2}+3 x+x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 22, normalized size = 0.79 \begin {gather*} -\frac {x}{-3+e^{x+17 x^2}+3 x+x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 2*x^3 + E^(x + 17*x^2)*(-1 + x + 34*x^2))/(9 + E^(2*x + 34*x^2) - 18*x + 9*x^2 - 6*x^3 + 6*x^4
+ x^6 + E^(x + 17*x^2)*(-6 + 6*x + 2*x^3)),x]

[Out]

-(x/(-3 + E^(x + 17*x^2) + 3*x + x^3))

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fricas [A]  time = 0.56, size = 21, normalized size = 0.75 \begin {gather*} -\frac {x}{x^{3} + 3 \, x + e^{\left (17 \, x^{2} + x\right )} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((34*x^2+x-1)*exp(x)*exp(17*x^2)+2*x^3+3)/(exp(x)^2*exp(17*x^2)^2+(2*x^3+6*x-6)*exp(x)*exp(17*x^2)+x
^6+6*x^4-6*x^3+9*x^2-18*x+9),x, algorithm="fricas")

[Out]

-x/(x^3 + 3*x + e^(17*x^2 + x) - 3)

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giac [A]  time = 0.37, size = 21, normalized size = 0.75 \begin {gather*} -\frac {x}{x^{3} + 3 \, x + e^{\left (17 \, x^{2} + x\right )} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((34*x^2+x-1)*exp(x)*exp(17*x^2)+2*x^3+3)/(exp(x)^2*exp(17*x^2)^2+(2*x^3+6*x-6)*exp(x)*exp(17*x^2)+x
^6+6*x^4-6*x^3+9*x^2-18*x+9),x, algorithm="giac")

[Out]

-x/(x^3 + 3*x + e^(17*x^2 + x) - 3)

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maple [A]  time = 0.04, size = 22, normalized size = 0.79

method	result	size
risch	\(-\frac {x}{x^{3}+{\mathrm e}^{x \left (17 x +1\right )}+3 x -3}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((34*x^2+x-1)*exp(x)*exp(17*x^2)+2*x^3+3)/(exp(x)^2*exp(17*x^2)^2+(2*x^3+6*x-6)*exp(x)*exp(17*x^2)+x^6+6*x
^4-6*x^3+9*x^2-18*x+9),x,method=_RETURNVERBOSE)

[Out]

-x/(x^3+exp(x*(17*x+1))+3*x-3)

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maxima [A]  time = 0.40, size = 21, normalized size = 0.75 \begin {gather*} -\frac {x}{x^{3} + 3 \, x + e^{\left (17 \, x^{2} + x\right )} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((34*x^2+x-1)*exp(x)*exp(17*x^2)+2*x^3+3)/(exp(x)^2*exp(17*x^2)^2+(2*x^3+6*x-6)*exp(x)*exp(17*x^2)+x
^6+6*x^4-6*x^3+9*x^2-18*x+9),x, algorithm="maxima")

[Out]

-x/(x^3 + 3*x + e^(17*x^2 + x) - 3)

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mupad [B]  time = 0.82, size = 21, normalized size = 0.75 \begin {gather*} -\frac {x}{3\,x+{\mathrm {e}}^{17\,x^2+x}+x^3-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^3 + exp(17*x^2)*exp(x)*(x + 34*x^2 - 1) + 3)/(exp(2*x)*exp(34*x^2) - 18*x + 9*x^2 - 6*x^3 + 6*x^4 + x
^6 + exp(17*x^2)*exp(x)*(6*x + 2*x^3 - 6) + 9),x)

[Out]

-x/(3*x + exp(x + 17*x^2) + x^3 - 3)

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sympy [A]  time = 0.21, size = 20, normalized size = 0.71 \begin {gather*} - \frac {x}{x^{3} + 3 x + e^{x} e^{17 x^{2}} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((34*x**2+x-1)*exp(x)*exp(17*x**2)+2*x**3+3)/(exp(x)**2*exp(17*x**2)**2+(2*x**3+6*x-6)*exp(x)*exp(17
*x**2)+x**6+6*x**4-6*x**3+9*x**2-18*x+9),x)

[Out]

-x/(x**3 + 3*x + exp(x)*exp(17*x**2) - 3)

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