3.99.11 \(\int \frac {5+30 x^2+45 x^4+e^{\frac {81 x^3}{5+15 x^2}} (-5-30 x^2-243 x^3-45 x^4-243 x^5)}{5+30 x^2+45 x^4} \, dx\)

Optimal. Leaf size=22 \[ \left (1-e^{\frac {81 x^3}{5+15 x^2}}\right ) x \]

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Rubi [B]  time = 0.33, antiderivative size = 68, normalized size of antiderivative = 3.09, number of steps used = 4, number of rules used = 3, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {28, 6742, 2288} \begin {gather*} \frac {e^{\frac {81 x^3}{5 \left (3 x^2+1\right )}} \left (x^5+x^3\right )}{\left (3 x^2+1\right )^2 \left (\frac {2 x^4}{\left (3 x^2+1\right )^2}-\frac {x^2}{3 x^2+1}\right )}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 + 30*x^2 + 45*x^4 + E^((81*x^3)/(5 + 15*x^2))*(-5 - 30*x^2 - 243*x^3 - 45*x^4 - 243*x^5))/(5 + 30*x^2 +
45*x^4),x]

[Out]

x + (E^((81*x^3)/(5*(1 + 3*x^2)))*(x^3 + x^5))/((1 + 3*x^2)^2*((2*x^4)/(1 + 3*x^2)^2 - x^2/(1 + 3*x^2)))

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=45 \int \frac {5+30 x^2+45 x^4+e^{\frac {81 x^3}{5+15 x^2}} \left (-5-30 x^2-243 x^3-45 x^4-243 x^5\right )}{\left (15+45 x^2\right )^2} \, dx\\ &=45 \int \left (\frac {1}{45}-\frac {e^{\frac {81 x^3}{5+15 x^2}} \left (5+30 x^2+243 x^3+45 x^4+243 x^5\right )}{225 \left (1+3 x^2\right )^2}\right ) \, dx\\ &=x-\frac {1}{5} \int \frac {e^{\frac {81 x^3}{5+15 x^2}} \left (5+30 x^2+243 x^3+45 x^4+243 x^5\right )}{\left (1+3 x^2\right )^2} \, dx\\ &=x+\frac {e^{\frac {81 x^3}{5 \left (1+3 x^2\right )}} \left (x^3+x^5\right )}{\left (1+3 x^2\right )^2 \left (\frac {2 x^4}{\left (1+3 x^2\right )^2}-\frac {x^2}{1+3 x^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 21, normalized size = 0.95 \begin {gather*} x-e^{\frac {81 x^3}{5+15 x^2}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 + 30*x^2 + 45*x^4 + E^((81*x^3)/(5 + 15*x^2))*(-5 - 30*x^2 - 243*x^3 - 45*x^4 - 243*x^5))/(5 + 30
*x^2 + 45*x^4),x]

[Out]

x - E^((81*x^3)/(5 + 15*x^2))*x

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fricas [A]  time = 1.00, size = 20, normalized size = 0.91 \begin {gather*} -x e^{\left (\frac {81 \, x^{3}}{5 \, {\left (3 \, x^{2} + 1\right )}}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-243*x^5-45*x^4-243*x^3-30*x^2-5)*exp(81*x^3/(15*x^2+5))+45*x^4+30*x^2+5)/(45*x^4+30*x^2+5),x, alg
orithm="fricas")

[Out]

-x*e^(81/5*x^3/(3*x^2 + 1)) + x

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giac [A]  time = 0.23, size = 20, normalized size = 0.91 \begin {gather*} -x e^{\left (\frac {81 \, x^{3}}{5 \, {\left (3 \, x^{2} + 1\right )}}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-243*x^5-45*x^4-243*x^3-30*x^2-5)*exp(81*x^3/(15*x^2+5))+45*x^4+30*x^2+5)/(45*x^4+30*x^2+5),x, alg
orithm="giac")

[Out]

-x*e^(81/5*x^3/(3*x^2 + 1)) + x

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maple [A]  time = 0.20, size = 21, normalized size = 0.95

method	result	size
risch	\(-x \,{\mathrm e}^{\frac {81 x^{3}}{5 \left (3 x^{2}+1\right )}}+x\)	\(21\)
norman	\(\frac {x +3 x^{3}-x \,{\mathrm e}^{\frac {81 x^{3}}{15 x^{2}+5}}-3 x^{3} {\mathrm e}^{\frac {81 x^{3}}{15 x^{2}+5}}}{3 x^{2}+1}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-243*x^5-45*x^4-243*x^3-30*x^2-5)*exp(81*x^3/(15*x^2+5))+45*x^4+30*x^2+5)/(45*x^4+30*x^2+5),x,method=_RE
TURNVERBOSE)

[Out]

-x*exp(81/5*x^3/(3*x^2+1))+x

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maxima [A]  time = 0.50, size = 22, normalized size = 1.00 \begin {gather*} -x e^{\left (\frac {27}{5} \, x - \frac {27 \, x}{5 \, {\left (3 \, x^{2} + 1\right )}}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-243*x^5-45*x^4-243*x^3-30*x^2-5)*exp(81*x^3/(15*x^2+5))+45*x^4+30*x^2+5)/(45*x^4+30*x^2+5),x, alg
orithm="maxima")

[Out]

-x*e^(27/5*x - 27/5*x/(3*x^2 + 1)) + x

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mupad [B]  time = 0.19, size = 20, normalized size = 0.91 \begin {gather*} -x\,\left ({\mathrm {e}}^{\frac {81\,x^3}{15\,x^2+5}}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((30*x^2 - exp((81*x^3)/(15*x^2 + 5))*(30*x^2 + 243*x^3 + 45*x^4 + 243*x^5 + 5) + 45*x^4 + 5)/(30*x^2 + 45*
x^4 + 5),x)

[Out]

-x*(exp((81*x^3)/(15*x^2 + 5)) - 1)

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sympy [A]  time = 0.20, size = 15, normalized size = 0.68 \begin {gather*} - x e^{\frac {81 x^{3}}{15 x^{2} + 5}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-243*x**5-45*x**4-243*x**3-30*x**2-5)*exp(81*x**3/(15*x**2+5))+45*x**4+30*x**2+5)/(45*x**4+30*x**2
+5),x)

[Out]

-x*exp(81*x**3/(15*x**2 + 5)) + x

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