3.12.100 \(\int \frac {(5-x) (2+3 x^2)^{3/2}}{(3+2 x)^2} \, dx\)

Optimal. Leaf size=97 \[ -\frac {(x+21) \left (3 x^2+2\right )^{3/2}}{6 (2 x+3)}-\frac {1}{8} (193-63 x) \sqrt {3 x^2+2}+\frac {193}{16} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )+\frac {663}{16} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {813, 815, 844, 215, 725, 206} \begin {gather*} -\frac {(x+21) \left (3 x^2+2\right )^{3/2}}{6 (2 x+3)}-\frac {1}{8} (193-63 x) \sqrt {3 x^2+2}+\frac {193}{16} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )+\frac {663}{16} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^2,x]

[Out]

-((193 - 63*x)*Sqrt[2 + 3*x^2])/8 - ((21 + x)*(2 + 3*x^2)^(3/2))/(6*(3 + 2*x)) + (663*Sqrt[3]*ArcSinh[Sqrt[3/2
]*x])/16 + (193*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/16

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 815

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*(a + c*x^2)^p)/(c*e^2*(m + 2*p + 1)*(m
+ 2*p + 2)), x] + Dist[(2*p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*
m, 2*p])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx &=-\frac {(21+x) \left (2+3 x^2\right )^{3/2}}{6 (3+2 x)}-\frac {1}{8} \int \frac {(8-252 x) \sqrt {2+3 x^2}}{3+2 x} \, dx\\ &=-\frac {1}{8} (193-63 x) \sqrt {2+3 x^2}-\frac {(21+x) \left (2+3 x^2\right )^{3/2}}{6 (3+2 x)}-\frac {1}{192} \int \frac {9456-47736 x}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=-\frac {1}{8} (193-63 x) \sqrt {2+3 x^2}-\frac {(21+x) \left (2+3 x^2\right )^{3/2}}{6 (3+2 x)}+\frac {1989}{16} \int \frac {1}{\sqrt {2+3 x^2}} \, dx-\frac {6755}{16} \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=-\frac {1}{8} (193-63 x) \sqrt {2+3 x^2}-\frac {(21+x) \left (2+3 x^2\right )^{3/2}}{6 (3+2 x)}+\frac {663}{16} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )+\frac {6755}{16} \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )\\ &=-\frac {1}{8} (193-63 x) \sqrt {2+3 x^2}-\frac {(21+x) \left (2+3 x^2\right )^{3/2}}{6 (3+2 x)}+\frac {663}{16} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )+\frac {193}{16} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 87, normalized size = 0.90 \begin {gather*} \frac {1}{48} \left (579 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {2 \sqrt {3 x^2+2} \left (12 x^3-126 x^2+599 x+1905\right )}{2 x+3}+1989 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^2,x]

[Out]

((-2*Sqrt[2 + 3*x^2]*(1905 + 599*x - 126*x^2 + 12*x^3))/(3 + 2*x) + 1989*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] + 579*Sq
rt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/48

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.58, size = 116, normalized size = 1.20 \begin {gather*} -\frac {663}{16} \sqrt {3} \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )-\frac {193}{8} \sqrt {35} \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )+\frac {\sqrt {3 x^2+2} \left (-12 x^3+126 x^2-599 x-1905\right )}{24 (2 x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^2,x]

[Out]

(Sqrt[2 + 3*x^2]*(-1905 - 599*x + 126*x^2 - 12*x^3))/(24*(3 + 2*x)) - (193*Sqrt[35]*ArcTanh[3*Sqrt[3/35] + 2*S
qrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/8 - (663*Sqrt[3]*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/16

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 121, normalized size = 1.25 \begin {gather*} \frac {1989 \, \sqrt {3} {\left (2 \, x + 3\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 579 \, \sqrt {35} {\left (2 \, x + 3\right )} \log \left (\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} - 93 \, x^{2} + 36 \, x - 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 4 \, {\left (12 \, x^{3} - 126 \, x^{2} + 599 \, x + 1905\right )} \sqrt {3 \, x^{2} + 2}}{96 \, {\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^2,x, algorithm="fricas")

[Out]

1/96*(1989*sqrt(3)*(2*x + 3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 579*sqrt(35)*(2*x + 3)*log((sqrt(35
)*sqrt(3*x^2 + 2)*(9*x - 4) - 93*x^2 + 36*x - 43)/(4*x^2 + 12*x + 9)) - 4*(12*x^3 - 126*x^2 + 599*x + 1905)*sq
rt(3*x^2 + 2))/(2*x + 3)

________________________________________________________________________________________

giac [B]  time = 0.73, size = 475, normalized size = 4.90 \begin {gather*} \frac {193}{16} \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {663}{16} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {35}}{2 \, x + 3} \right |}}{2 \, {\left (\sqrt {3} + \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {455}{32} \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {3 \, {\left (704 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 323 \, \sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{4} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 1944 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 1158 \, \sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{2} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 1872 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 1263 \, \sqrt {35} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{8 \, {\left ({\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{2} - 3\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^2,x, algorithm="giac")

[Out]

193/16*sqrt(35)*log(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)) - 9)*sgn(1/(2*x +
3)) - 663/16*sqrt(3)*log(1/2*abs(-2*sqrt(3) + 2*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + 2*sqrt(35)/(2*x +
3))/(sqrt(3) + sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)))*sgn(1/(2*x + 3)) - 455/32*sqrt(
-18/(2*x + 3) + 35/(2*x + 3)^2 + 3)*sgn(1/(2*x + 3)) + 3/8*(704*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sq
rt(35)/(2*x + 3))^5*sgn(1/(2*x + 3)) - 323*sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x
+ 3))^4*sgn(1/(2*x + 3)) - 1944*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^3*sgn(1/(2*x +
3)) + 1158*sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^2*sgn(1/(2*x + 3)) + 1872
*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))*sgn(1/(2*x + 3)) - 1263*sqrt(35)*sgn(1/(2*x +
3)))/((sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^2 - 3)^3

________________________________________________________________________________________

maple [A]  time = 0.05, size = 131, normalized size = 1.35 \begin {gather*} \frac {63 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{8}+\frac {39 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}} x}{70}+\frac {663 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{16}+\frac {193 \sqrt {35}\, \arctanh \left (\frac {2 \left (-9 x +4\right ) \sqrt {35}}{35 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{16}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{70 \left (x +\frac {3}{2}\right )}-\frac {193 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{210}-\frac {193 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+2)^(3/2)/(2*x+3)^2,x)

[Out]

-13/70/(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(5/2)-193/210*(-9*x+3*(x+3/2)^2-19/4)^(3/2)+63/8*(-9*x+3*(x+3/2)^2-19/4
)^(1/2)*x+663/16*arcsinh(1/2*6^(1/2)*x)*3^(1/2)-193/16*(-36*x+12*(x+3/2)^2-19)^(1/2)+193/16*35^(1/2)*arctanh(2
/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2))+39/70*x*(-9*x+3*(x+3/2)^2-19/4)^(3/2)

________________________________________________________________________________________

maxima [A]  time = 1.34, size = 99, normalized size = 1.02 \begin {gather*} -\frac {1}{12} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} + \frac {63}{8} \, \sqrt {3 \, x^{2} + 2} x + \frac {663}{16} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {193}{16} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {193}{8} \, \sqrt {3 \, x^{2} + 2} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{4 \, {\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^2,x, algorithm="maxima")

[Out]

-1/12*(3*x^2 + 2)^(3/2) + 63/8*sqrt(3*x^2 + 2)*x + 663/16*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 193/16*sqrt(35)*arc
sinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 193/8*sqrt(3*x^2 + 2) - 13/4*(3*x^2 + 2)^(3/2)/(
2*x + 3)

________________________________________________________________________________________

mupad [B]  time = 0.12, size = 108, normalized size = 1.11 \begin {gather*} \frac {663\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{16}-\frac {815\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{48}-\frac {193\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{16}+\frac {193\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{16}-\frac {\sqrt {3}\,x^2\,\sqrt {x^2+\frac {2}{3}}}{4}-\frac {455\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{32\,\left (x+\frac {3}{2}\right )}+3\,\sqrt {3}\,x\,\sqrt {x^2+\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((3*x^2 + 2)^(3/2)*(x - 5))/(2*x + 3)^2,x)

[Out]

(663*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/16 - (815*3^(1/2)*(x^2 + 2/3)^(1/2))/48 - (193*35^(1/2)*log(x + 3/2
))/16 + (193*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/16 - (3^(1/2)*x^2*(x^2 + 2/3)^(1/
2))/4 - (455*3^(1/2)*(x^2 + 2/3)^(1/2))/(32*(x + 3/2)) + 3*3^(1/2)*x*(x^2 + 2/3)^(1/2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {10 \sqrt {3 x^{2} + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \frac {2 x \sqrt {3 x^{2} + 2}}{4 x^{2} + 12 x + 9}\, dx - \int \left (- \frac {15 x^{2} \sqrt {3 x^{2} + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 2}}{4 x^{2} + 12 x + 9}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+2)**(3/2)/(3+2*x)**2,x)

[Out]

-Integral(-10*sqrt(3*x**2 + 2)/(4*x**2 + 12*x + 9), x) - Integral(2*x*sqrt(3*x**2 + 2)/(4*x**2 + 12*x + 9), x)
- Integral(-15*x**2*sqrt(3*x**2 + 2)/(4*x**2 + 12*x + 9), x) - Integral(3*x**3*sqrt(3*x**2 + 2)/(4*x**2 + 12*
x + 9), x)

________________________________________________________________________________________