### 3.127 $$\int \frac{1+3 x+4 x^2}{(1+2 x) (2+3 x^2)^{3/2}} \, dx$$

Optimal. Leaf size=53 $-\frac{38-21 x}{66 \sqrt{3 x^2+2}}-\frac{2 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{11 \sqrt{11}}$

[Out]

-(38 - 21*x)/(66*Sqrt[2 + 3*x^2]) - (2*ArcTanh[(4 - 3*x)/(Sqrt[11]*Sqrt[2 + 3*x^2])])/(11*Sqrt[11])

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Rubi [A]  time = 0.0599846, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.138, Rules used = {1647, 12, 725, 206} $-\frac{38-21 x}{66 \sqrt{3 x^2+2}}-\frac{2 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{11 \sqrt{11}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(1 + 3*x + 4*x^2)/((1 + 2*x)*(2 + 3*x^2)^(3/2)),x]

[Out]

-(38 - 21*x)/(66*Sqrt[2 + 3*x^2]) - (2*ArcTanh[(4 - 3*x)/(Sqrt[11]*Sqrt[2 + 3*x^2])])/(11*Sqrt[11])

Rule 1647

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[((a*g - c*f*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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])

Rubi steps

\begin{align*} \int \frac{1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{3/2}} \, dx &=-\frac{38-21 x}{66 \sqrt{2+3 x^2}}-\frac{1}{6} \int -\frac{12}{11 (1+2 x) \sqrt{2+3 x^2}} \, dx\\ &=-\frac{38-21 x}{66 \sqrt{2+3 x^2}}+\frac{2}{11} \int \frac{1}{(1+2 x) \sqrt{2+3 x^2}} \, dx\\ &=-\frac{38-21 x}{66 \sqrt{2+3 x^2}}-\frac{2}{11} \operatorname{Subst}\left (\int \frac{1}{11-x^2} \, dx,x,\frac{4-3 x}{\sqrt{2+3 x^2}}\right )\\ &=-\frac{38-21 x}{66 \sqrt{2+3 x^2}}-\frac{2 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{2+3 x^2}}\right )}{11 \sqrt{11}}\\ \end{align*}

Mathematica [A]  time = 0.0238863, size = 51, normalized size = 0.96 $\frac{-12 \sqrt{33 x^2+22} \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{33 x^2+22}}\right )+231 x-418}{726 \sqrt{3 x^2+2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)*(2 + 3*x^2)^(3/2)),x]

[Out]

(-418 + 231*x - 12*Sqrt[22 + 33*x^2]*ArcTanh[(4 - 3*x)/Sqrt[22 + 33*x^2]])/(726*Sqrt[2 + 3*x^2])

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Maple [B]  time = 0.053, size = 88, normalized size = 1.7 \begin{align*} -{\frac{2}{3}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{x}{4}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{1}{11}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}+{\frac{3\,x}{44}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}-{\frac{2\,\sqrt{11}}{121}{\it Artanh} \left ({\frac{ \left ( 8-6\,x \right ) \sqrt{11}}{11}{\frac{1}{\sqrt{12\, \left ( x+1/2 \right ) ^{2}-12\,x+5}}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2+3*x+1)/(1+2*x)/(3*x^2+2)^(3/2),x)

[Out]

-2/3/(3*x^2+2)^(1/2)+1/4*x/(3*x^2+2)^(1/2)+1/11/(3*(x+1/2)^2-3*x+5/4)^(1/2)+3/44*x/(3*(x+1/2)^2-3*x+5/4)^(1/2)
-2/121*11^(1/2)*arctanh(2/11*(4-3*x)*11^(1/2)/(12*(x+1/2)^2-12*x+5)^(1/2))

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Maxima [A]  time = 1.50389, size = 78, normalized size = 1.47 \begin{align*} \frac{2}{121} \, \sqrt{11} \operatorname{arsinh}\left (\frac{\sqrt{6} x}{2 \,{\left | 2 \, x + 1 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 1 \right |}}\right ) + \frac{7 \, x}{22 \, \sqrt{3 \, x^{2} + 2}} - \frac{19}{33 \, \sqrt{3 \, x^{2} + 2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+3*x+1)/(1+2*x)/(3*x^2+2)^(3/2),x, algorithm="maxima")

[Out]

2/121*sqrt(11)*arcsinh(1/2*sqrt(6)*x/abs(2*x + 1) - 2/3*sqrt(6)/abs(2*x + 1)) + 7/22*x/sqrt(3*x^2 + 2) - 19/33
/sqrt(3*x^2 + 2)

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Fricas [A]  time = 1.81235, size = 215, normalized size = 4.06 \begin{align*} \frac{6 \, \sqrt{11}{\left (3 \, x^{2} + 2\right )} \log \left (-\frac{\sqrt{11} \sqrt{3 \, x^{2} + 2}{\left (3 \, x - 4\right )} + 21 \, x^{2} - 12 \, x + 19}{4 \, x^{2} + 4 \, x + 1}\right ) + 11 \, \sqrt{3 \, x^{2} + 2}{\left (21 \, x - 38\right )}}{726 \,{\left (3 \, x^{2} + 2\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+3*x+1)/(1+2*x)/(3*x^2+2)^(3/2),x, algorithm="fricas")

[Out]

1/726*(6*sqrt(11)*(3*x^2 + 2)*log(-(sqrt(11)*sqrt(3*x^2 + 2)*(3*x - 4) + 21*x^2 - 12*x + 19)/(4*x^2 + 4*x + 1)
) + 11*sqrt(3*x^2 + 2)*(21*x - 38))/(3*x^2 + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{4 x^{2} + 3 x + 1}{\left (2 x + 1\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**2+3*x+1)/(1+2*x)/(3*x**2+2)**(3/2),x)

[Out]

Integral((4*x**2 + 3*x + 1)/((2*x + 1)*(3*x**2 + 2)**(3/2)), x)

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Giac [A]  time = 1.20817, size = 111, normalized size = 2.09 \begin{align*} \frac{2}{121} \, \sqrt{11} \log \left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{11} - \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{11} + \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) + \frac{21 \, x - 38}{66 \, \sqrt{3 \, x^{2} + 2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+3*x+1)/(1+2*x)/(3*x^2+2)^(3/2),x, algorithm="giac")

[Out]

2/121*sqrt(11)*log(-abs(-2*sqrt(3)*x - sqrt(11) - sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(11) + sqrt(
3) - 2*sqrt(3*x^2 + 2))) + 1/66*(21*x - 38)/sqrt(3*x^2 + 2)