3.251 \(\int \frac{1+3 x+4 x^2}{(1+2 x)^3 (2-x+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=112 \[ \frac{2 (3693 x+2363)}{50531 \sqrt{3 x^2-x+2}}-\frac{4 \sqrt{3 x^2-x+2}}{2197 (2 x+1)}-\frac{2 \sqrt{3 x^2-x+2}}{169 (2 x+1)^2}-\frac{487 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{2197 \sqrt{13}} \]

[Out]

(2*(2363 + 3693*x))/(50531*Sqrt[2 - x + 3*x^2]) - (2*Sqrt[2 - x + 3*x^2])/(169*(1 + 2*x)^2) - (4*Sqrt[2 - x +
3*x^2])/(2197*(1 + 2*x)) - (487*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(2197*Sqrt[13])

________________________________________________________________________________________

Rubi [A]  time = 0.154798, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1646, 1650, 806, 724, 206} \[ \frac{2 (3693 x+2363)}{50531 \sqrt{3 x^2-x+2}}-\frac{4 \sqrt{3 x^2-x+2}}{2197 (2 x+1)}-\frac{2 \sqrt{3 x^2-x+2}}{169 (2 x+1)^2}-\frac{487 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{2197 \sqrt{13}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(2363 + 3693*x))/(50531*Sqrt[2 - x + 3*x^2]) - (2*Sqrt[2 - x + 3*x^2])/(169*(1 + 2*x)^2) - (4*Sqrt[2 - x +
3*x^2])/(2197*(1 + 2*x)) - (487*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(2197*Sqrt[13])

Rule 1646

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1650

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*
x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^
(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

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)^3 \left (2-x+3 x^2\right )^{3/2}} \, dx &=\frac{2 (2363+3693 x)}{50531 \sqrt{2-x+3 x^2}}+\frac{2}{23} \int \frac{\frac{8349}{2197}+\frac{20838 x}{2197}+\frac{23828 x^2}{2197}}{(1+2 x)^3 \sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2 (2363+3693 x)}{50531 \sqrt{2-x+3 x^2}}-\frac{2 \sqrt{2-x+3 x^2}}{169 (1+2 x)^2}-\frac{1}{299} \int \frac{-\frac{11615}{169}-\frac{22034 x}{169}}{(1+2 x)^2 \sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2 (2363+3693 x)}{50531 \sqrt{2-x+3 x^2}}-\frac{2 \sqrt{2-x+3 x^2}}{169 (1+2 x)^2}-\frac{4 \sqrt{2-x+3 x^2}}{2197 (1+2 x)}+\frac{487 \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx}{2197}\\ &=\frac{2 (2363+3693 x)}{50531 \sqrt{2-x+3 x^2}}-\frac{2 \sqrt{2-x+3 x^2}}{169 (1+2 x)^2}-\frac{4 \sqrt{2-x+3 x^2}}{2197 (1+2 x)}-\frac{974 \operatorname{Subst}\left (\int \frac{1}{52-x^2} \, dx,x,\frac{9-8 x}{\sqrt{2-x+3 x^2}}\right )}{2197}\\ &=\frac{2 (2363+3693 x)}{50531 \sqrt{2-x+3 x^2}}-\frac{2 \sqrt{2-x+3 x^2}}{169 (1+2 x)^2}-\frac{4 \sqrt{2-x+3 x^2}}{2197 (1+2 x)}-\frac{487 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{2-x+3 x^2}}\right )}{2197 \sqrt{13}}\\ \end{align*}

Mathematica [A]  time = 0.0584065, size = 79, normalized size = 0.71 \[ \frac{2 \left (14496 x^3+23281 x^2+13306 x+1673\right )}{50531 (2 x+1)^2 \sqrt{3 x^2-x+2}}-\frac{487 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{2197 \sqrt{13}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1673 + 13306*x + 23281*x^2 + 14496*x^3))/(50531*(1 + 2*x)^2*Sqrt[2 - x + 3*x^2]) - (487*ArcTanh[(9 - 8*x)/
(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(2197*Sqrt[13])

________________________________________________________________________________________

Maple [A]  time = 0.057, size = 111, normalized size = 1. \begin{align*} -{\frac{1}{104} \left ( x+{\frac{1}{2}} \right ) ^{-2}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}}+{\frac{3}{338} \left ( x+{\frac{1}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}}+{\frac{487}{4394}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}}+{\frac{-1208+7248\,x}{50531}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}}-{\frac{487\,\sqrt{13}}{28561}{\it Artanh} \left ({\frac{2\,\sqrt{13}}{13} \left ({\frac{9}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,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/(3*x^2-x+2)^(3/2),x)

[Out]

-1/104/(x+1/2)^2/(3*(x+1/2)^2-4*x+5/4)^(1/2)+3/338/(x+1/2)/(3*(x+1/2)^2-4*x+5/4)^(1/2)+487/4394/(3*(x+1/2)^2-4
*x+5/4)^(1/2)+1208/50531*(-1+6*x)/(3*(x+1/2)^2-4*x+5/4)^(1/2)-487/28561*13^(1/2)*arctanh(2/13*(9/2-4*x)*13^(1/
2)/(12*(x+1/2)^2-16*x+5)^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.50668, size = 196, normalized size = 1.75 \begin{align*} \frac{487}{28561} \, \sqrt{13} \operatorname{arsinh}\left (\frac{8 \, \sqrt{23} x}{23 \,{\left | 2 \, x + 1 \right |}} - \frac{9 \, \sqrt{23}}{23 \,{\left | 2 \, x + 1 \right |}}\right ) + \frac{7248 \, x}{50531 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{8785}{101062 \, \sqrt{3 \, x^{2} - x + 2}} - \frac{1}{26 \,{\left (4 \, \sqrt{3 \, x^{2} - x + 2} x^{2} + 4 \, \sqrt{3 \, x^{2} - x + 2} x + \sqrt{3 \, x^{2} - x + 2}\right )}} + \frac{3}{169 \,{\left (2 \, \sqrt{3 \, x^{2} - x + 2} x + \sqrt{3 \, x^{2} - x + 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/(3*x^2-x+2)^(3/2),x, algorithm="maxima")

[Out]

487/28561*sqrt(13)*arcsinh(8/23*sqrt(23)*x/abs(2*x + 1) - 9/23*sqrt(23)/abs(2*x + 1)) + 7248/50531*x/sqrt(3*x^
2 - x + 2) + 8785/101062/sqrt(3*x^2 - x + 2) - 1/26/(4*sqrt(3*x^2 - x + 2)*x^2 + 4*sqrt(3*x^2 - x + 2)*x + sqr
t(3*x^2 - x + 2)) + 3/169/(2*sqrt(3*x^2 - x + 2)*x + sqrt(3*x^2 - x + 2))

________________________________________________________________________________________

Fricas [A]  time = 1.13226, size = 344, normalized size = 3.07 \begin{align*} \frac{11201 \, \sqrt{13}{\left (12 \, x^{4} + 8 \, x^{3} + 7 \, x^{2} + 7 \, x + 2\right )} \log \left (-\frac{4 \, \sqrt{13} \sqrt{3 \, x^{2} - x + 2}{\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 52 \,{\left (14496 \, x^{3} + 23281 \, x^{2} + 13306 \, x + 1673\right )} \sqrt{3 \, x^{2} - x + 2}}{1313806 \,{\left (12 \, x^{4} + 8 \, x^{3} + 7 \, x^{2} + 7 \, x + 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/(3*x^2-x+2)^(3/2),x, algorithm="fricas")

[Out]

1/1313806*(11201*sqrt(13)*(12*x^4 + 8*x^3 + 7*x^2 + 7*x + 2)*log(-(4*sqrt(13)*sqrt(3*x^2 - x + 2)*(8*x - 9) +
220*x^2 - 196*x + 185)/(4*x^2 + 4*x + 1)) + 52*(14496*x^3 + 23281*x^2 + 13306*x + 1673)*sqrt(3*x^2 - x + 2))/(
12*x^4 + 8*x^3 + 7*x^2 + 7*x + 2)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.24475, size = 301, normalized size = 2.69 \begin{align*} \frac{487}{28561} \, \sqrt{13} \log \left (-\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{13} - 2 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} - x + 2} \right |}}{2 \,{\left (2 \, \sqrt{3} x - \sqrt{13} + \sqrt{3} - 2 \, \sqrt{3 \, x^{2} - x + 2}\right )}}\right ) + \frac{2 \,{\left (3693 \, x + 2363\right )}}{50531 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{2 \,{\left (62 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )}^{3} - 37 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )}^{2} + 263 \, \sqrt{3} x - 71 \, \sqrt{3} - 263 \, \sqrt{3 \, x^{2} - x + 2}\right )}}{2197 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )}^{2} + 2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )} - 5\right )}^{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/(3*x^2-x+2)^(3/2),x, algorithm="giac")

[Out]

487/28561*sqrt(13)*log(-1/2*abs(-4*sqrt(3)*x - 2*sqrt(13) - 2*sqrt(3) + 4*sqrt(3*x^2 - x + 2))/(2*sqrt(3)*x -
sqrt(13) + sqrt(3) - 2*sqrt(3*x^2 - x + 2))) + 2/50531*(3693*x + 2363)/sqrt(3*x^2 - x + 2) + 2/2197*(62*(sqrt(
3)*x - sqrt(3*x^2 - x + 2))^3 - 37*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2))^2 + 263*sqrt(3)*x - 71*sqrt(3) -
263*sqrt(3*x^2 - x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 - x + 2))^2 + 2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2))
- 5)^2