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

Optimal. Leaf size=71 \[ \frac{70-47 x}{18 \sqrt{3 x^2+2}}+\frac{8}{9} x \sqrt{3 x^2+2}+\frac{28}{9} \sqrt{3 x^2+2}+\frac{4 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

[Out]

(70 - 47*x)/(18*Sqrt[2 + 3*x^2]) + (28*Sqrt[2 + 3*x^2])/9 + (8*x*Sqrt[2 + 3*x^2])/9 + (4*ArcSinh[Sqrt[3/2]*x])
/(3*Sqrt[3])

________________________________________________________________________________________

Rubi [A]  time = 0.0840017, antiderivative size = 71, 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 = {1814, 1815, 641, 215} \[ \frac{70-47 x}{18 \sqrt{3 x^2+2}}+\frac{8}{9} x \sqrt{3 x^2+2}+\frac{28}{9} \sqrt{3 x^2+2}+\frac{4 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(70 - 47*x)/(18*Sqrt[2 + 3*x^2]) + (28*Sqrt[2 + 3*x^2])/9 + (8*x*Sqrt[2 + 3*x^2])/9 + (4*ArcSinh[Sqrt[3/2]*x])
/(3*Sqrt[3])

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 1815

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Si
mp[(e*x^(q - 1)*(a + b*x^2)^(p + 1))/(b*(q + 2*p + 1)), x] + Dist[1/(b*(q + 2*p + 1)), Int[(a + b*x^2)^p*Expan
dToSum[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x]
&& PolyQ[Pq, x] &&  !LeQ[p, -1]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

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]

Rubi steps

\begin{align*} \int \frac{(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{3/2}} \, dx &=\frac{70-47 x}{18 \sqrt{2+3 x^2}}-\frac{1}{2} \int \frac{-\frac{56}{9}-\frac{56 x}{3}-\frac{32 x^2}{3}}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{70-47 x}{18 \sqrt{2+3 x^2}}+\frac{8}{9} x \sqrt{2+3 x^2}-\frac{1}{12} \int \frac{-16-112 x}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{70-47 x}{18 \sqrt{2+3 x^2}}+\frac{28}{9} \sqrt{2+3 x^2}+\frac{8}{9} x \sqrt{2+3 x^2}+\frac{4}{3} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{70-47 x}{18 \sqrt{2+3 x^2}}+\frac{28}{9} \sqrt{2+3 x^2}+\frac{8}{9} x \sqrt{2+3 x^2}+\frac{4 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0525299, size = 53, normalized size = 0.75 \[ \frac{48 x^3+168 x^2+8 \sqrt{9 x^2+6} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-15 x+182}{18 \sqrt{3 x^2+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(182 - 15*x + 168*x^2 + 48*x^3 + 8*Sqrt[6 + 9*x^2]*ArcSinh[Sqrt[3/2]*x])/(18*Sqrt[2 + 3*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.054, size = 65, normalized size = 0.9 \begin{align*}{\frac{8\,{x}^{3}}{3}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{5\,x}{6}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{4\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{28\,{x}^{2}}{3}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{91}{9}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/3*x^3/(3*x^2+2)^(1/2)-5/6*x/(3*x^2+2)^(1/2)+4/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)+28/3*x^2/(3*x^2+2)^(1/2)+91/9
/(3*x^2+2)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.53049, size = 86, normalized size = 1.21 \begin{align*} \frac{8 \, x^{3}}{3 \, \sqrt{3 \, x^{2} + 2}} + \frac{28 \, x^{2}}{3 \, \sqrt{3 \, x^{2} + 2}} + \frac{4}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) - \frac{5 \, x}{6 \, \sqrt{3 \, x^{2} + 2}} + \frac{91}{9 \, \sqrt{3 \, x^{2} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/3*x^3/sqrt(3*x^2 + 2) + 28/3*x^2/sqrt(3*x^2 + 2) + 4/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 5/6*x/sqrt(3*x^2 + 2
) + 91/9/sqrt(3*x^2 + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.55454, size = 184, normalized size = 2.59 \begin{align*} \frac{4 \, \sqrt{3}{\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) +{\left (48 \, x^{3} + 168 \, x^{2} - 15 \, x + 182\right )} \sqrt{3 \, x^{2} + 2}}{18 \,{\left (3 \, x^{2} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(4*sqrt(3)*(3*x^2 + 2)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + (48*x^3 + 168*x^2 - 15*x + 182)*sqrt
(3*x^2 + 2))/(3*x^2 + 2)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.26012, size = 66, normalized size = 0.93 \begin{align*} -\frac{4}{9} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{3 \,{\left (8 \,{\left (2 \, x + 7\right )} x - 5\right )} x + 182}{18 \, \sqrt{3 \, x^{2} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 1/18*(3*(8*(2*x + 7)*x - 5)*x + 182)/sqrt(3*x^2 + 2)