### 3.131 $$\int \frac{(1+2 x)^2 (1+3 x+4 x^2)}{(2+3 x^2)^{5/2}} \, dx$$

Optimal. Leaf size=60 $\frac{70-47 x}{54 \left (3 x^2+2\right )^{3/2}}-\frac{59 x+168}{54 \sqrt{3 x^2+2}}+\frac{16 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}}$

[Out]

(70 - 47*x)/(54*(2 + 3*x^2)^(3/2)) - (168 + 59*x)/(54*Sqrt[2 + 3*x^2]) + (16*ArcSinh[Sqrt[3/2]*x])/(9*Sqrt[3])

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Rubi [A]  time = 0.0752021, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.103, Rules used = {1814, 12, 215} $\frac{70-47 x}{54 \left (3 x^2+2\right )^{3/2}}-\frac{59 x+168}{54 \sqrt{3 x^2+2}}+\frac{16 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

Int[((1 + 2*x)^2*(1 + 3*x + 4*x^2))/(2 + 3*x^2)^(5/2),x]

[Out]

(70 - 47*x)/(54*(2 + 3*x^2)^(3/2)) - (168 + 59*x)/(54*Sqrt[2 + 3*x^2]) + (16*ArcSinh[Sqrt[3/2]*x])/(9*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 12

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

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 )^{5/2}} \, dx &=\frac{70-47 x}{54 \left (2+3 x^2\right )^{3/2}}-\frac{1}{6} \int \frac{-\frac{74}{9}-56 x-32 x^2}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac{70-47 x}{54 \left (2+3 x^2\right )^{3/2}}-\frac{168+59 x}{54 \sqrt{2+3 x^2}}+\frac{1}{12} \int \frac{64}{3 \sqrt{2+3 x^2}} \, dx\\ &=\frac{70-47 x}{54 \left (2+3 x^2\right )^{3/2}}-\frac{168+59 x}{54 \sqrt{2+3 x^2}}+\frac{16}{9} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{70-47 x}{54 \left (2+3 x^2\right )^{3/2}}-\frac{168+59 x}{54 \sqrt{2+3 x^2}}+\frac{16 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0527414, size = 58, normalized size = 0.97 $\frac{-177 x^3-504 x^2+32 \sqrt{3} \left (3 x^2+2\right )^{3/2} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-165 x-266}{54 \left (3 x^2+2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[((1 + 2*x)^2*(1 + 3*x + 4*x^2))/(2 + 3*x^2)^(5/2),x]

[Out]

(-266 - 165*x - 504*x^2 - 177*x^3 + 32*Sqrt[3]*(2 + 3*x^2)^(3/2)*ArcSinh[Sqrt[3/2]*x])/(54*(2 + 3*x^2)^(3/2))

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Maple [A]  time = 0.057, size = 77, normalized size = 1.3 \begin{align*} -{\frac{16\,{x}^{3}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{x}{2}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{16\,\sqrt{3}}{27}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{28\,{x}^{2}}{3} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{133}{27} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{37\,x}{18} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{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)^(5/2),x)

[Out]

-16/9*x^3/(3*x^2+2)^(3/2)-1/2*x/(3*x^2+2)^(1/2)+16/27*arcsinh(1/2*x*6^(1/2))*3^(1/2)-28/3*x^2/(3*x^2+2)^(3/2)-
133/27/(3*x^2+2)^(3/2)-37/18*x/(3*x^2+2)^(3/2)

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Maxima [B]  time = 1.46374, size = 123, normalized size = 2.05 \begin{align*} -\frac{16}{27} \, x{\left (\frac{9 \, x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{4}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}\right )} + \frac{16}{27} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{37 \, x}{54 \, \sqrt{3 \, x^{2} + 2}} - \frac{28 \, x^{2}}{3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{37 \, x}{18 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{133}{27 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{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)^(5/2),x, algorithm="maxima")

[Out]

-16/27*x*(9*x^2/(3*x^2 + 2)^(3/2) + 4/(3*x^2 + 2)^(3/2)) + 16/27*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 37/54*x/sqrt
(3*x^2 + 2) - 28/3*x^2/(3*x^2 + 2)^(3/2) - 37/18*x/(3*x^2 + 2)^(3/2) - 133/27/(3*x^2 + 2)^(3/2)

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Fricas [A]  time = 1.58438, size = 212, normalized size = 3.53 \begin{align*} \frac{16 \, \sqrt{3}{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) -{\left (177 \, x^{3} + 504 \, x^{2} + 165 \, x + 266\right )} \sqrt{3 \, x^{2} + 2}}{54 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\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)^(5/2),x, algorithm="fricas")

[Out]

1/54*(16*sqrt(3)*(9*x^4 + 12*x^2 + 4)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) - (177*x^3 + 504*x^2 + 165*x
+ 266)*sqrt(3*x^2 + 2))/(9*x^4 + 12*x^2 + 4)

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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{5}{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)**(5/2),x)

[Out]

Integral((2*x + 1)**2*(4*x**2 + 3*x + 1)/(3*x**2 + 2)**(5/2), x)

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Giac [A]  time = 1.27473, size = 65, normalized size = 1.08 \begin{align*} -\frac{16}{27} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{3 \,{\left ({\left (59 \, x + 168\right )} x + 55\right )} x + 266}{54 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{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)^(5/2),x, algorithm="giac")

[Out]

-16/27*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/54*(3*((59*x + 168)*x + 55)*x + 266)/(3*x^2 + 2)^(3/2)