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

Optimal. Leaf size=55 \[ \frac {2-51 x}{18 \sqrt {3 x^2+2}}+\frac {8}{9} \sqrt {3 x^2+2}+\frac {10 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]

[Out]

10/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)+1/18*(2-51*x)/(3*x^2+2)^(1/2)+8/9*(3*x^2+2)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1814, 641, 215} \[ \frac {2-51 x}{18 \sqrt {3 x^2+2}}+\frac {8}{9} \sqrt {3 x^2+2}+\frac {10 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2 - 51*x)/(18*Sqrt[2 + 3*x^2]) + (8*Sqrt[2 + 3*x^2])/9 + (10*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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 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 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]

Rubi steps

\begin {align*} \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{3/2}} \, dx &=\frac {2-51 x}{18 \sqrt {2+3 x^2}}-\frac {1}{2} \int \frac {-\frac {20}{3}-\frac {16 x}{3}}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {2-51 x}{18 \sqrt {2+3 x^2}}+\frac {8}{9} \sqrt {2+3 x^2}+\frac {10}{3} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {2-51 x}{18 \sqrt {2+3 x^2}}+\frac {8}{9} \sqrt {2+3 x^2}+\frac {10 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 48, normalized size = 0.87 \[ \frac {48 x^2+20 \sqrt {9 x^2+6} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-51 x+34}{18 \sqrt {3 x^2+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(34 - 51*x + 48*x^2 + 20*Sqrt[6 + 9*x^2]*ArcSinh[Sqrt[3/2]*x])/(18*Sqrt[2 + 3*x^2])

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fricas [A]  time = 0.76, size = 67, normalized size = 1.22 \[ \frac {10 \, \sqrt {3} {\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + {\left (48 \, x^{2} - 51 \, x + 34\right )} \sqrt {3 \, x^{2} + 2}}{18 \, {\left (3 \, x^{2} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(10*sqrt(3)*(3*x^2 + 2)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + (48*x^2 - 51*x + 34)*sqrt(3*x^2 + 2
))/(3*x^2 + 2)

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giac [A]  time = 0.23, size = 44, normalized size = 0.80 \[ -\frac {10}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {3 \, {\left (16 \, x - 17\right )} x + 34}{18 \, \sqrt {3 \, x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 1/18*(3*(16*x - 17)*x + 34)/sqrt(3*x^2 + 2)

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maple [A]  time = 0.00, size = 51, normalized size = 0.93 \[ \frac {8 x^{2}}{3 \sqrt {3 x^{2}+2}}-\frac {17 x}{6 \sqrt {3 x^{2}+2}}+\frac {10 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{9}+\frac {17}{9 \sqrt {3 x^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/3/(3*x^2+2)^(1/2)*x^2+17/9/(3*x^2+2)^(1/2)-17/6/(3*x^2+2)^(1/2)*x+10/9*arcsinh(1/2*6^(1/2)*x)*3^(1/2)

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maxima [A]  time = 0.96, size = 50, normalized size = 0.91 \[ \frac {8 \, x^{2}}{3 \, \sqrt {3 \, x^{2} + 2}} + \frac {10}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {17 \, x}{6 \, \sqrt {3 \, x^{2} + 2}} + \frac {17}{9 \, \sqrt {3 \, x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/3*x^2/sqrt(3*x^2 + 2) + 10/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 17/6*x/sqrt(3*x^2 + 2) + 17/9/sqrt(3*x^2 + 2)

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mupad [B]  time = 0.04, size = 100, normalized size = 1.82 \[ \frac {8\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{9}+\frac {10\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{9}+\frac {\sqrt {3}\,\sqrt {6}\,\left (-6+\sqrt {6}\,51{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{648\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\left (6+\sqrt {6}\,51{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{648\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*3^(1/2)*(x^2 + 2/3)^(1/2))/9 + (10*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/9 + (3^(1/2)*6^(1/2)*(6^(1/2)*51i
- 6)*(x^2 + 2/3)^(1/2)*1i)/(648*(x - (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1/2)*(6^(1/2)*51i + 6)*(x^2 + 2/3)^(1/2)*1
i)/(648*(x + (6^(1/2)*1i)/3))

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sympy [B]  time = 15.96, size = 114, normalized size = 2.07 \[ \frac {30 \sqrt {3} x^{2} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{27 x^{2} + 18} + \frac {8 x^{2}}{3 \sqrt {3 x^{2} + 2}} - \frac {30 x \sqrt {3 x^{2} + 2}}{27 x^{2} + 18} + \frac {x}{2 \sqrt {3 x^{2} + 2}} + \frac {20 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{27 x^{2} + 18} + \frac {17}{9 \sqrt {3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

30*sqrt(3)*x**2*asinh(sqrt(6)*x/2)/(27*x**2 + 18) + 8*x**2/(3*sqrt(3*x**2 + 2)) - 30*x*sqrt(3*x**2 + 2)/(27*x*
*2 + 18) + x/(2*sqrt(3*x**2 + 2)) + 20*sqrt(3)*asinh(sqrt(6)*x/2)/(27*x**2 + 18) + 17/(9*sqrt(3*x**2 + 2))

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