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]

4/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)+1/18*(70-47*x)/(3*x^2+2)^(1/2)+28/9*(3*x^2+2)^(1/2)+8/9*x*(3*x^2+2)^(1/2)

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Rubi [A]  time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.05, 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])

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fricas [A]  time = 0.86, size = 72, normalized size = 1.01 \[ \frac {4 \, \sqrt {3} {\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + {\left (48 \, x^{3} + 168 \, x^{2} - 15 \, x + 182\right )} \sqrt {3 \, x^{2} + 2}}{18 \, {\left (3 \, x^{2} + 2\right )}} \]

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)

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giac [A]  time = 0.20, size = 49, normalized size = 0.69 \[ -\frac {4}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {3 \, {\left (8 \, {\left (2 \, x + 7\right )} x - 5\right )} x + 182}{18 \, \sqrt {3 \, x^{2} + 2}} \]

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)

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maple [A]  time = 0.01, size = 65, normalized size = 0.92 \[ \frac {8 x^{3}}{3 \sqrt {3 x^{2}+2}}+\frac {28 x^{2}}{3 \sqrt {3 x^{2}+2}}-\frac {5 x}{6 \sqrt {3 x^{2}+2}}+\frac {4 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{9}+\frac {91}{9 \sqrt {3 x^{2}+2}} \]

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

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maxima [A]  time = 0.96, size = 64, normalized size = 0.90 \[ \frac {8 \, x^{3}}{3 \, \sqrt {3 \, x^{2} + 2}} + \frac {28 \, x^{2}}{3 \, \sqrt {3 \, x^{2} + 2}} + \frac {4}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {5 \, x}{6 \, \sqrt {3 \, x^{2} + 2}} + \frac {91}{9 \, \sqrt {3 \, x^{2} + 2}} \]

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)

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mupad [B]  time = 4.07, size = 105, normalized size = 1.48 \[ \frac {4\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{9}+\frac {\sqrt {3}\,\left (\frac {8\,x}{3}+\frac {28}{3}\right )\,\sqrt {x^2+\frac {2}{3}}}{3}+\frac {\sqrt {3}\,\sqrt {6}\,\left (-630+\sqrt {6}\,141{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1944\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\left (630+\sqrt {6}\,141{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1944\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/9 + (3^(1/2)*((8*x)/3 + 28/3)*(x^2 + 2/3)^(1/2))/3 + (3^(1/2)*6^(1/2)
*(6^(1/2)*141i - 630)*(x^2 + 2/3)^(1/2)*1i)/(1944*(x - (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1/2)*(6^(1/2)*141i + 630
)*(x^2 + 2/3)^(1/2)*1i)/(1944*(x + (6^(1/2)*1i)/3))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (2 x + 1\right )^{2} \left (4 x^{2} + 3 x + 1\right )}{\left (3 x^{2} + 2\right )^{\frac {3}{2}}}\, dx \]

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)

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