### 3.246 $$\int \frac{(1+2 x)^3 (1+3 x+4 x^2)}{(2-x+3 x^2)^{3/2}} \, dx$$

Optimal. Leaf size=103 $\frac{32}{27} \sqrt{3 x^2-x+2} x^2+\frac{412}{81} \sqrt{3 x^2-x+2} x+\frac{746}{81} \sqrt{3 x^2-x+2}+\frac{2 (12839-3871 x)}{1863 \sqrt{3 x^2-x+2}}+\frac{353 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{81 \sqrt{3}}$

[Out]

(2*(12839 - 3871*x))/(1863*Sqrt[2 - x + 3*x^2]) + (746*Sqrt[2 - x + 3*x^2])/81 + (412*x*Sqrt[2 - x + 3*x^2])/8
1 + (32*x^2*Sqrt[2 - x + 3*x^2])/27 + (353*ArcSinh[(1 - 6*x)/Sqrt[23]])/(81*Sqrt[3])

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Rubi [A]  time = 0.12421, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 32, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.156, Rules used = {1660, 1661, 640, 619, 215} $\frac{32}{27} \sqrt{3 x^2-x+2} x^2+\frac{412}{81} \sqrt{3 x^2-x+2} x+\frac{746}{81} \sqrt{3 x^2-x+2}+\frac{2 (12839-3871 x)}{1863 \sqrt{3 x^2-x+2}}+\frac{353 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{81 \sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(12839 - 3871*x))/(1863*Sqrt[2 - x + 3*x^2]) + (746*Sqrt[2 - x + 3*x^2])/81 + (412*x*Sqrt[2 - x + 3*x^2])/8
1 + (32*x^2*Sqrt[2 - x + 3*x^2])/27 + (353*ArcSinh[(1 - 6*x)/Sqrt[23]])/(81*Sqrt[3])

Rule 1660

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

Rule 1661

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[(e*x^(q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(q + 2*p + 1)), x] + Dist[1/(c*(q + 2*p + 1)), Int
[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q +
2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 640

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

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

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)^3 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{3/2}} \, dx &=\frac{2 (12839-3871 x)}{1863 \sqrt{2-x+3 x^2}}+\frac{2}{23} \int \frac{\frac{1127}{81}+\frac{7682 x}{27}+\frac{2852 x^2}{9}+\frac{368 x^3}{3}}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2 (12839-3871 x)}{1863 \sqrt{2-x+3 x^2}}+\frac{32}{27} x^2 \sqrt{2-x+3 x^2}+\frac{2}{207} \int \frac{\frac{1127}{9}+2070 x+\frac{9476 x^2}{3}}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2 (12839-3871 x)}{1863 \sqrt{2-x+3 x^2}}+\frac{412}{81} x \sqrt{2-x+3 x^2}+\frac{32}{27} x^2 \sqrt{2-x+3 x^2}+\frac{1}{621} \int \frac{-5566+17158 x}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2 (12839-3871 x)}{1863 \sqrt{2-x+3 x^2}}+\frac{746}{81} \sqrt{2-x+3 x^2}+\frac{412}{81} x \sqrt{2-x+3 x^2}+\frac{32}{27} x^2 \sqrt{2-x+3 x^2}-\frac{353}{81} \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2 (12839-3871 x)}{1863 \sqrt{2-x+3 x^2}}+\frac{746}{81} \sqrt{2-x+3 x^2}+\frac{412}{81} x \sqrt{2-x+3 x^2}+\frac{32}{27} x^2 \sqrt{2-x+3 x^2}-\frac{353 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{81 \sqrt{69}}\\ &=\frac{2 (12839-3871 x)}{1863 \sqrt{2-x+3 x^2}}+\frac{746}{81} \sqrt{2-x+3 x^2}+\frac{412}{81} x \sqrt{2-x+3 x^2}+\frac{32}{27} x^2 \sqrt{2-x+3 x^2}+\frac{353 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{81 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0405456, size = 69, normalized size = 0.67 $\frac{6 \left (3312 x^4+13110 x^3+23207 x^2-2974 x+29997\right )-8119 \sqrt{9 x^2-3 x+6} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{5589 \sqrt{3 x^2-x+2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(6*(29997 - 2974*x + 23207*x^2 + 13110*x^3 + 3312*x^4) - 8119*Sqrt[6 - 3*x + 9*x^2]*ArcSinh[(-1 + 6*x)/Sqrt[23
]])/(5589*Sqrt[2 - x + 3*x^2])

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Maple [A]  time = 0.053, size = 115, normalized size = 1.1 \begin{align*}{\frac{32\,{x}^{4}}{9}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{380\,{x}^{3}}{27}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{2018\,{x}^{2}}{81}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}-{\frac{353\,\sqrt{3}}{243}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }+{\frac{353\,x}{81}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}-{\frac{-521+3126\,x}{414}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{557}{18}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

32/9*x^4/(3*x^2-x+2)^(1/2)+380/27*x^3/(3*x^2-x+2)^(1/2)+2018/81*x^2/(3*x^2-x+2)^(1/2)-353/243*3^(1/2)*arcsinh(
6/23*23^(1/2)*(x-1/6))+353/81*x/(3*x^2-x+2)^(1/2)-521/414*(-1+6*x)/(3*x^2-x+2)^(1/2)+557/18/(3*x^2-x+2)^(1/2)

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Maxima [A]  time = 1.5473, size = 131, normalized size = 1.27 \begin{align*} \frac{32 \, x^{4}}{9 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{380 \, x^{3}}{27 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{2018 \, x^{2}}{81 \, \sqrt{3 \, x^{2} - x + 2}} - \frac{353}{243} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) - \frac{5948 \, x}{1863 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{2222}{69 \, \sqrt{3 \, x^{2} - x + 2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

32/9*x^4/sqrt(3*x^2 - x + 2) + 380/27*x^3/sqrt(3*x^2 - x + 2) + 2018/81*x^2/sqrt(3*x^2 - x + 2) - 353/243*sqrt
(3)*arcsinh(1/23*sqrt(23)*(6*x - 1)) - 5948/1863*x/sqrt(3*x^2 - x + 2) + 2222/69/sqrt(3*x^2 - x + 2)

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Fricas [A]  time = 1.09804, size = 269, normalized size = 2.61 \begin{align*} \frac{8119 \, \sqrt{3}{\left (3 \, x^{2} - x + 2\right )} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 12 \,{\left (3312 \, x^{4} + 13110 \, x^{3} + 23207 \, x^{2} - 2974 \, x + 29997\right )} \sqrt{3 \, x^{2} - x + 2}}{11178 \,{\left (3 \, x^{2} - x + 2\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11178*(8119*sqrt(3)*(3*x^2 - x + 2)*log(4*sqrt(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^2 + 24*x - 25) + 12*(
3312*x^4 + 13110*x^3 + 23207*x^2 - 2974*x + 29997)*sqrt(3*x^2 - x + 2))/(3*x^2 - x + 2)

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

[Out]

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

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Giac [A]  time = 1.19926, size = 90, normalized size = 0.87 \begin{align*} \frac{353}{243} \, \sqrt{3} \log \left (-2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )} + 1\right ) + \frac{2 \,{\left ({\left (23 \,{\left (6 \,{\left (24 \, x + 95\right )} x + 1009\right )} x - 2974\right )} x + 29997\right )}}{1863 \, \sqrt{3 \, x^{2} - x + 2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

353/243*sqrt(3)*log(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) + 1) + 2/1863*((23*(6*(24*x + 95)*x + 1009)*x
- 2974)*x + 29997)/sqrt(3*x^2 - x + 2)