### 3.217 $$\int \frac{(2-x+3 x^2)^{3/2} (1+3 x+4 x^2)}{1+2 x} \, dx$$

Optimal. Leaf size=124 $\frac{2}{15} \left (3 x^2-x+2\right )^{5/2}+\frac{1}{144} (30 x+7) \left (3 x^2-x+2\right )^{3/2}+\frac{(402 x+869) \sqrt{3 x^2-x+2}}{1152}-\frac{13}{32} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )+\frac{2203 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{2304 \sqrt{3}}$

[Out]

((869 + 402*x)*Sqrt[2 - x + 3*x^2])/1152 + ((7 + 30*x)*(2 - x + 3*x^2)^(3/2))/144 + (2*(2 - x + 3*x^2)^(5/2))/
15 + (2203*ArcSinh[(1 - 6*x)/Sqrt[23]])/(2304*Sqrt[3]) - (13*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x
+ 3*x^2])])/32

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Rubi [A]  time = 0.144169, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 32, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.219, Rules used = {1653, 814, 843, 619, 215, 724, 206} $\frac{2}{15} \left (3 x^2-x+2\right )^{5/2}+\frac{1}{144} (30 x+7) \left (3 x^2-x+2\right )^{3/2}+\frac{(402 x+869) \sqrt{3 x^2-x+2}}{1152}-\frac{13}{32} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )+\frac{2203 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{2304 \sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((869 + 402*x)*Sqrt[2 - x + 3*x^2])/1152 + ((7 + 30*x)*(2 - x + 3*x^2)^(3/2))/144 + (2*(2 - x + 3*x^2)^(5/2))/
15 + (2203*ArcSinh[(1 - 6*x)/Sqrt[23]])/(2304*Sqrt[3]) - (13*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x
+ 3*x^2])])/32

Rule 1653

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

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]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx &=\frac{2}{15} \left (2-x+3 x^2\right )^{5/2}+\frac{1}{60} \int \frac{(80+100 x) \left (2-x+3 x^2\right )^{3/2}}{1+2 x} \, dx\\ &=\frac{1}{144} (7+30 x) \left (2-x+3 x^2\right )^{3/2}+\frac{2}{15} \left (2-x+3 x^2\right )^{5/2}-\frac{\int \frac{(-13380-8040 x) \sqrt{2-x+3 x^2}}{1+2 x} \, dx}{5760}\\ &=\frac{(869+402 x) \sqrt{2-x+3 x^2}}{1152}+\frac{1}{144} (7+30 x) \left (2-x+3 x^2\right )^{3/2}+\frac{2}{15} \left (2-x+3 x^2\right )^{5/2}+\frac{\int \frac{1195800-528720 x}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx}{276480}\\ &=\frac{(869+402 x) \sqrt{2-x+3 x^2}}{1152}+\frac{1}{144} (7+30 x) \left (2-x+3 x^2\right )^{3/2}+\frac{2}{15} \left (2-x+3 x^2\right )^{5/2}-\frac{2203 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{2304}+\frac{169}{32} \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx\\ &=\frac{(869+402 x) \sqrt{2-x+3 x^2}}{1152}+\frac{1}{144} (7+30 x) \left (2-x+3 x^2\right )^{3/2}+\frac{2}{15} \left (2-x+3 x^2\right )^{5/2}-\frac{169}{16} \operatorname{Subst}\left (\int \frac{1}{52-x^2} \, dx,x,\frac{9-8 x}{\sqrt{2-x+3 x^2}}\right )-\frac{2203 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{2304 \sqrt{69}}\\ &=\frac{(869+402 x) \sqrt{2-x+3 x^2}}{1152}+\frac{1}{144} (7+30 x) \left (2-x+3 x^2\right )^{3/2}+\frac{2}{15} \left (2-x+3 x^2\right )^{5/2}+\frac{2203 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{2304 \sqrt{3}}-\frac{13}{32} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{2-x+3 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0577476, size = 96, normalized size = 0.77 $\frac{6 \sqrt{3 x^2-x+2} \left (6912 x^4-1008 x^3+9624 x^2+1058 x+7977\right )-14040 \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )-11015 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{34560}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(6*Sqrt[2 - x + 3*x^2]*(7977 + 1058*x + 9624*x^2 - 1008*x^3 + 6912*x^4) - 11015*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqr
t[23]] - 14040*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/34560

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Maple [A]  time = 0.051, size = 151, normalized size = 1.2 \begin{align*}{\frac{2}{15} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{-5+30\,x}{144} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{-115+690\,x}{1152}\sqrt{3\,{x}^{2}-x+2}}-{\frac{2203\,\sqrt{3}}{6912}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }+{\frac{1}{12} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{-1+6\,x}{24}\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}+{\frac{13}{32}\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,x+5}}-{\frac{13\,\sqrt{13}}{32}{\it Artanh} \left ({\frac{2\,\sqrt{13}}{13} \left ({\frac{9}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,x+5}}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*x^2-x+2)^(5/2)+5/144*(-1+6*x)*(3*x^2-x+2)^(3/2)+115/1152*(-1+6*x)*(3*x^2-x+2)^(1/2)-2203/6912*3^(1/2)*
arcsinh(6/23*23^(1/2)*(x-1/6))+1/12*(3*(x+1/2)^2-4*x+5/4)^(3/2)-1/24*(-1+6*x)*(3*(x+1/2)^2-4*x+5/4)^(1/2)+13/3
2*(12*(x+1/2)^2-16*x+5)^(1/2)-13/32*13^(1/2)*arctanh(2/13*(9/2-4*x)*13^(1/2)/(12*(x+1/2)^2-16*x+5)^(1/2))

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Maxima [A]  time = 1.48406, size = 169, normalized size = 1.36 \begin{align*} \frac{2}{15} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} + \frac{5}{24} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x + \frac{7}{144} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} + \frac{67}{192} \, \sqrt{3 \, x^{2} - x + 2} x - \frac{2203}{6912} \, \sqrt{3} \operatorname{arsinh}\left (\frac{6}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) + \frac{13}{32} \, \sqrt{13} \operatorname{arsinh}\left (\frac{8 \, \sqrt{23} x}{23 \,{\left | 2 \, x + 1 \right |}} - \frac{9 \, \sqrt{23}}{23 \,{\left | 2 \, x + 1 \right |}}\right ) + \frac{869}{1152} \, \sqrt{3 \, x^{2} - x + 2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*x^2 - x + 2)^(5/2) + 5/24*(3*x^2 - x + 2)^(3/2)*x + 7/144*(3*x^2 - x + 2)^(3/2) + 67/192*sqrt(3*x^2 -
x + 2)*x - 2203/6912*sqrt(3)*arcsinh(6/23*sqrt(23)*x - 1/23*sqrt(23)) + 13/32*sqrt(13)*arcsinh(8/23*sqrt(23)*x
/abs(2*x + 1) - 9/23*sqrt(23)/abs(2*x + 1)) + 869/1152*sqrt(3*x^2 - x + 2)

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Fricas [A]  time = 1.6081, size = 367, normalized size = 2.96 \begin{align*} \frac{1}{5760} \,{\left (6912 \, x^{4} - 1008 \, x^{3} + 9624 \, x^{2} + 1058 \, x + 7977\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{2203}{13824} \, \sqrt{3} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + \frac{13}{64} \, \sqrt{13} \log \left (-\frac{4 \, \sqrt{13} \sqrt{3 \, x^{2} - x + 2}{\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5760*(6912*x^4 - 1008*x^3 + 9624*x^2 + 1058*x + 7977)*sqrt(3*x^2 - x + 2) + 2203/13824*sqrt(3)*log(4*sqrt(3)
*sqrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^2 + 24*x - 25) + 13/64*sqrt(13)*log(-(4*sqrt(13)*sqrt(3*x^2 - x + 2)*(8*
x - 9) + 220*x^2 - 196*x + 185)/(4*x^2 + 4*x + 1))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.28475, size = 184, normalized size = 1.48 \begin{align*} \frac{1}{5760} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (48 \, x - 7\right )} x + 401\right )} x + 529\right )} x + 7977\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{2203}{6912} \, \sqrt{3} \log \left (-6 \, \sqrt{3} x + \sqrt{3} + 6 \, \sqrt{3 \, x^{2} - x + 2}\right ) + \frac{13}{32} \, \sqrt{13} \log \left (-\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{13} - 2 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} - x + 2} \right |}}{2 \,{\left (2 \, \sqrt{3} x - \sqrt{13} + \sqrt{3} - 2 \, \sqrt{3 \, x^{2} - x + 2}\right )}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5760*(2*(12*(6*(48*x - 7)*x + 401)*x + 529)*x + 7977)*sqrt(3*x^2 - x + 2) + 2203/6912*sqrt(3)*log(-6*sqrt(3)
*x + sqrt(3) + 6*sqrt(3*x^2 - x + 2)) + 13/32*sqrt(13)*log(-1/2*abs(-4*sqrt(3)*x - 2*sqrt(13) - 2*sqrt(3) + 4*
sqrt(3*x^2 - x + 2))/(2*sqrt(3)*x - sqrt(13) + sqrt(3) - 2*sqrt(3*x^2 - x + 2)))