3.211 \(\int \frac {\sqrt {2-x+3 x^2} (1+3 x+4 x^2)}{1+2 x} \, dx\)

Optimal. Leaf size=101 \[ \frac {2}{9} \left (3 x^2-x+2\right )^{3/2}+\frac {1}{72} (30 x+13) \sqrt {3 x^2-x+2}-\frac {1}{8} \sqrt {13} \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )-\frac {43 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{144 \sqrt {3}} \]

[Out]

2/9*(3*x^2-x+2)^(3/2)-43/432*arcsinh(1/23*(1-6*x)*23^(1/2))*3^(1/2)-1/8*arctanh(1/26*(9-8*x)*13^(1/2)/(3*x^2-x
+2)^(1/2))*13^(1/2)+1/72*(13+30*x)*(3*x^2-x+2)^(1/2)

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Rubi [A]  time = 0.12, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {2}{9} \left (3 x^2-x+2\right )^{3/2}+\frac {1}{72} (30 x+13) \sqrt {3 x^2-x+2}-\frac {1}{8} \sqrt {13} \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )-\frac {43 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{144 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((13 + 30*x)*Sqrt[2 - x + 3*x^2])/72 + (2*(2 - x + 3*x^2)^(3/2))/9 - (43*ArcSinh[(1 - 6*x)/Sqrt[23]])/(144*Sqr
t[3]) - (Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/8

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])

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 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 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 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 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]))

Rubi steps

\begin {align*} \int \frac {\sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx &=\frac {2}{9} \left (2-x+3 x^2\right )^{3/2}+\frac {1}{36} \int \frac {(48+60 x) \sqrt {2-x+3 x^2}}{1+2 x} \, dx\\ &=\frac {1}{72} (13+30 x) \sqrt {2-x+3 x^2}+\frac {2}{9} \left (2-x+3 x^2\right )^{3/2}-\frac {\int \frac {-3324-1032 x}{(1+2 x) \sqrt {2-x+3 x^2}} \, dx}{1728}\\ &=\frac {1}{72} (13+30 x) \sqrt {2-x+3 x^2}+\frac {2}{9} \left (2-x+3 x^2\right )^{3/2}+\frac {43}{144} \int \frac {1}{\sqrt {2-x+3 x^2}} \, dx+\frac {13}{8} \int \frac {1}{(1+2 x) \sqrt {2-x+3 x^2}} \, dx\\ &=\frac {1}{72} (13+30 x) \sqrt {2-x+3 x^2}+\frac {2}{9} \left (2-x+3 x^2\right )^{3/2}-\frac {13}{4} \operatorname {Subst}\left (\int \frac {1}{52-x^2} \, dx,x,\frac {9-8 x}{\sqrt {2-x+3 x^2}}\right )+\frac {43 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+6 x\right )}{144 \sqrt {69}}\\ &=\frac {1}{72} (13+30 x) \sqrt {2-x+3 x^2}+\frac {2}{9} \left (2-x+3 x^2\right )^{3/2}-\frac {43 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{144 \sqrt {3}}-\frac {1}{8} \sqrt {13} \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 86, normalized size = 0.85 \[ \frac {1}{432} \left (6 \sqrt {3 x^2-x+2} \left (48 x^2+14 x+45\right )-54 \sqrt {13} \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )+43 \sqrt {3} \sinh ^{-1}\left (\frac {6 x-1}{\sqrt {23}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Sqrt[2 - x + 3*x^2]*(45 + 14*x + 48*x^2) + 43*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]] - 54*Sqrt[13]*ArcTanh[(9
- 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/432

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fricas [A]  time = 0.91, size = 115, normalized size = 1.14 \[ \frac {1}{72} \, {\left (48 \, x^{2} + 14 \, x + 45\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {43}{864} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + \frac {1}{16} \, \sqrt {13} \log \left (-\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(48*x^2 + 14*x + 45)*sqrt(3*x^2 - x + 2) + 43/864*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) -
72*x^2 + 24*x - 25) + 1/16*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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giac [A]  time = 0.39, size = 126, normalized size = 1.25 \[ \frac {1}{72} \, {\left (2 \, {\left (24 \, x + 7\right )} x + 45\right )} \sqrt {3 \, x^{2} - x + 2} - \frac {43}{432} \, \sqrt {3} \log \left (-6 \, \sqrt {3} x + \sqrt {3} + 6 \, \sqrt {3 \, x^{2} - x + 2}\right ) + \frac {1}{8} \, \sqrt {13} \log \left (-\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {13} - 2 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} - x + 2} \right |}}{2 \, {\left (2 \, \sqrt {3} x - \sqrt {13} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} - x + 2}\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(2*(24*x + 7)*x + 45)*sqrt(3*x^2 - x + 2) - 43/432*sqrt(3)*log(-6*sqrt(3)*x + sqrt(3) + 6*sqrt(3*x^2 - x
+ 2)) + 1/8*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)))

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maple [A]  time = 0.01, size = 95, normalized size = 0.94 \[ \frac {43 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{432}-\frac {\sqrt {13}\, \arctanh \left (\frac {2 \left (-4 x +\frac {9}{2}\right ) \sqrt {13}}{13 \sqrt {-16 x +12 \left (x +\frac {1}{2}\right )^{2}+5}}\right )}{8}+\frac {2 \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{9}+\frac {5 \left (6 x -1\right ) \sqrt {3 x^{2}-x +2}}{72}+\frac {\sqrt {-16 x +12 \left (x +\frac {1}{2}\right )^{2}+5}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(3*x^2-x+2)^(3/2)+5/72*(6*x-1)*(3*x^2-x+2)^(1/2)+43/432*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))+1/8*(12*(x+
1/2)^2-16*x+5)^(1/2)-1/8*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 = 0.96, size = 96, normalized size = 0.95 \[ \frac {2}{9} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} + \frac {5}{12} \, \sqrt {3 \, x^{2} - x + 2} x + \frac {43}{432} \, \sqrt {3} \operatorname {arsinh}\left (\frac {6}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) + \frac {1}{8} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {13}{72} \, \sqrt {3 \, x^{2} - x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(3*x^2 - x + 2)^(3/2) + 5/12*sqrt(3*x^2 - x + 2)*x + 43/432*sqrt(3)*arcsinh(6/23*sqrt(23)*x - 1/23*sqrt(23
)) + 1/8*sqrt(13)*arcsinh(8/23*sqrt(23)*x/abs(2*x + 1) - 9/23*sqrt(23)/abs(2*x + 1)) + 13/72*sqrt(3*x^2 - x +
2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {3\,x^2-x+2}\,\left (4\,x^2+3\,x+1\right )}{2\,x+1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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