### 3.223 $$\int \frac{(2-x+3 x^2)^{5/2} (1+3 x+4 x^2)}{1+2 x} \, dx$$

Optimal. Leaf size=147 $\frac{2}{21} \left (3 x^2-x+2\right )^{7/2}+\frac{(150 x+29) \left (3 x^2-x+2\right )^{5/2}}{1080}+\frac{(2154 x+2449) \left (3 x^2-x+2\right )^{3/2}}{10368}+\frac{(221999-17850 x) \sqrt{3 x^2-x+2}}{82944}-\frac{169}{128} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )+\frac{944521 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{165888 \sqrt{3}}$

[Out]

((221999 - 17850*x)*Sqrt[2 - x + 3*x^2])/82944 + ((2449 + 2154*x)*(2 - x + 3*x^2)^(3/2))/10368 + ((29 + 150*x)
*(2 - x + 3*x^2)^(5/2))/1080 + (2*(2 - x + 3*x^2)^(7/2))/21 + (944521*ArcSinh[(1 - 6*x)/Sqrt[23]])/(165888*Sqr
t[3]) - (169*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/128

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Rubi [A]  time = 0.159525, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 9, 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} $\frac{2}{21} \left (3 x^2-x+2\right )^{7/2}+\frac{(150 x+29) \left (3 x^2-x+2\right )^{5/2}}{1080}+\frac{(2154 x+2449) \left (3 x^2-x+2\right )^{3/2}}{10368}+\frac{(221999-17850 x) \sqrt{3 x^2-x+2}}{82944}-\frac{169}{128} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )+\frac{944521 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{165888 \sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

Int[((2 - x + 3*x^2)^(5/2)*(1 + 3*x + 4*x^2))/(1 + 2*x),x]

[Out]

((221999 - 17850*x)*Sqrt[2 - x + 3*x^2])/82944 + ((2449 + 2154*x)*(2 - x + 3*x^2)^(3/2))/10368 + ((29 + 150*x)
*(2 - x + 3*x^2)^(5/2))/1080 + (2*(2 - x + 3*x^2)^(7/2))/21 + (944521*ArcSinh[(1 - 6*x)/Sqrt[23]])/(165888*Sqr
t[3]) - (169*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/128

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 )^{5/2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx &=\frac{2}{21} \left (2-x+3 x^2\right )^{7/2}+\frac{1}{84} \int \frac{(112+140 x) \left (2-x+3 x^2\right )^{5/2}}{1+2 x} \, dx\\ &=\frac{(29+150 x) \left (2-x+3 x^2\right )^{5/2}}{1080}+\frac{2}{21} \left (2-x+3 x^2\right )^{7/2}-\frac{\int \frac{(-29708-20104 x) \left (2-x+3 x^2\right )^{3/2}}{1+2 x} \, dx}{12096}\\ &=\frac{(2449+2154 x) \left (2-x+3 x^2\right )^{3/2}}{10368}+\frac{(29+150 x) \left (2-x+3 x^2\right )^{5/2}}{1080}+\frac{2}{21} \left (2-x+3 x^2\right )^{7/2}+\frac{\int \frac{(5632872-999600 x) \sqrt{2-x+3 x^2}}{1+2 x} \, dx}{1161216}\\ &=\frac{(221999-17850 x) \sqrt{2-x+3 x^2}}{82944}+\frac{(2449+2154 x) \left (2-x+3 x^2\right )^{3/2}}{10368}+\frac{(29+150 x) \left (2-x+3 x^2\right )^{5/2}}{1080}+\frac{2}{21} \left (2-x+3 x^2\right )^{7/2}-\frac{\int \frac{-639337776+634718112 x}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx}{55738368}\\ &=\frac{(221999-17850 x) \sqrt{2-x+3 x^2}}{82944}+\frac{(2449+2154 x) \left (2-x+3 x^2\right )^{3/2}}{10368}+\frac{(29+150 x) \left (2-x+3 x^2\right )^{5/2}}{1080}+\frac{2}{21} \left (2-x+3 x^2\right )^{7/2}-\frac{944521 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{165888}+\frac{2197}{128} \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx\\ &=\frac{(221999-17850 x) \sqrt{2-x+3 x^2}}{82944}+\frac{(2449+2154 x) \left (2-x+3 x^2\right )^{3/2}}{10368}+\frac{(29+150 x) \left (2-x+3 x^2\right )^{5/2}}{1080}+\frac{2}{21} \left (2-x+3 x^2\right )^{7/2}-\frac{2197}{64} \operatorname{Subst}\left (\int \frac{1}{52-x^2} \, dx,x,\frac{9-8 x}{\sqrt{2-x+3 x^2}}\right )-\frac{944521 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{165888 \sqrt{69}}\\ &=\frac{(221999-17850 x) \sqrt{2-x+3 x^2}}{82944}+\frac{(2449+2154 x) \left (2-x+3 x^2\right )^{3/2}}{10368}+\frac{(29+150 x) \left (2-x+3 x^2\right )^{5/2}}{1080}+\frac{2}{21} \left (2-x+3 x^2\right )^{7/2}+\frac{944521 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{165888 \sqrt{3}}-\frac{169}{128} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{2-x+3 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0735996, size = 106, normalized size = 0.72 $\frac{6 \sqrt{3 x^2-x+2} \left (7464960 x^6-3836160 x^5+15700608 x^4-3646512 x^3+12466776 x^2-2120998 x+11665053\right )-22997520 \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )-33058235 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{17418240}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[((2 - x + 3*x^2)^(5/2)*(1 + 3*x + 4*x^2))/(1 + 2*x),x]

[Out]

(6*Sqrt[2 - x + 3*x^2]*(11665053 - 2120998*x + 12466776*x^2 - 3646512*x^3 + 15700608*x^4 - 3836160*x^5 + 74649
60*x^6) - 33058235*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]] - 22997520*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt
[2 - x + 3*x^2])])/17418240

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Maple [A]  time = 0.052, size = 207, normalized size = 1.4 \begin{align*}{\frac{2}{21} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{-5+30\,x}{216} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{-575+3450\,x}{10368} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{-13225+79350\,x}{82944}\sqrt{3\,{x}^{2}-x+2}}-{\frac{944521\,\sqrt{3}}{497664}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }+{\frac{1}{20} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{-1+6\,x}{48} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{-25+150\,x}{128}\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}+{\frac{13}{48} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{169}{128}\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,x+5}}-{\frac{169\,\sqrt{13}}{128}{\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)^(5/2)*(4*x^2+3*x+1)/(1+2*x),x)

[Out]

2/21*(3*x^2-x+2)^(7/2)+5/216*(-1+6*x)*(3*x^2-x+2)^(5/2)+575/10368*(-1+6*x)*(3*x^2-x+2)^(3/2)+13225/82944*(-1+6
*x)*(3*x^2-x+2)^(1/2)-944521/497664*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))+1/20*(3*(x+1/2)^2-4*x+5/4)^(5/2)-1/
48*(-1+6*x)*(3*(x+1/2)^2-4*x+5/4)^(3/2)-25/128*(-1+6*x)*(3*(x+1/2)^2-4*x+5/4)^(1/2)+13/48*(3*(x+1/2)^2-4*x+5/4
)^(3/2)+169/128*(12*(x+1/2)^2-16*x+5)^(1/2)-169/128*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 = 1.51275, size = 208, normalized size = 1.41 \begin{align*} \frac{2}{21} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} + \frac{5}{36} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x + \frac{29}{1080} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} + \frac{359}{1728} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x + \frac{2449}{10368} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} - \frac{2975}{13824} \, \sqrt{3 \, x^{2} - x + 2} x - \frac{944521}{497664} \, \sqrt{3} \operatorname{arsinh}\left (\frac{6}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) + \frac{169}{128} \, \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{221999}{82944} \, \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)^(5/2)*(4*x^2+3*x+1)/(1+2*x),x, algorithm="maxima")

[Out]

2/21*(3*x^2 - x + 2)^(7/2) + 5/36*(3*x^2 - x + 2)^(5/2)*x + 29/1080*(3*x^2 - x + 2)^(5/2) + 359/1728*(3*x^2 -
x + 2)^(3/2)*x + 2449/10368*(3*x^2 - x + 2)^(3/2) - 2975/13824*sqrt(3*x^2 - x + 2)*x - 944521/497664*sqrt(3)*a
rcsinh(6/23*sqrt(23)*x - 1/23*sqrt(23)) + 169/128*sqrt(13)*arcsinh(8/23*sqrt(23)*x/abs(2*x + 1) - 9/23*sqrt(23
)/abs(2*x + 1)) + 221999/82944*sqrt(3*x^2 - x + 2)

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Fricas [A]  time = 1.37156, size = 440, normalized size = 2.99 \begin{align*} \frac{1}{2903040} \,{\left (7464960 \, x^{6} - 3836160 \, x^{5} + 15700608 \, x^{4} - 3646512 \, x^{3} + 12466776 \, x^{2} - 2120998 \, x + 11665053\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{944521}{995328} \, \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{169}{256} \, \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)^(5/2)*(4*x^2+3*x+1)/(1+2*x),x, algorithm="fricas")

[Out]

1/2903040*(7464960*x^6 - 3836160*x^5 + 15700608*x^4 - 3646512*x^3 + 12466776*x^2 - 2120998*x + 11665053)*sqrt(
3*x^2 - x + 2) + 944521/995328*sqrt(3)*log(4*sqrt(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^2 + 24*x - 25) + 169
/256*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{5}{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)**(5/2)*(4*x**2+3*x+1)/(1+2*x),x)

[Out]

Integral((3*x**2 - x + 2)**(5/2)*(4*x**2 + 3*x + 1)/(2*x + 1), x)

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Giac [A]  time = 1.34667, size = 197, normalized size = 1.34 \begin{align*} \frac{1}{2903040} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (8 \,{\left (30 \,{\left (72 \, x - 37\right )} x + 4543\right )} x - 8441\right )} x + 519449\right )} x - 1060499\right )} x + 11665053\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{944521}{497664} \, \sqrt{3} \log \left (-6 \, \sqrt{3} x + \sqrt{3} + 6 \, \sqrt{3 \, x^{2} - x + 2}\right ) + \frac{169}{128} \, \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)^(5/2)*(4*x^2+3*x+1)/(1+2*x),x, algorithm="giac")

[Out]

1/2903040*(2*(12*(18*(8*(30*(72*x - 37)*x + 4543)*x - 8441)*x + 519449)*x - 1060499)*x + 11665053)*sqrt(3*x^2
- x + 2) + 944521/497664*sqrt(3)*log(-6*sqrt(3)*x + sqrt(3) + 6*sqrt(3*x^2 - x + 2)) + 169/128*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*sq
rt(3*x^2 - x + 2)))