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]

1/144*(7+30*x)*(3*x^2-x+2)^(3/2)+2/15*(3*x^2-x+2)^(5/2)+2203/6912*arcsinh(1/23*(1-6*x)*23^(1/2))*3^(1/2)-13/32
*arctanh(1/26*(9-8*x)*13^(1/2)/(3*x^2-x+2)^(1/2))*13^(1/2)+1/1152*(869+402*x)*(3*x^2-x+2)^(1/2)

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Rubi [A]  time = 0.14, antiderivative size = 124, normalized size of antiderivative = 1.00, 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 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 {\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*}

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Mathematica [A]  time = 0.06, size = 96, normalized size = 0.77 \[ \frac {-14040 \sqrt {13} \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )+6 \sqrt {3 x^2-x+2} \left (6912 x^4-1008 x^3+9624 x^2+1058 x+7977\right )-11015 \sqrt {3} \sinh ^{-1}\left (\frac {6 x-1}{\sqrt {23}}\right )}{34560} \]

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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fricas [A]  time = 0.91, size = 125, normalized size = 1.01 \[ \frac {1}{5760} \, {\left (6912 \, x^{4} - 1008 \, x^{3} + 9624 \, x^{2} + 1058 \, x + 7977\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {2203}{13824} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + \frac {13}{64} \, \sqrt {13} \log \left (-\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) \]

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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giac [A]  time = 0.28, size = 136, normalized size = 1.10 \[ \frac {1}{5760} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (48 \, x - 7\right )} x + 401\right )} x + 529\right )} x + 7977\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {2203}{6912} \, \sqrt {3} \log \left (-6 \, \sqrt {3} x + \sqrt {3} + 6 \, \sqrt {3 \, x^{2} - x + 2}\right ) + \frac {13}{32} \, \sqrt {13} \log \left (-\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {13} - 2 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} - x + 2} \right |}}{2 \, {\left (2 \, \sqrt {3} x - \sqrt {13} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} - x + 2}\right )}}\right ) \]

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)))

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maple [A]  time = 0.01, size = 151, normalized size = 1.22 \[ -\frac {2203 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{6912}-\frac {13 \sqrt {13}\, \arctanh \left (\frac {2 \left (-4 x +\frac {9}{2}\right ) \sqrt {13}}{13 \sqrt {-16 x +12 \left (x +\frac {1}{2}\right )^{2}+5}}\right )}{32}+\frac {2 \left (3 x^{2}-x +2\right )^{\frac {5}{2}}}{15}+\frac {5 \left (6 x -1\right ) \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{144}+\frac {115 \left (6 x -1\right ) \sqrt {3 x^{2}-x +2}}{1152}+\frac {\left (-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}\right )^{\frac {3}{2}}}{12}-\frac {\left (6 x -1\right ) \sqrt {-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}}}{24}+\frac {13 \sqrt {-16 x +12 \left (x +\frac {1}{2}\right )^{2}+5}}{32} \]

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)/(2*x+1),x)

[Out]

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

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maxima [A]  time = 0.99, size = 125, normalized size = 1.01 \[ \frac {2}{15} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} + \frac {5}{24} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x + \frac {7}{144} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} + \frac {67}{192} \, \sqrt {3 \, x^{2} - x + 2} x - \frac {2203}{6912} \, \sqrt {3} \operatorname {arsinh}\left (\frac {6}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) + \frac {13}{32} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {869}{1152} \, \sqrt {3 \, x^{2} - x + 2} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((3*x^2 - x + 2)^(3/2)*(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 \[ \int \frac {\left (3 x^{2} - x + 2\right )^{\frac {3}{2}} \left (4 x^{2} + 3 x + 1\right )}{2 x + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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