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

Optimal. Leaf size=193 \[ -\frac {2 (14 x+15)}{51 \left (-2 x^2+3 x+1\right )^{3/2}}-\frac {2 (4814 x+291)}{867 \sqrt {-2 x^2+3 x+1}}+\frac {9}{2} \sqrt {\frac {1}{5} \left (17 \sqrt {10}-53\right )} \tan ^{-1}\left (\frac {\left (1+4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {1+\sqrt {10}} \sqrt {-2 x^2+3 x+1}}\right )+\frac {9}{2} \sqrt {\frac {1}{5} \left (53+17 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {\sqrt {10}-1} \sqrt {-2 x^2+3 x+1}}\right ) \]

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Rubi [A]  time = 0.27, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1016, 1060, 1032, 724, 204, 206} \begin {gather*} -\frac {2 (14 x+15)}{51 \left (-2 x^2+3 x+1\right )^{3/2}}-\frac {2 (4814 x+291)}{867 \sqrt {-2 x^2+3 x+1}}+\frac {9}{2} \sqrt {\frac {1}{5} \left (17 \sqrt {10}-53\right )} \tan ^{-1}\left (\frac {\left (1+4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {1+\sqrt {10}} \sqrt {-2 x^2+3 x+1}}\right )+\frac {9}{2} \sqrt {\frac {1}{5} \left (53+17 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {\sqrt {10}-1} \sqrt {-2 x^2+3 x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(15 + 14*x))/(51*(1 + 3*x - 2*x^2)^(3/2)) - (2*(291 + 4814*x))/(867*Sqrt[1 + 3*x - 2*x^2]) + (9*Sqrt[(-53
+ 17*Sqrt[10])/5]*ArcTan[(3*(4 - Sqrt[10]) + (1 + 4*Sqrt[10])*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])]
)/2 + (9*Sqrt[(53 + 17*Sqrt[10])/5]*ArcTanh[(3*(4 + Sqrt[10]) + (1 - 4*Sqrt[10])*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqr
t[1 + 3*x - 2*x^2])])/2

Rule 204

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

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 1016

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Sy
mbol] :> Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*(g*c*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h
)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - h*(b*c*d - 2*a*c*e + a*b*f)
)*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*
f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*h - 2*g*c)
*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*
f) - a*(-(h*c*e))))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d +
b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*g*f - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a*(-(h*c*e))))*(b
*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) + a*h*c*e
))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2
- 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1
])

Rule 1032

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbo
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*g - h*(b - q))/q, Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2])
, x], x] - Dist[(2*c*g - h*(b + q))/q, Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b,
c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]

Rule 1060

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*((A*c - a*C)*(2*a*c*e - b*(c
*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b
*c*d - 2*a*c*e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)
*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a
+ b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f)
)*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C
*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c
*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(
c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*
e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b,
c, d, e, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 -
(b*d - a*e)*(c*e - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rubi steps

\begin {align*} \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{5/2}} \, dx &=-\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}+\frac {2}{51} \int \frac {-56+\frac {235 x}{2}+84 x^2}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {2 (291+4814 x)}{867 \sqrt {1+3 x-2 x^2}}+\frac {4}{867} \int \frac {\frac {7803}{2}+\frac {23409 x}{4}}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x-2 x^2}} \, dx\\ &=-\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {2 (291+4814 x)}{867 \sqrt {1+3 x-2 x^2}}+\frac {1}{5} \left (27 \left (5-2 \sqrt {10}\right )\right ) \int \frac {1}{\left (4-2 \sqrt {10}-6 x\right ) \sqrt {1+3 x-2 x^2}} \, dx+\frac {1}{5} \left (27 \left (5+2 \sqrt {10}\right )\right ) \int \frac {1}{\left (4+2 \sqrt {10}-6 x\right ) \sqrt {1+3 x-2 x^2}} \, dx\\ &=-\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {2 (291+4814 x)}{867 \sqrt {1+3 x-2 x^2}}-\frac {1}{5} \left (54 \left (5-2 \sqrt {10}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{144+72 \left (4-2 \sqrt {10}\right )-8 \left (4-2 \sqrt {10}\right )^2-x^2} \, dx,x,\frac {-12-3 \left (4-2 \sqrt {10}\right )-\left (18-4 \left (4-2 \sqrt {10}\right )\right ) x}{\sqrt {1+3 x-2 x^2}}\right )-\frac {1}{5} \left (54 \left (5+2 \sqrt {10}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{144+72 \left (4+2 \sqrt {10}\right )-8 \left (4+2 \sqrt {10}\right )^2-x^2} \, dx,x,\frac {-12-3 \left (4+2 \sqrt {10}\right )-\left (18-4 \left (4+2 \sqrt {10}\right )\right ) x}{\sqrt {1+3 x-2 x^2}}\right )\\ &=-\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {2 (291+4814 x)}{867 \sqrt {1+3 x-2 x^2}}+\frac {9}{2} \sqrt {\frac {1}{5} \left (-53+17 \sqrt {10}\right )} \tan ^{-1}\left (\frac {3 \left (4-\sqrt {10}\right )+\left (1+4 \sqrt {10}\right ) x}{2 \sqrt {1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right )+\frac {9}{2} \sqrt {\frac {1}{5} \left (53+17 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (1-4 \sqrt {10}\right ) x}{2 \sqrt {-1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.60, size = 185, normalized size = 0.96 \begin {gather*} \frac {3}{10} \sqrt {1+\sqrt {10}} \left (7 \sqrt {10}-25\right ) \tan ^{-1}\left (\frac {3 \left (\sqrt {10}-4\right )-\left (1+4 \sqrt {10}\right ) x}{2 \sqrt {1+\sqrt {10}} \sqrt {-2 x^2+3 x+1}}\right )-\frac {3}{10} \sqrt {\sqrt {10}-1} \left (25+7 \sqrt {10}\right ) \tanh ^{-1}\left (\frac {\left (4 \sqrt {10}-1\right ) x-3 \left (4+\sqrt {10}\right )}{2 \sqrt {\sqrt {10}-1} \sqrt {-2 x^2+3 x+1}}\right )-\frac {2 \left (-9628 x^3+13860 x^2+5925 x+546\right )}{867 \left (-2 x^2+3 x+1\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(546 + 5925*x + 13860*x^2 - 9628*x^3))/(867*(1 + 3*x - 2*x^2)^(3/2)) + (3*Sqrt[1 + Sqrt[10]]*(-25 + 7*Sqrt
[10])*ArcTan[(3*(-4 + Sqrt[10]) - (1 + 4*Sqrt[10])*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/10 - (3*S
qrt[-1 + Sqrt[10]]*(25 + 7*Sqrt[10])*ArcTanh[(-3*(4 + Sqrt[10]) + (-1 + 4*Sqrt[10])*x)/(2*Sqrt[-1 + Sqrt[10]]*
Sqrt[1 + 3*x - 2*x^2])])/10

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IntegrateAlgebraic [C]  time = 0.54, size = 195, normalized size = 1.01 \begin {gather*} \frac {2 \sqrt {-2 x^2+3 x+1} \left (9628 x^3-13860 x^2-5925 x-546\right )}{867 \left (2 x^2-3 x-1\right )^2}-\frac {9}{2} \text {RootSum}\left [2 \text {$\#$1}^4-8 \text {$\#$1}^3+8 \text {$\#$1}^2+20 \text {$\#$1}+5\&,\frac {2 \text {$\#$1}^2 \log \left (\text {$\#$1} (-x)+\sqrt {-2 x^2+3 x+1}-1\right )-2 \text {$\#$1}^2 \log (x)-6 \text {$\#$1} \log \left (\text {$\#$1} (-x)+\sqrt {-2 x^2+3 x+1}-1\right )+13 \log \left (\text {$\#$1} (-x)+\sqrt {-2 x^2+3 x+1}-1\right )+6 \text {$\#$1} \log (x)-13 \log (x)}{2 \text {$\#$1}^3-6 \text {$\#$1}^2+4 \text {$\#$1}+5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 + 3*x - 2*x^2]*(-546 - 5925*x - 13860*x^2 + 9628*x^3))/(867*(-1 - 3*x + 2*x^2)^2) - (9*RootSum[5 + 2
0*#1 + 8*#1^2 - 8*#1^3 + 2*#1^4 & , (-13*Log[x] + 13*Log[-1 + Sqrt[1 + 3*x - 2*x^2] - x*#1] + 6*Log[x]*#1 - 6*
Log[-1 + Sqrt[1 + 3*x - 2*x^2] - x*#1]*#1 - 2*Log[x]*#1^2 + 2*Log[-1 + Sqrt[1 + 3*x - 2*x^2] - x*#1]*#1^2)/(5
+ 4*#1 - 6*#1^2 + 2*#1^3) & ])/2

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fricas [B]  time = 0.44, size = 439, normalized size = 2.27 \begin {gather*} -\frac {43680 \, x^{4} - 131040 \, x^{3} - 31212 \, \sqrt {5} {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt {17 \, \sqrt {10} - 53} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {10} \sqrt {5} x + 10 \, \sqrt {5} x\right )} \sqrt {17 \, \sqrt {10} - 53} \sqrt {\frac {6 \, x^{2} + \sqrt {10} {\left (3 \, x^{2} + 2 \, x\right )} - 2 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} {\left (\sqrt {10} x + 2 \, x + 2\right )} + 10 \, x + 4}{x^{2}}} + 2 \, {\left (\sqrt {10} \sqrt {5} {\left (6 \, x + 1\right )} - \sqrt {-2 \, x^{2} + 3 \, x + 1} {\left (\sqrt {10} \sqrt {5} + 10 \, \sqrt {5}\right )} + 5 \, \sqrt {5} {\left (3 \, x + 2\right )}\right )} \sqrt {17 \, \sqrt {10} - 53}}{90 \, x}\right ) - 7803 \, \sqrt {5} {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt {17 \, \sqrt {10} + 53} \log \left (\frac {9 \, {\left (45 \, \sqrt {10} x + {\left (13 \, \sqrt {10} \sqrt {5} x - 40 \, \sqrt {5} x\right )} \sqrt {17 \, \sqrt {10} + 53} - 90 \, x + 90 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} - 90\right )}}{x}\right ) + 7803 \, \sqrt {5} {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt {17 \, \sqrt {10} + 53} \log \left (\frac {9 \, {\left (45 \, \sqrt {10} x - {\left (13 \, \sqrt {10} \sqrt {5} x - 40 \, \sqrt {5} x\right )} \sqrt {17 \, \sqrt {10} + 53} - 90 \, x + 90 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} - 90\right )}}{x}\right ) + 54600 \, x^{2} - 20 \, {\left (9628 \, x^{3} - 13860 \, x^{2} - 5925 \, x - 546\right )} \sqrt {-2 \, x^{2} + 3 \, x + 1} + 65520 \, x + 10920}{8670 \, {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(-3*x^2+4*x+2)/(-2*x^2+3*x+1)^(5/2),x, algorithm="fricas")

[Out]

-1/8670*(43680*x^4 - 131040*x^3 - 31212*sqrt(5)*(4*x^4 - 12*x^3 + 5*x^2 + 6*x + 1)*sqrt(17*sqrt(10) - 53)*arct
an(1/90*(sqrt(2)*(sqrt(10)*sqrt(5)*x + 10*sqrt(5)*x)*sqrt(17*sqrt(10) - 53)*sqrt((6*x^2 + sqrt(10)*(3*x^2 + 2*
x) - 2*sqrt(-2*x^2 + 3*x + 1)*(sqrt(10)*x + 2*x + 2) + 10*x + 4)/x^2) + 2*(sqrt(10)*sqrt(5)*(6*x + 1) - sqrt(-
2*x^2 + 3*x + 1)*(sqrt(10)*sqrt(5) + 10*sqrt(5)) + 5*sqrt(5)*(3*x + 2))*sqrt(17*sqrt(10) - 53))/x) - 7803*sqrt
(5)*(4*x^4 - 12*x^3 + 5*x^2 + 6*x + 1)*sqrt(17*sqrt(10) + 53)*log(9*(45*sqrt(10)*x + (13*sqrt(10)*sqrt(5)*x -
40*sqrt(5)*x)*sqrt(17*sqrt(10) + 53) - 90*x + 90*sqrt(-2*x^2 + 3*x + 1) - 90)/x) + 7803*sqrt(5)*(4*x^4 - 12*x^
3 + 5*x^2 + 6*x + 1)*sqrt(17*sqrt(10) + 53)*log(9*(45*sqrt(10)*x - (13*sqrt(10)*sqrt(5)*x - 40*sqrt(5)*x)*sqrt
(17*sqrt(10) + 53) - 90*x + 90*sqrt(-2*x^2 + 3*x + 1) - 90)/x) + 54600*x^2 - 20*(9628*x^3 - 13860*x^2 - 5925*x
- 546)*sqrt(-2*x^2 + 3*x + 1) + 65520*x + 10920)/(4*x^4 - 12*x^3 + 5*x^2 + 6*x + 1)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(-3*x^2+4*x+2)/(-2*x^2+3*x+1)^(5/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.02, size = 1560, normalized size = 8.08 \begin {gather*} \text {Expression too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

248/2601/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/
2))^(1/2)+248/2601/(-1/9+1/9*10^(1/2))/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+
1/9*10^(1/2))^(1/2)+7/153/(-1/9+1/9*10^(1/2))/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2
))-1/9+1/9*10^(1/2))^(3/2)+7/153/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*
10^(1/2))-1/9-1/9*10^(1/2))^(3/2)+7/51/(-1/9+1/9*10^(1/2))^2/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-
2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(1/2)+7/51/(-1/9-1/9*10^(1/2))^2/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1
/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(1/2)-32/765/(-1/9+1/9*10^(1/2))^2/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3
-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(1/2)*10^(1/2)*x-32/2295/(-1/9+1/9*10^(1/2))*10^(1/2)/(-
2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(3/2)*x+2/5*10^(1/2)/(-1/9-
1/9*10^(1/2))^2/(1+10^(1/2))^(1/2)*arctan(9/2*(-2/9-2/9*10^(1/2)+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2)))/(1+1
0^(1/2))^(1/2)/(-18*(x-2/3+1/3*10^(1/2))^2+9*(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1-10^(1/2))^(1/2))+32/229
5/(-1/9-1/9*10^(1/2))*10^(1/2)/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(
1/2))^(3/2)*x+32/765/(-1/9-1/9*10^(1/2))^2/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-
1/9-1/9*10^(1/2))^(1/2)*10^(1/2)*x+2/5*10^(1/2)/(-1/9+1/9*10^(1/2))^2/(-1+10^(1/2))^(1/2)*arctanh(9/2*(-2/9+2/
9*10^(1/2)+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2)))/(-1+10^(1/2))^(1/2)/(-18*(x-2/3-1/3*10^(1/2))^2+9*(1/3-4/3
*10^(1/2))*(x-2/3-1/3*10^(1/2))-1+10^(1/2))^(1/2))+512/39015/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+(1
/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(1/2)*10^(1/2)*x-512/39015/(-1/9+1/9*10^(1/2))/(-2*(x-
2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(1/2)*10^(1/2)*x-62/153/(-1/9-1/
9*10^(1/2))^2/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(1/2)*x+26/
255/(-1/9-1/9*10^(1/2))^2/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))
^(1/2)*10^(1/2)-1/2/(-1/9-1/9*10^(1/2))^2/(1+10^(1/2))^(1/2)*arctan(9/2*(-2/9-2/9*10^(1/2)+(1/3+4/3*10^(1/2))*
(x-2/3+1/3*10^(1/2)))/(1+10^(1/2))^(1/2)/(-18*(x-2/3+1/3*10^(1/2))^2+9*(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))
-1-10^(1/2))^(1/2))-62/459/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/
2))-1/9-1/9*10^(1/2))^(3/2)*x+26/765/(-1/9-1/9*10^(1/2))*10^(1/2)/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2)
)*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(3/2)-26/255/(-1/9+1/9*10^(1/2))^2/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/
3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(1/2)*10^(1/2)+1/2/(-1/9+1/9*10^(1/2))^2/(-1+10^(1/2))^(1/2
)*arctanh(9/2*(-2/9+2/9*10^(1/2)+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2)))/(-1+10^(1/2))^(1/2)/(-18*(x-2/3-1/3*
10^(1/2))^2+9*(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1+10^(1/2))^(1/2))-26/765/(-1/9+1/9*10^(1/2))*10^(1/2)/(
-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(3/2)-62/153/(-1/9+1/9*10^
(1/2))^2/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(1/2)*x-62/459/(
-1/9+1/9*10^(1/2))/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(3/2)*
x+128/13005*10^(1/2)/(-1/9+1/9*10^(1/2))/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/
9+1/9*10^(1/2))^(1/2)-992/7803/(-1/9+1/9*10^(1/2))/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10
^(1/2))-1/9+1/9*10^(1/2))^(1/2)*x-128/13005*10^(1/2)/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*1
0^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(1/2)-992/7803/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+
(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(1/2)*x

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maxima [B]  time = 1.26, size = 1276, normalized size = 6.61

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(-3*x^2+4*x+2)/(-2*x^2+3*x+1)^(5/2),x, algorithm="maxima")

[Out]

1/17340*sqrt(10)*(2108*sqrt(10)*x/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) + (-2*x^2 + 3*x + 1)^(3/2)) - 2108*sqrt(1
0)*x/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) - (-2*x^2 + 3*x + 1)^(3/2)) - 56916*sqrt(10)*x/(2*sqrt(10)*sqrt(-2*x^2
+ 3*x + 1) + 11*sqrt(-2*x^2 + 3*x + 1)) + 56916*sqrt(10)*x/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - 11*sqrt(-2*x^
2 + 3*x + 1)) + 1984*sqrt(10)*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 + 3*x + 1)) - 1984*sqrt(10)*x/(
sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1)) - 70227*sqrt(10)*arcsin(8/17*sqrt(17)*sqrt(10)*x/abs
(6*x + 2*sqrt(10) - 4) + 2/17*sqrt(17)*x/abs(6*x + 2*sqrt(10) - 4) - 6/17*sqrt(17)*sqrt(10)/abs(6*x + 2*sqrt(1
0) - 4) + 24/17*sqrt(17)/abs(6*x + 2*sqrt(10) - 4))/(2*sqrt(10)*sqrt(sqrt(10) + 1) + 11*sqrt(sqrt(10) + 1)) -
2176*x/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) + (-2*x^2 + 3*x + 1)^(3/2)) - 2176*x/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3
/2) - (-2*x^2 + 3*x + 1)^(3/2)) + 58752*x/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + 11*sqrt(-2*x^2 + 3*x + 1)) + 58
752*x/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - 11*sqrt(-2*x^2 + 3*x + 1)) - 2048*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1
) + sqrt(-2*x^2 + 3*x + 1)) - 2048*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1)) + 561816*arcsi
n(8/17*sqrt(17)*sqrt(10)*x/abs(6*x + 2*sqrt(10) - 4) + 2/17*sqrt(17)*x/abs(6*x + 2*sqrt(10) - 4) - 6/17*sqrt(1
7)*sqrt(10)/abs(6*x + 2*sqrt(10) - 4) + 24/17*sqrt(17)/abs(6*x + 2*sqrt(10) - 4))/(2*sqrt(10)*sqrt(sqrt(10) +
1) + 11*sqrt(sqrt(10) + 1)) - 714*sqrt(10)/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) + (-2*x^2 + 3*x + 1)^(3/2)) + 71
4*sqrt(10)/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) - (-2*x^2 + 3*x + 1)^(3/2)) + 19278*sqrt(10)/(2*sqrt(10)*sqrt(-2
*x^2 + 3*x + 1) + 11*sqrt(-2*x^2 + 3*x + 1)) - 19278*sqrt(10)/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - 11*sqrt(-2*
x^2 + 3*x + 1)) - 1488*sqrt(10)/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 + 3*x + 1)) + 1488*sqrt(10)/(sq
rt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1)) - 5304/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) + (-2*x^2 +
3*x + 1)^(3/2)) - 5304/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) - (-2*x^2 + 3*x + 1)^(3/2)) + 143208/(2*sqrt(10)*sqr
t(-2*x^2 + 3*x + 1) + 11*sqrt(-2*x^2 + 3*x + 1)) + 143208/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - 11*sqrt(-2*x^2
+ 3*x + 1)) + 1536/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 + 3*x + 1)) + 1536/(sqrt(10)*sqrt(-2*x^2 + 3
*x + 1) - sqrt(-2*x^2 + 3*x + 1)) + 70227*sqrt(10)*log(-2/9*sqrt(10) + 2/3*sqrt(-2*x^2 + 3*x + 1)*sqrt(sqrt(10
) - 1)/abs(6*x - 2*sqrt(10) - 4) + 2/9*sqrt(10)/abs(6*x - 2*sqrt(10) - 4) - 2/9/abs(6*x - 2*sqrt(10) - 4) + 1/
18)/(sqrt(10) - 1)^(5/2) + 561816*log(-2/9*sqrt(10) + 2/3*sqrt(-2*x^2 + 3*x + 1)*sqrt(sqrt(10) - 1)/abs(6*x -
2*sqrt(10) - 4) + 2/9*sqrt(10)/abs(6*x - 2*sqrt(10) - 4) - 2/9/abs(6*x - 2*sqrt(10) - 4) + 1/18)/(sqrt(10) - 1
)^(5/2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x+2}{{\left (-2\,x^2+3\,x+1\right )}^{5/2}\,\left (-3\,x^2+4\,x+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{12 x^{6} \sqrt {- 2 x^{2} + 3 x + 1} - 52 x^{5} \sqrt {- 2 x^{2} + 3 x + 1} + 55 x^{4} \sqrt {- 2 x^{2} + 3 x + 1} + 22 x^{3} \sqrt {- 2 x^{2} + 3 x + 1} - 31 x^{2} \sqrt {- 2 x^{2} + 3 x + 1} - 16 x \sqrt {- 2 x^{2} + 3 x + 1} - 2 \sqrt {- 2 x^{2} + 3 x + 1}}\, dx - \int \frac {2}{12 x^{6} \sqrt {- 2 x^{2} + 3 x + 1} - 52 x^{5} \sqrt {- 2 x^{2} + 3 x + 1} + 55 x^{4} \sqrt {- 2 x^{2} + 3 x + 1} + 22 x^{3} \sqrt {- 2 x^{2} + 3 x + 1} - 31 x^{2} \sqrt {- 2 x^{2} + 3 x + 1} - 16 x \sqrt {- 2 x^{2} + 3 x + 1} - 2 \sqrt {- 2 x^{2} + 3 x + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x/(12*x**6*sqrt(-2*x**2 + 3*x + 1) - 52*x**5*sqrt(-2*x**2 + 3*x + 1) + 55*x**4*sqrt(-2*x**2 + 3*x +
1) + 22*x**3*sqrt(-2*x**2 + 3*x + 1) - 31*x**2*sqrt(-2*x**2 + 3*x + 1) - 16*x*sqrt(-2*x**2 + 3*x + 1) - 2*sqrt
(-2*x**2 + 3*x + 1)), x) - Integral(2/(12*x**6*sqrt(-2*x**2 + 3*x + 1) - 52*x**5*sqrt(-2*x**2 + 3*x + 1) + 55*
x**4*sqrt(-2*x**2 + 3*x + 1) + 22*x**3*sqrt(-2*x**2 + 3*x + 1) - 31*x**2*sqrt(-2*x**2 + 3*x + 1) - 16*x*sqrt(-
2*x**2 + 3*x + 1) - 2*sqrt(-2*x**2 + 3*x + 1)), x)

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