3.390 \(\int \frac{2+5 x+x^2}{(1+4 x-7 x^2)^2 \sqrt{3+2 x+5 x^2}} \, dx\)

Optimal. Leaf size=178 \[ -\frac{3 \sqrt{5 x^2+2 x+3} (40-371 x)}{5588 \left (-7 x^2+4 x+1\right )}-\frac{\sqrt{\frac{3027900955+14035681 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{11176}+\frac{\sqrt{\frac{3027900955-14035681 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{11176} \]

[Out]

(-3*(40 - 371*x)*Sqrt[3 + 2*x + 5*x^2])/(5588*(1 + 4*x - 7*x^2)) - (Sqrt[(3027900955 + 14035681*Sqrt[11])/2794
]*ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/11176 +
(Sqrt[(3027900955 - 14035681*Sqrt[11])/2794]*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*S
qrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/11176

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Rubi [A]  time = 0.195938, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {1060, 1032, 724, 206} \[ -\frac{3 \sqrt{5 x^2+2 x+3} (40-371 x)}{5588 \left (-7 x^2+4 x+1\right )}-\frac{\sqrt{\frac{3027900955+14035681 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{11176}+\frac{\sqrt{\frac{3027900955-14035681 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{11176} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)^2*Sqrt[3 + 2*x + 5*x^2]),x]

[Out]

(-3*(40 - 371*x)*Sqrt[3 + 2*x + 5*x^2])/(5588*(1 + 4*x - 7*x^2)) - (Sqrt[(3027900955 + 14035681*Sqrt[11])/2794
]*ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/11176 +
(Sqrt[(3027900955 - 14035681*Sqrt[11])/2794]*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*S
qrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/11176

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]

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 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{2+5 x+x^2}{\left (1+4 x-7 x^2\right )^2 \sqrt{3+2 x+5 x^2}} \, dx &=-\frac{3 (40-371 x) \sqrt{3+2 x+5 x^2}}{5588 \left (1+4 x-7 x^2\right )}-\frac{\int \frac{-52136-29544 x}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx}{44704}\\ &=-\frac{3 (40-371 x) \sqrt{3+2 x+5 x^2}}{5588 \left (1+4 x-7 x^2\right )}-\frac{\left (-40623+53005 \sqrt{11}\right ) \int \frac{1}{\left (4-2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{61468}+\frac{\left (40623+53005 \sqrt{11}\right ) \int \frac{1}{\left (4+2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{61468}\\ &=-\frac{3 (40-371 x) \sqrt{3+2 x+5 x^2}}{5588 \left (1+4 x-7 x^2\right )}-\frac{\left (40623-53005 \sqrt{11}\right ) \operatorname{Subst}\left (\int \frac{1}{2352+112 \left (4-2 \sqrt{11}\right )+20 \left (4-2 \sqrt{11}\right )^2-x^2} \, dx,x,\frac{-84-2 \left (4-2 \sqrt{11}\right )-\left (28+10 \left (4-2 \sqrt{11}\right )\right ) x}{\sqrt{3+2 x+5 x^2}}\right )}{30734}-\frac{\left (40623+53005 \sqrt{11}\right ) \operatorname{Subst}\left (\int \frac{1}{2352+112 \left (4+2 \sqrt{11}\right )+20 \left (4+2 \sqrt{11}\right )^2-x^2} \, dx,x,\frac{-84-2 \left (4+2 \sqrt{11}\right )-\left (28+10 \left (4+2 \sqrt{11}\right )\right ) x}{\sqrt{3+2 x+5 x^2}}\right )}{30734}\\ &=-\frac{3 (40-371 x) \sqrt{3+2 x+5 x^2}}{5588 \left (1+4 x-7 x^2\right )}-\frac{\sqrt{\frac{3027900955+14035681 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{23-\sqrt{11}+\left (17-5 \sqrt{11}\right ) x}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{3+2 x+5 x^2}}\right )}{11176}+\frac{\sqrt{\frac{3027900955-14035681 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{23+\sqrt{11}+\left (17+5 \sqrt{11}\right ) x}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{3+2 x+5 x^2}}\right )}{11176}\\ \end{align*}

Mathematica [A]  time = 1.02412, size = 313, normalized size = 1.76 \[ \frac{\frac{48972 \sqrt{5 x^2+2 x+3} x}{-7 x^2+4 x+1}+\frac{5280 \sqrt{5 x^2+2 x+3}}{7 x^2-4 x-1}+53005 \sqrt{\frac{22}{125+17 \sqrt{11}}} \log \left (\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left (55+17 \sqrt{11}\right ) x+23 \sqrt{11}+11\right )+40623 \sqrt{\frac{2}{125+17 \sqrt{11}}} \log \left (\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left (55+17 \sqrt{11}\right ) x+23 \sqrt{11}+11\right )+\sqrt{\frac{2}{125-17 \sqrt{11}}} \left (53005 \sqrt{11}-40623\right ) \tanh ^{-1}\left (\frac{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}{\left (5 \sqrt{11}-17\right ) x+\sqrt{11}-23}\right )-\sqrt{\frac{2}{125+17 \sqrt{11}}} \left (40623+53005 \sqrt{11}\right ) \log \left (-7 x+\sqrt{11}+2\right )}{245872} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)^2*Sqrt[3 + 2*x + 5*x^2]),x]

[Out]

((48972*x*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2) + (5280*Sqrt[3 + 2*x + 5*x^2])/(-1 - 4*x + 7*x^2) + Sqrt[2/
(125 - 17*Sqrt[11])]*(-40623 + 53005*Sqrt[11])*ArcTanh[(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])/(-23 +
Sqrt[11] + (-17 + 5*Sqrt[11])*x)] - Sqrt[2/(125 + 17*Sqrt[11])]*(40623 + 53005*Sqrt[11])*Log[2 + Sqrt[11] - 7*
x] + 40623*Sqrt[2/(125 + 17*Sqrt[11])]*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]
*Sqrt[3 + 2*x + 5*x^2]] + 53005*Sqrt[22/(125 + 17*Sqrt[11])]*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqr
t[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]])/245872

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Maple [B]  time = 0.117, size = 510, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+5*x+2)/(-7*x^2+4*x+1)^2/(5*x^2+2*x+3)^(1/2),x)

[Out]

161/484*11^(1/2)/(250+34*11^(1/2))^(1/2)*arctanh(49/2*(500/49+68/49*11^(1/2)+(34/7+10/7*11^(1/2))*(x-2/7-1/7*1
1^(1/2)))/(250+34*11^(1/2))^(1/2)/(245*(x-2/7-1/7*11^(1/2))^2+49*(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/2))+250
+34*11^(1/2))^(1/2))+(183/44+39/44*11^(1/2))*(-1/49/(250/49+34/49*11^(1/2))/(x-2/7-1/7*11^(1/2))*(5*(x-2/7-1/7
*11^(1/2))^2+(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/2))+250/49+34/49*11^(1/2))^(1/2)+1/14*(34/7+10/7*11^(1/2))/
(250/49+34/49*11^(1/2))/(250+34*11^(1/2))^(1/2)*arctanh(49/2*(500/49+68/49*11^(1/2)+(34/7+10/7*11^(1/2))*(x-2/
7-1/7*11^(1/2)))/(250+34*11^(1/2))^(1/2)/(245*(x-2/7-1/7*11^(1/2))^2+49*(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/
2))+250+34*11^(1/2))^(1/2)))-161/484*11^(1/2)/(250-34*11^(1/2))^(1/2)*arctanh(49/2*(500/49-68/49*11^(1/2)+(34/
7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2)))/(250-34*11^(1/2))^(1/2)/(245*(x-2/7+1/7*11^(1/2))^2+49*(34/7-10/7*11^(1
/2))*(x-2/7+1/7*11^(1/2))+250-34*11^(1/2))^(1/2))+(183/44-39/44*11^(1/2))*(-1/49/(250/49-34/49*11^(1/2))/(x-2/
7+1/7*11^(1/2))*(5*(x-2/7+1/7*11^(1/2))^2+(34/7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2))+250/49-34/49*11^(1/2))^(1/
2)+1/14*(34/7-10/7*11^(1/2))/(250/49-34/49*11^(1/2))/(250-34*11^(1/2))^(1/2)*arctanh(49/2*(500/49-68/49*11^(1/
2)+(34/7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2)))/(250-34*11^(1/2))^(1/2)/(245*(x-2/7+1/7*11^(1/2))^2+49*(34/7-10/
7*11^(1/2))*(x-2/7+1/7*11^(1/2))+250-34*11^(1/2))^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} + 5 \, x + 2}{{\left (7 \, x^{2} - 4 \, x - 1\right )}^{2} \sqrt{5 \, x^{2} + 2 \, x + 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+5*x+2)/(-7*x^2+4*x+1)^2/(5*x^2+2*x+3)^(1/2),x, algorithm="maxima")

[Out]

integrate((x^2 + 5*x + 2)/((7*x^2 - 4*x - 1)^2*sqrt(5*x^2 + 2*x + 3)), x)

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Fricas [B]  time = 1.36634, size = 1393, normalized size = 7.83 \begin{align*} -\frac{\sqrt{2794}{\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt{14035681 \, \sqrt{11} + 3027900955} \log \left (-\frac{\sqrt{2794} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{14035681 \, \sqrt{11} + 3027900955}{\left (71796 \, \sqrt{11} + 567523\right )} + 265381033753 \, \sqrt{11}{\left (x + 3\right )} - 796143101259 \, x + 1326905168765}{x}\right ) - \sqrt{2794}{\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt{14035681 \, \sqrt{11} + 3027900955} \log \left (\frac{\sqrt{2794} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{14035681 \, \sqrt{11} + 3027900955}{\left (71796 \, \sqrt{11} + 567523\right )} - 265381033753 \, \sqrt{11}{\left (x + 3\right )} + 796143101259 \, x - 1326905168765}{x}\right ) + \sqrt{2794}{\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt{-14035681 \, \sqrt{11} + 3027900955} \log \left (\frac{\sqrt{2794} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (71796 \, \sqrt{11} - 567523\right )} \sqrt{-14035681 \, \sqrt{11} + 3027900955} + 265381033753 \, \sqrt{11}{\left (x + 3\right )} + 796143101259 \, x - 1326905168765}{x}\right ) - \sqrt{2794}{\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt{-14035681 \, \sqrt{11} + 3027900955} \log \left (-\frac{\sqrt{2794} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (71796 \, \sqrt{11} - 567523\right )} \sqrt{-14035681 \, \sqrt{11} + 3027900955} - 265381033753 \, \sqrt{11}{\left (x + 3\right )} - 796143101259 \, x + 1326905168765}{x}\right ) + 33528 \, \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (371 \, x - 40\right )}}{62451488 \,{\left (7 \, x^{2} - 4 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+5*x+2)/(-7*x^2+4*x+1)^2/(5*x^2+2*x+3)^(1/2),x, algorithm="fricas")

[Out]

-1/62451488*(sqrt(2794)*(7*x^2 - 4*x - 1)*sqrt(14035681*sqrt(11) + 3027900955)*log(-(sqrt(2794)*sqrt(5*x^2 + 2
*x + 3)*sqrt(14035681*sqrt(11) + 3027900955)*(71796*sqrt(11) + 567523) + 265381033753*sqrt(11)*(x + 3) - 79614
3101259*x + 1326905168765)/x) - sqrt(2794)*(7*x^2 - 4*x - 1)*sqrt(14035681*sqrt(11) + 3027900955)*log((sqrt(27
94)*sqrt(5*x^2 + 2*x + 3)*sqrt(14035681*sqrt(11) + 3027900955)*(71796*sqrt(11) + 567523) - 265381033753*sqrt(1
1)*(x + 3) + 796143101259*x - 1326905168765)/x) + sqrt(2794)*(7*x^2 - 4*x - 1)*sqrt(-14035681*sqrt(11) + 30279
00955)*log((sqrt(2794)*sqrt(5*x^2 + 2*x + 3)*(71796*sqrt(11) - 567523)*sqrt(-14035681*sqrt(11) + 3027900955) +
265381033753*sqrt(11)*(x + 3) + 796143101259*x - 1326905168765)/x) - sqrt(2794)*(7*x^2 - 4*x - 1)*sqrt(-14035
681*sqrt(11) + 3027900955)*log(-(sqrt(2794)*sqrt(5*x^2 + 2*x + 3)*(71796*sqrt(11) - 567523)*sqrt(-14035681*sqr
t(11) + 3027900955) - 265381033753*sqrt(11)*(x + 3) - 796143101259*x + 1326905168765)/x) + 33528*sqrt(5*x^2 +
2*x + 3)*(371*x - 40))/(7*x^2 - 4*x - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} + 5 x + 2}{\sqrt{5 x^{2} + 2 x + 3} \left (7 x^{2} - 4 x - 1\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+5*x+2)/(-7*x**2+4*x+1)**2/(5*x**2+2*x+3)**(1/2),x)

[Out]

Integral((x**2 + 5*x + 2)/(sqrt(5*x**2 + 2*x + 3)*(7*x**2 - 4*x - 1)**2), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+5*x+2)/(-7*x^2+4*x+1)^2/(5*x^2+2*x+3)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError