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]

-3/5588*(40-371*x)*(5*x^2+2*x+3)^(1/2)/(-7*x^2+4*x+1)+1/31225744*arctanh((23+11^(1/2)+x*(17+5*11^(1/2)))/(5*x^
2+2*x+3)^(1/2)/(250+34*11^(1/2))^(1/2))*(8459955268270-39215692714*11^(1/2))^(1/2)-1/31225744*arctanh((23+x*(1
7-5*11^(1/2))-11^(1/2))/(5*x^2+2*x+3)^(1/2)/(250-34*11^(1/2))^(1/2))*(8459955268270+39215692714*11^(1/2))^(1/2
)

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Rubi [A]  time = 0.20, antiderivative size = 178, normalized size of antiderivative = 1.00, 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 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 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+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*}

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Mathematica [A]  time = 1.00, 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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fricas [B]  time = 0.87, size = 330, normalized size = 1.85 \[ -\frac {\sqrt {2794} {\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt {14035681 \, \sqrt {11} + 3027900955} \log \left (-\frac {\sqrt {2794} \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {14035681 \, \sqrt {11} + 3027900955} {\left (71796 \, \sqrt {11} + 567523\right )} + 265381033753 \, \sqrt {11} {\left (x + 3\right )} - 796143101259 \, x + 1326905168765}{x}\right ) - \sqrt {2794} {\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt {14035681 \, \sqrt {11} + 3027900955} \log \left (\frac {\sqrt {2794} \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {14035681 \, \sqrt {11} + 3027900955} {\left (71796 \, \sqrt {11} + 567523\right )} - 265381033753 \, \sqrt {11} {\left (x + 3\right )} + 796143101259 \, x - 1326905168765}{x}\right ) + \sqrt {2794} {\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt {-14035681 \, \sqrt {11} + 3027900955} \log \left (\frac {\sqrt {2794} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (71796 \, \sqrt {11} - 567523\right )} \sqrt {-14035681 \, \sqrt {11} + 3027900955} + 265381033753 \, \sqrt {11} {\left (x + 3\right )} + 796143101259 \, x - 1326905168765}{x}\right ) - \sqrt {2794} {\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt {-14035681 \, \sqrt {11} + 3027900955} \log \left (-\frac {\sqrt {2794} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (71796 \, \sqrt {11} - 567523\right )} \sqrt {-14035681 \, \sqrt {11} + 3027900955} - 265381033753 \, \sqrt {11} {\left (x + 3\right )} - 796143101259 \, x + 1326905168765}{x}\right ) + 33528 \, \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (371 \, x - 40\right )}}{62451488 \, {\left (7 \, x^{2} - 4 \, x - 1\right )}} \]

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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giac [B]  time = 0.28, size = 276, normalized size = 1.55 \[ \frac {3 \, {\left (1231 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{3} + 1735 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{2} - 3913 \, \sqrt {5} x - 3989 \, \sqrt {5} + 3913 \, \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}}{2794 \, {\left (7 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{4} - 8 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{3} - 70 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{2} + 16 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} + 83\right )}} + 0.0924287071106453 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} + 4.41924736459000\right ) - 0.0938608034604765 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} + 1.25295163054000\right ) - 0.0924287071106453 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} - 1.02258038113000\right ) + 0.0938608034604765 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} - 2.09411235400000\right ) \]

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]

3/2794*(1231*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^3 + 1735*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^2 - 3913
*sqrt(5)*x - 3989*sqrt(5) + 3913*sqrt(5*x^2 + 2*x + 3))/(7*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^4 - 8*sqrt(5)*(
sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^3 - 70*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^2 + 16*sqrt(5)*(sqrt(5)*x - sqrt
(5*x^2 + 2*x + 3)) + 83) + 0.0924287071106453*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) + 4.41924736459000) - 0.0
938608034604765*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) + 1.25295163054000) - 0.0924287071106453*log(-sqrt(5)*x
+ sqrt(5*x^2 + 2*x + 3) - 1.02258038113000) + 0.0938608034604765*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) - 2.0
9411235400000)

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maple [B]  time = 0.02, size = 510, normalized size = 2.87 \[ -\frac {161 \sqrt {11}\, \arctanh \left (\frac {250-34 \sqrt {11}+\frac {49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250-34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )+250-34 \sqrt {11}}}\right )}{484 \sqrt {250-34 \sqrt {11}}}+\frac {161 \sqrt {11}\, \arctanh \left (\frac {250+34 \sqrt {11}+\frac {49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250+34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )+250+34 \sqrt {11}}}\right )}{484 \sqrt {250+34 \sqrt {11}}}+\left (\frac {183}{44}-\frac {39 \sqrt {11}}{44}\right ) \left (\frac {\left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \arctanh \left (\frac {250-34 \sqrt {11}+\frac {49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250-34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )+250-34 \sqrt {11}}}\right )}{14 \left (\frac {250}{49}-\frac {34 \sqrt {11}}{49}\right ) \sqrt {250-34 \sqrt {11}}}-\frac {\sqrt {5 \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )^{2}+\left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )+\frac {250}{49}-\frac {34 \sqrt {11}}{49}}}{49 \left (\frac {250}{49}-\frac {34 \sqrt {11}}{49}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )}\right )+\left (\frac {183}{44}+\frac {39 \sqrt {11}}{44}\right ) \left (\frac {\left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \arctanh \left (\frac {250+34 \sqrt {11}+\frac {49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250+34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )+250+34 \sqrt {11}}}\right )}{14 \left (\frac {250}{49}+\frac {34 \sqrt {11}}{49}\right ) \sqrt {250+34 \sqrt {11}}}-\frac {\sqrt {5 \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )^{2}+\left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )+\frac {250}{49}+\frac {34 \sqrt {11}}{49}}}{49 \left (\frac {250}{49}+\frac {34 \sqrt {11}}{49}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )}\right ) \]

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*
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))+25
0-34*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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} + 5 \, x + 2}{{\left (7 \, x^{2} - 4 \, x - 1\right )}^{2} \sqrt {5 \, x^{2} + 2 \, x + 3}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2+5\,x+2}{\sqrt {5\,x^2+2\,x+3}\,{\left (-7\,x^2+4\,x+1\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} + 5 x + 2}{\sqrt {5 x^{2} + 2 x + 3} \left (7 x^{2} - 4 x - 1\right )^{2}}\, dx \]

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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