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

Optimal. Leaf size=234 \[ \frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{308 \left (-7 x^2+4 x+1\right )^2}-\frac {(9495-37088 x) \sqrt {5 x^2+2 x+3}}{23716 \left (-7 x^2+4 x+1\right )}-\frac {\sqrt {\frac {62294197250171-2085440742055 \sqrt {11}}{2794}} \tanh ^{-1}\left (\frac {\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{332024}+\frac {\sqrt {\frac {62294197250171+2085440742055 \sqrt {11}}{2794}} \tanh ^{-1}\left (\frac {\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{332024}-\frac {5}{343} \sqrt {5} \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right ) \]

[Out]

3/308*(3+61*x)*(5*x^2+2*x+3)^(3/2)/(-7*x^2+4*x+1)^2-5/343*arcsinh(1/14*(1+5*x)*14^(1/2))*5^(1/2)-1/23716*(9495
-37088*x)*(5*x^2+2*x+3)^(1/2)/(-7*x^2+4*x+1)-1/927675056*arctanh((23+x*(17-5*11^(1/2))-11^(1/2))/(5*x^2+2*x+3)
^(1/2)/(250-34*11^(1/2))^(1/2))*(174049987116977774-5826721433301670*11^(1/2))^(1/2)+1/927675056*arctanh((23+1
1^(1/2)+x*(17+5*11^(1/2)))/(5*x^2+2*x+3)^(1/2)/(250+34*11^(1/2))^(1/2))*(174049987116977774+5826721433301670*1
1^(1/2))^(1/2)

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Rubi [A]  time = 0.32, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1054, 1076, 619, 215, 1032, 724, 206} \[ \frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{308 \left (-7 x^2+4 x+1\right )^2}-\frac {(9495-37088 x) \sqrt {5 x^2+2 x+3}}{23716 \left (-7 x^2+4 x+1\right )}-\frac {\sqrt {\frac {62294197250171-2085440742055 \sqrt {11}}{2794}} \tanh ^{-1}\left (\frac {\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{332024}+\frac {\sqrt {\frac {62294197250171+2085440742055 \sqrt {11}}{2794}} \tanh ^{-1}\left (\frac {\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{332024}-\frac {5}{343} \sqrt {5} \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((9495 - 37088*x)*Sqrt[3 + 2*x + 5*x^2])/(23716*(1 + 4*x - 7*x^2)) + (3*(3 + 61*x)*(3 + 2*x + 5*x^2)^(3/2))/(
308*(1 + 4*x - 7*x^2)^2) - (5*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/343 - (Sqrt[(62294197250171 - 2085440742055
*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])])/332024 + (Sqrt[(62294197250171 + 2085440742055*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])])/332024

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 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 1054

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*c - 2*a*B*c + a*b*C - (c*(b*B - 2*A*c) - C*(b^2 - 2*a*c))*x)*(a + b*x + c*x
^2)^(p + 1)*(d + e*x + f*x^2)^q)/(c*(b^2 - 4*a*c)*(p + 1)), x] - Dist[1/(c*(b^2 - 4*a*c)*(p + 1)), Int[(a + b*
x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[e*q*(A*b*c - 2*a*B*c + a*b*C) - d*(c*(b*B - 2*A*c)*(2*p + 3)
+ C*(2*a*c - b^2*(p + 2))) + (2*f*q*(A*b*c - 2*a*B*c + a*b*C) - e*(c*(b*B - 2*A*c)*(2*p + q + 3) + C*(2*a*c*(
q + 1) - b^2*(p + q + 2))))*x - f*(c*(b*B - 2*A*c)*(2*p + 2*q + 3) + C*(2*a*c*(2*q + 1) - b^2*(p + 2*q + 2)))*
x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[
p, -1] && GtQ[q, 0] &&  !IGtQ[q, 0]

Rule 1076

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x
_)^2]), x_Symbol] :> Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + (B*c - b*C)*x)
/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c
, 0] && NeQ[e^2 - 4*d*f, 0]

Rubi steps

\begin {align*} \int \frac {\left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}}{\left (1+4 x-7 x^2\right )^3} \, dx &=\frac {3 (3+61 x) \left (3+2 x+5 x^2\right )^{3/2}}{308 \left (1+4 x-7 x^2\right )^2}-\frac {1}{616} \int \frac {\sqrt {3+2 x+5 x^2} \left (-2976-652 x+440 x^2\right )}{\left (1+4 x-7 x^2\right )^2} \, dx\\ &=-\frac {(9495-37088 x) \sqrt {3+2 x+5 x^2}}{23716 \left (1+4 x-7 x^2\right )}+\frac {3 (3+61 x) \left (3+2 x+5 x^2\right )^{3/2}}{308 \left (1+4 x-7 x^2\right )^2}+\frac {\int \frac {1024152+715224 x+96800 x^2}{\left (1+4 x-7 x^2\right ) \sqrt {3+2 x+5 x^2}} \, dx}{189728}\\ &=-\frac {(9495-37088 x) \sqrt {3+2 x+5 x^2}}{23716 \left (1+4 x-7 x^2\right )}+\frac {3 (3+61 x) \left (3+2 x+5 x^2\right )^{3/2}}{308 \left (1+4 x-7 x^2\right )^2}-\frac {\int \frac {-7265864-5393768 x}{\left (1+4 x-7 x^2\right ) \sqrt {3+2 x+5 x^2}} \, dx}{1328096}-\frac {25}{343} \int \frac {1}{\sqrt {3+2 x+5 x^2}} \, dx\\ &=-\frac {(9495-37088 x) \sqrt {3+2 x+5 x^2}}{23716 \left (1+4 x-7 x^2\right )}+\frac {3 (3+61 x) \left (3+2 x+5 x^2\right )^{3/2}}{308 \left (1+4 x-7 x^2\right )^2}-\frac {1}{686} \left (5 \sqrt {\frac {5}{14}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{56}}} \, dx,x,2+10 x\right )-\frac {\left (-7416431+7706073 \sqrt {11}\right ) \int \frac {1}{\left (4-2 \sqrt {11}-14 x\right ) \sqrt {3+2 x+5 x^2}} \, dx}{1826132}+\frac {\left (7416431+7706073 \sqrt {11}\right ) \int \frac {1}{\left (4+2 \sqrt {11}-14 x\right ) \sqrt {3+2 x+5 x^2}} \, dx}{1826132}\\ &=-\frac {(9495-37088 x) \sqrt {3+2 x+5 x^2}}{23716 \left (1+4 x-7 x^2\right )}+\frac {3 (3+61 x) \left (3+2 x+5 x^2\right )^{3/2}}{308 \left (1+4 x-7 x^2\right )^2}-\frac {5}{343} \sqrt {5} \sinh ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )-\frac {\left (7416431-7706073 \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 )}{913066}-\frac {\left (7416431+7706073 \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 )}{913066}\\ &=-\frac {(9495-37088 x) \sqrt {3+2 x+5 x^2}}{23716 \left (1+4 x-7 x^2\right )}+\frac {3 (3+61 x) \left (3+2 x+5 x^2\right )^{3/2}}{308 \left (1+4 x-7 x^2\right )^2}-\frac {5}{343} \sqrt {5} \sinh ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )-\frac {\left (7706073-674221 \sqrt {11}\right ) \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 )}{332024 \sqrt {22 \left (125-17 \sqrt {11}\right )}}+\frac {\left (7706073+674221 \sqrt {11}\right ) \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 )}{332024 \sqrt {22 \left (125+17 \sqrt {11}\right )}}\\ \end {align*}

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Mathematica [A]  time = 2.24, size = 376, normalized size = 1.61 \[ \frac {\frac {88 \sqrt {5 x^2+2 x+3} (138372-189161 x)}{7 x^2-4 x-1}+\frac {11616 (5028 x+655) \sqrt {5 x^2+2 x+3}}{\left (-7 x^2+4 x+1\right )^2}-\sqrt {\frac {22}{125-17 \sqrt {11}}} \left (674221 \sqrt {11}-7706073\right ) \log \left (49 x^2+14 \left (\sqrt {11}-2\right ) x-4 \sqrt {11}+15\right )+2 \sqrt {\frac {22}{125+17 \sqrt {11}}} \left (7706073+674221 \sqrt {11}\right ) \log \left (\sqrt {2750+374 \sqrt {11}} \sqrt {5 x^2+2 x+3}+\left (55+17 \sqrt {11}\right ) x+23 \sqrt {11}+11\right )-2 \sqrt {\frac {22}{125-17 \sqrt {11}}} \left (674221 \sqrt {11}-7706073\right ) \tanh ^{-1}\left (\frac {\sqrt {250-34 \sqrt {11}} \sqrt {5 x^2+2 x+3}}{\left (5 \sqrt {11}-17\right ) x+\sqrt {11}-23}\right )-2 \sqrt {\frac {22}{125+17 \sqrt {11}}} \left (7706073+674221 \sqrt {11}\right ) \log \left (-7 x+\sqrt {11}+2\right )+\sqrt {\frac {22}{125-17 \sqrt {11}}} \left (674221 \sqrt {11}-7706073\right ) \log \left (\left (7 x+\sqrt {11}-2\right )^2\right )-212960 \sqrt {5} \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{14609056} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((11616*(655 + 5028*x)*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2)^2 + (88*(138372 - 189161*x)*Sqrt[3 + 2*x + 5*x
^2])/(-1 - 4*x + 7*x^2) - 212960*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]] - 2*Sqrt[22/(125 - 17*Sqrt[11])]*(-770607
3 + 674221*Sqrt[11])*ArcTanh[(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])/(-23 + Sqrt[11] + (-17 + 5*Sqrt[1
1])*x)] - 2*Sqrt[22/(125 + 17*Sqrt[11])]*(7706073 + 674221*Sqrt[11])*Log[2 + Sqrt[11] - 7*x] + Sqrt[22/(125 -
17*Sqrt[11])]*(-7706073 + 674221*Sqrt[11])*Log[(-2 + Sqrt[11] + 7*x)^2] - Sqrt[22/(125 - 17*Sqrt[11])]*(-77060
73 + 674221*Sqrt[11])*Log[15 - 4*Sqrt[11] + 14*(-2 + Sqrt[11])*x + 49*x^2] + 2*Sqrt[22/(125 + 17*Sqrt[11])]*(7
706073 + 674221*Sqrt[11])*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x
+ 5*x^2]])/14609056

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fricas [B]  time = 0.99, size = 447, normalized size = 1.91 \[ -\frac {\sqrt {2794} {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {2085440742055 \, \sqrt {11} + 62294197250171} \log \left (\frac {\sqrt {2794} \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {2085440742055 \, \sqrt {11} + 62294197250171} {\left (11840590 \, \sqrt {11} - 83479737\right )} + 5426671202560069 \, \sqrt {11} {\left (x + 3\right )} + 16280013607680207 \, x - 27133356012800345}{x}\right ) - \sqrt {2794} {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {2085440742055 \, \sqrt {11} + 62294197250171} \log \left (-\frac {\sqrt {2794} \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {2085440742055 \, \sqrt {11} + 62294197250171} {\left (11840590 \, \sqrt {11} - 83479737\right )} - 5426671202560069 \, \sqrt {11} {\left (x + 3\right )} - 16280013607680207 \, x + 27133356012800345}{x}\right ) + \sqrt {2794} {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {-2085440742055 \, \sqrt {11} + 62294197250171} \log \left (-\frac {\sqrt {2794} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (11840590 \, \sqrt {11} + 83479737\right )} \sqrt {-2085440742055 \, \sqrt {11} + 62294197250171} + 5426671202560069 \, \sqrt {11} {\left (x + 3\right )} - 16280013607680207 \, x + 27133356012800345}{x}\right ) - \sqrt {2794} {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {-2085440742055 \, \sqrt {11} + 62294197250171} \log \left (\frac {\sqrt {2794} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (11840590 \, \sqrt {11} + 83479737\right )} \sqrt {-2085440742055 \, \sqrt {11} + 62294197250171} - 5426671202560069 \, \sqrt {11} {\left (x + 3\right )} + 16280013607680207 \, x - 27133356012800345}{x}\right ) - 13522960 \, \sqrt {5} {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \log \left (\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) + 78232 \, {\left (189161 \, x^{3} - 246464 \, x^{2} - 42767 \, x + 7416\right )} \sqrt {5 \, x^{2} + 2 \, x + 3}}{1855350112 \, {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1855350112*(sqrt(2794)*(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(2085440742055*sqrt(11) + 62294197250171)*lo
g((sqrt(2794)*sqrt(5*x^2 + 2*x + 3)*sqrt(2085440742055*sqrt(11) + 62294197250171)*(11840590*sqrt(11) - 8347973
7) + 5426671202560069*sqrt(11)*(x + 3) + 16280013607680207*x - 27133356012800345)/x) - sqrt(2794)*(49*x^4 - 56
*x^3 + 2*x^2 + 8*x + 1)*sqrt(2085440742055*sqrt(11) + 62294197250171)*log(-(sqrt(2794)*sqrt(5*x^2 + 2*x + 3)*s
qrt(2085440742055*sqrt(11) + 62294197250171)*(11840590*sqrt(11) - 83479737) - 5426671202560069*sqrt(11)*(x + 3
) - 16280013607680207*x + 27133356012800345)/x) + sqrt(2794)*(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(-2085440
742055*sqrt(11) + 62294197250171)*log(-(sqrt(2794)*sqrt(5*x^2 + 2*x + 3)*(11840590*sqrt(11) + 83479737)*sqrt(-
2085440742055*sqrt(11) + 62294197250171) + 5426671202560069*sqrt(11)*(x + 3) - 16280013607680207*x + 271333560
12800345)/x) - sqrt(2794)*(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(-2085440742055*sqrt(11) + 62294197250171)*l
og((sqrt(2794)*sqrt(5*x^2 + 2*x + 3)*(11840590*sqrt(11) + 83479737)*sqrt(-2085440742055*sqrt(11) + 62294197250
171) - 5426671202560069*sqrt(11)*(x + 3) + 16280013607680207*x - 27133356012800345)/x) - 13522960*sqrt(5)*(49*
x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*log(sqrt(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8) + 78232*(1891
61*x^3 - 246464*x^2 - 42767*x + 7416)*sqrt(5*x^2 + 2*x + 3))/(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{-1771684761728,[12]%%%}+%%%{%%{[-6074347754496,0]:[1,0,-5]%%},[
11]%%%}+%%%{18439984254720,[10]%%%}+%%%{%%{[120412580657152,0]:[1,0,-5]%%},[9]%%%}+%%%{-108578966111616,[8]%%%
}+%%%{%%{[-915119084156928,0]:[1,0,-5]%%},[7]%%%}+%%%{1093279290575360,[6]%%%}+%%%{%%{[2784778529734656,0]:[1,
0,-5]%%},[5]%%%}+%%%{-4014694487954304,[4]%%%}+%%%{%%{[-3629195511796736,0]:[1,0,-5]%%},[3]%%%}+%%%{5826260235
237120,[2]%%%}+%%%{%%{[1708007415539712,0]:[1,0,-5]%%},[1]%%%}+%%%{-2953429489370752,[0]%%%} / %%%{%%{[1715,0]
:[1,0,-5]%%},[12]%%%}+%%%{29400,[11]%%%}+%%%{%%{[-17850,0]:[1,0,-5]%%},[10]%%%}+%%%{-582800,[9]%%%}+%%%{%%{[10
5105,0]:[1,0,-5]%%},[8]%%%}+%%%{4429200,[7]%%%}+%%%{%%{[-1058300,0]:[1,0,-5]%%},[6]%%%}+%%%{-13478400,[5]%%%}+
%%%{%%{[3886245,0]:[1,0,-5]%%},[4]%%%}+%%%{17565400,[3]%%%}+%%%{%%{[-5639850,0]:[1,0,-5]%%},[2]%%%}+%%%{-82668
00,[1]%%%}+%%%{%%{[2858935,0]:[1,0,-5]%%},[0]%%%} Error: Bad Argument Value

________________________________________________________________________________________

maple [B]  time = 0.02, size = 3828, normalized size = 16.36 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3535/21296*11^(1/2)*(1/21*(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))^(3/2)+1/14*(34/7-10/7*11^(1/2))*(1/20*(10*x+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)+1/200*(5000/49-680/49*11^(1/2)-(34/7-10/7*11^(1/2))^2)*5^(1/2)*arcsin
h(5^(1/2)/(250/49-34/49*11^(1/2)-1/20*(34/7-10/7*11^(1/2))^2)^(1/2)*(x+1/5)))+1/7*(250/49-34/49*11^(1/2))*(1/7
*(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)+1/10*(34/7-10
/7*11^(1/2))*5^(1/2)*arcsinh(5^(1/2)/(250/49-34/49*11^(1/2)-1/20*(34/7-10/7*11^(1/2))^2)^(1/2)*(x+1/5))-7*(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))))-21/968*(-61+13*11^(1/2))*11^(1/2)*(-1/686/(250/49-34/49*11^(1/2))/(x-2/7+1/7*11^(1/2
))^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))^(5/2)+1/1372*(
34/7-10/7*11^(1/2))/(250/49-34/49*11^(1/2))*(-1/(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))^(5/2)+3/2*(34/7-10/7*11^(1/2))/(250/
49-34/49*11^(1/2))*(1/3*(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))^(3/2)+1/2*(34/7-10/7*11^(1/2))*(1/20*(10*x+2)*(5*(x-2/7+1/7*11^(1/2))^2+(34/7-10/7*11^(1/2))*(x-2/7+1/7*1
1^(1/2))+250/49-34/49*11^(1/2))^(1/2)+1/200*(5000/49-680/49*11^(1/2)-(34/7-10/7*11^(1/2))^2)*5^(1/2)*arcsinh(5
^(1/2)/(250/49-34/49*11^(1/2)-1/20*(34/7-10/7*11^(1/2))^2)^(1/2)*(x+1/5)))+(250/49-34/49*11^(1/2))*(1/7*(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)+1/10*(34/7-10/7*11^(
1/2))*5^(1/2)*arcsinh(5^(1/2)/(250/49-34/49*11^(1/2)-1/20*(34/7-10/7*11^(1/2))^2)^(1/2)*(x+1/5))-7*(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))))+20/(250/49-34/49*11^(1/2))*(1/40*(10*x+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))^(3/2)+3/80*(5000/49-680/49*11^(1/2)-(34/7-10/7*11^(1/2))^2)*(1/20*(1
0*x+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)+1/200*
(5000/49-680/49*11^(1/2)-(34/7-10/7*11^(1/2))^2)*5^(1/2)*arcsinh(5^(1/2)/(250/49-34/49*11^(1/2)-1/20*(34/7-10/
7*11^(1/2))^2)^(1/2)*(x+1/5)))))+15/686/(250/49-34/49*11^(1/2))*(1/3*(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))^(3/2)+1/2*(34/7-10/7*11^(1/2))*(1/20*(10*x+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)+1/200*(5000/49-680/49*11^(1
/2)-(34/7-10/7*11^(1/2))^2)*5^(1/2)*arcsinh(5^(1/2)/(250/49-34/49*11^(1/2)-1/20*(34/7-10/7*11^(1/2))^2)^(1/2)*
(x+1/5)))+(250/49-34/49*11^(1/2))*(1/7*(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)+1/10*(34/7-10/7*11^(1/2))*5^(1/2)*arcsinh(5^(1/2)/(250/49-34/49*11^(1/2)-1/20*(34/7-1
0/7*11^(1/2))^2)^(1/2)*(x+1/5))-7*(250/49-34/49*11^(1/2))/(250-34*11^(1/2))^(1/2)*arctanh(49/2*(500/49-68/49*1
1^(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)))))-(-3535/1936-273/1936*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))^(5/2)+3/98*(34/7+10/7*11^(1/2))/(250/49+34/49*11^(1/2))*(1/3*(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))^(3/2)+1/2*(34/7+10/7*11^(1/2))*(1/20*(10*x+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)+1/200*(5000/49+6
80/49*11^(1/2)-(34/7+10/7*11^(1/2))^2)*5^(1/2)*arcsinh(5^(1/2)/(250/49+34/49*11^(1/2)-1/20*(34/7+10/7*11^(1/2)
)^2)^(1/2)*(x+1/5)))+(250/49+34/49*11^(1/2))*(1/7*(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)+1/10*(34/7+10/7*11^(1/2))*5^(1/2)*arcsinh(5^(1/2)/(250/49+34/49*11^(1/2)-1
/20*(34/7+10/7*11^(1/2))^2)^(1/2)*(x+1/5))-7*(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))))+20/49/(250/49+34/49*11^(1/2))*(1/40
*(10*x+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))^(3/2)+3/8
0*(5000/49+680/49*11^(1/2)-(34/7+10/7*11^(1/2))^2)*(1/20*(10*x+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)+1/200*(5000/49+680/49*11^(1/2)-(34/7+10/7*11^(1/2))^2)*5^
(1/2)*arcsinh(5^(1/2)/(250/49+34/49*11^(1/2)-1/20*(34/7+10/7*11^(1/2))^2)^(1/2)*(x+1/5)))))-(-3535/1936+273/19
36*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))^(5/2)+3/98*(34/7-10/7*11^(1/2))/(250/49-34/49*11^(1/2))*(1/3*(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))^(3/2)+1/2*(34/7-10/7*1
1^(1/2))*(1/20*(10*x+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)+1/200*(5000/49-680/49*11^(1/2)-(34/7-10/7*11^(1/2))^2)*5^(1/2)*arcsinh(5^(1/2)/(250/49-34/49*11^(1/
2)-1/20*(34/7-10/7*11^(1/2))^2)^(1/2)*(x+1/5)))+(250/49-34/49*11^(1/2))*(1/7*(245*(x-2/7+1/7*11^(1/2))^2+49*(3
4/7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2))+250-34*11^(1/2))^(1/2)+1/10*(34/7-10/7*11^(1/2))*5^(1/2)*arcsinh(5^(1/
2)/(250/49-34/49*11^(1/2)-1/20*(34/7-10/7*11^(1/2))^2)^(1/2)*(x+1/5))-7*(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))))+20/49/(2
50/49-34/49*11^(1/2))*(1/40*(10*x+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))+250/4
9-34/49*11^(1/2))^(3/2)+3/80*(5000/49-680/49*11^(1/2)-(34/7-10/7*11^(1/2))^2)*(1/20*(10*x+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)+1/200*(5000/49-680/49*11^(1/2)
-(34/7-10/7*11^(1/2))^2)*5^(1/2)*arcsinh(5^(1/2)/(250/49-34/49*11^(1/2)-1/20*(34/7-10/7*11^(1/2))^2)^(1/2)*(x+
1/5)))))-3535/21296*11^(1/2)*(1/21*(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))^(3/2)+1/14*(34/7+10/7*11^(1/2))*(1/20*(10*x+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)+1/200*(5000/49+680/49*11^(1/2)-(34/7+10/7*11^(1/2))^2)*5^(1/
2)*arcsinh(5^(1/2)/(250/49+34/49*11^(1/2)-1/20*(34/7+10/7*11^(1/2))^2)^(1/2)*(x+1/5)))+1/7*(250/49+34/49*11^(1
/2))*(1/7*(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)+1/10
*(34/7+10/7*11^(1/2))*5^(1/2)*arcsinh(5^(1/2)/(250/49+34/49*11^(1/2)-1/20*(34/7+10/7*11^(1/2))^2)^(1/2)*(x+1/5
))-7*(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*1
1^(1/2))+250+34*11^(1/2))^(1/2))))-21/968*(61+13*11^(1/2))*11^(1/2)*(-1/686/(250/49+34/49*11^(1/2))/(x-2/7-1/7
*11^(1/2))^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))^(5/2)+
1/1372*(34/7+10/7*11^(1/2))/(250/49+34/49*11^(1/2))*(-1/(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))^(5/2)+3/2*(34/7+10/7*11^(1/2
))/(250/49+34/49*11^(1/2))*(1/3*(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))^(3/2)+1/2*(34/7+10/7*11^(1/2))*(1/20*(10*x+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)+1/200*(5000/49+680/49*11^(1/2)-(34/7+10/7*11^(1/2))^2)*5^(1/2)*a
rcsinh(5^(1/2)/(250/49+34/49*11^(1/2)-1/20*(34/7+10/7*11^(1/2))^2)^(1/2)*(x+1/5)))+(250/49+34/49*11^(1/2))*(1/
7*(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)+1/10*(34/7+1
0/7*11^(1/2))*5^(1/2)*arcsinh(5^(1/2)/(250/49+34/49*11^(1/2)-1/20*(34/7+10/7*11^(1/2))^2)^(1/2)*(x+1/5))-7*(25
0/49+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))))+20/(250/49+34/49*11^(1/2))*(1/40*(10*x+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))^(3/2)+3/80*(5000/49+680/49*11^(1/2)-(34/7+10/7*11^(1/2))^2)*
(1/20*(10*x+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
)+1/200*(5000/49+680/49*11^(1/2)-(34/7+10/7*11^(1/2))^2)*5^(1/2)*arcsinh(5^(1/2)/(250/49+34/49*11^(1/2)-1/20*(
34/7+10/7*11^(1/2))^2)^(1/2)*(x+1/5)))))+15/686/(250/49+34/49*11^(1/2))*(1/3*(5*(x-2/7-1/7*11^(1/2))^2+(34/7+1
0/7*11^(1/2))*(x-2/7-1/7*11^(1/2))+250/49+34/49*11^(1/2))^(3/2)+1/2*(34/7+10/7*11^(1/2))*(1/20*(10*x+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)+1/200*(5000/49+680/
49*11^(1/2)-(34/7+10/7*11^(1/2))^2)*5^(1/2)*arcsinh(5^(1/2)/(250/49+34/49*11^(1/2)-1/20*(34/7+10/7*11^(1/2))^2
)^(1/2)*(x+1/5)))+(250/49+34/49*11^(1/2))*(1/7*(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)+1/10*(34/7+10/7*11^(1/2))*5^(1/2)*arcsinh(5^(1/2)/(250/49+34/49*11^(1/2)-1/20
*(34/7+10/7*11^(1/2))^2)^(1/2)*(x+1/5))-7*(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 \[ -\int \frac {{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} {\left (x^{2} + 5 \, x + 2\right )}}{{\left (7 \, x^{2} - 4 \, x - 1\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {6 \sqrt {5 x^{2} + 2 x + 3}}{343 x^{6} - 588 x^{5} + 189 x^{4} + 104 x^{3} - 27 x^{2} - 12 x - 1}\, dx - \int \frac {19 x \sqrt {5 x^{2} + 2 x + 3}}{343 x^{6} - 588 x^{5} + 189 x^{4} + 104 x^{3} - 27 x^{2} - 12 x - 1}\, dx - \int \frac {23 x^{2} \sqrt {5 x^{2} + 2 x + 3}}{343 x^{6} - 588 x^{5} + 189 x^{4} + 104 x^{3} - 27 x^{2} - 12 x - 1}\, dx - \int \frac {27 x^{3} \sqrt {5 x^{2} + 2 x + 3}}{343 x^{6} - 588 x^{5} + 189 x^{4} + 104 x^{3} - 27 x^{2} - 12 x - 1}\, dx - \int \frac {5 x^{4} \sqrt {5 x^{2} + 2 x + 3}}{343 x^{6} - 588 x^{5} + 189 x^{4} + 104 x^{3} - 27 x^{2} - 12 x - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(6*sqrt(5*x**2 + 2*x + 3)/(343*x**6 - 588*x**5 + 189*x**4 + 104*x**3 - 27*x**2 - 12*x - 1), x) - Inte
gral(19*x*sqrt(5*x**2 + 2*x + 3)/(343*x**6 - 588*x**5 + 189*x**4 + 104*x**3 - 27*x**2 - 12*x - 1), x) - Integr
al(23*x**2*sqrt(5*x**2 + 2*x + 3)/(343*x**6 - 588*x**5 + 189*x**4 + 104*x**3 - 27*x**2 - 12*x - 1), x) - Integ
ral(27*x**3*sqrt(5*x**2 + 2*x + 3)/(343*x**6 - 588*x**5 + 189*x**4 + 104*x**3 - 27*x**2 - 12*x - 1), x) - Inte
gral(5*x**4*sqrt(5*x**2 + 2*x + 3)/(343*x**6 - 588*x**5 + 189*x**4 + 104*x**3 - 27*x**2 - 12*x - 1), x)

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