### 3.383 $$\int \frac{(2+5 x+x^2) (3+2 x+5 x^2)^{3/2}}{1+4 x-7 x^2} \, dx$$

Optimal. Leaf size=210 $-\frac{1}{980} (35 x+267) \left (5 x^2+2 x+3\right )^{3/2}-\frac{3 (196105 x+571621) \sqrt{5 x^2+2 x+3}}{240100}-\frac{6 \sqrt{\frac{2}{11} \left (8098902607-2434122235 \sqrt{11}\right )} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{16807}+\frac{6 \sqrt{\frac{2}{11} \left (8098902607+2434122235 \sqrt{11}\right )} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{16807}-\frac{34425687 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{840350 \sqrt{5}}$

[Out]

(-3*(571621 + 196105*x)*Sqrt[3 + 2*x + 5*x^2])/240100 - ((267 + 35*x)*(3 + 2*x + 5*x^2)^(3/2))/980 - (34425687
*ArcSinh[(1 + 5*x)/Sqrt[14]])/(840350*Sqrt[5]) - (6*Sqrt[(2*(8098902607 - 2434122235*Sqrt[11]))/11]*ArcTanh[(2
3 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/16807 + (6*Sqrt[(2*(
8098902607 + 2434122235*Sqrt[11]))/11]*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11
])]*Sqrt[3 + 2*x + 5*x^2])])/16807

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Rubi [A]  time = 0.304076, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {1066, 1076, 619, 215, 1032, 724, 206} $-\frac{1}{980} (35 x+267) \left (5 x^2+2 x+3\right )^{3/2}-\frac{3 (196105 x+571621) \sqrt{5 x^2+2 x+3}}{240100}-\frac{6 \sqrt{\frac{2}{11} \left (8098902607-2434122235 \sqrt{11}\right )} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{16807}+\frac{6 \sqrt{\frac{2}{11} \left (8098902607+2434122235 \sqrt{11}\right )} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{16807}-\frac{34425687 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{840350 \sqrt{5}}$

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),x]

[Out]

(-3*(571621 + 196105*x)*Sqrt[3 + 2*x + 5*x^2])/240100 - ((267 + 35*x)*(3 + 2*x + 5*x^2)^(3/2))/980 - (34425687
*ArcSinh[(1 + 5*x)/Sqrt[14]])/(840350*Sqrt[5]) - (6*Sqrt[(2*(8098902607 - 2434122235*Sqrt[11]))/11]*ArcTanh[(2
3 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/16807 + (6*Sqrt[(2*(
8098902607 + 2434122235*Sqrt[11]))/11]*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11
])]*Sqrt[3 + 2*x + 5*x^2])])/16807

Rule 1066

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[((B*c*f*(2*p + 2*q + 3) + C*(b*f*p - c*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*
(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^(q + 1))/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), x] - Dist[1/(2*c*f^2*(p
+ q + 1)*(2*p + 2*q + 3)), Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[p*(b*d - a*e)*(C*(c*e - b*f)
*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*
(B*e - 2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (
p + q + 1)*(C*e*f*p*(b^2 - 4*a*c) - b*c*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q +
3))))*x + (p*(c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 -
4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x] /; F
reeQ[{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] && GtQ[p, 0] && NeQ[p +
q + 1, 0] && NeQ[2*p + 2*q + 3, 0] &&  !IGtQ[p, 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]

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 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 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{\left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}}{1+4 x-7 x^2} \, dx &=-\frac{1}{980} (267+35 x) \left (3+2 x+5 x^2\right )^{3/2}-\frac{\int \frac{\left (-20358-79272 x-100854 x^2\right ) \sqrt{3+2 x+5 x^2}}{1+4 x-7 x^2} \, dx}{2940}\\ &=-\frac{3 (571621+196105 x) \sqrt{3+2 x+5 x^2}}{240100}-\frac{1}{980} (267+35 x) \left (3+2 x+5 x^2\right )^{3/2}+\frac{\int \frac{50805108+282031632 x+413108244 x^2}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx}{1440600}\\ &=-\frac{3 (571621+196105 x) \sqrt{3+2 x+5 x^2}}{240100}-\frac{1}{980} (267+35 x) \left (3+2 x+5 x^2\right )^{3/2}-\frac{\int \frac{-768744000-3626654400 x}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx}{10084200}-\frac{34425687 \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx}{840350}\\ &=-\frac{3 (571621+196105 x) \sqrt{3+2 x+5 x^2}}{240100}-\frac{1}{980} (267+35 x) \left (3+2 x+5 x^2\right )^{3/2}-\frac{34425687 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )}{1680700 \sqrt{70}}+\frac{\left (24 \left (2770361-877397 \sqrt{11}\right )\right ) \int \frac{1}{\left (4-2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{184877}+\frac{\left (24 \left (2770361+877397 \sqrt{11}\right )\right ) \int \frac{1}{\left (4+2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{184877}\\ &=-\frac{3 (571621+196105 x) \sqrt{3+2 x+5 x^2}}{240100}-\frac{1}{980} (267+35 x) \left (3+2 x+5 x^2\right )^{3/2}-\frac{34425687 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{840350 \sqrt{5}}-\frac{\left (48 \left (2770361-877397 \sqrt{11}\right )\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 )}{184877}-\frac{\left (48 \left (2770361+877397 \sqrt{11}\right )\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 )}{184877}\\ &=-\frac{3 (571621+196105 x) \sqrt{3+2 x+5 x^2}}{240100}-\frac{1}{980} (267+35 x) \left (3+2 x+5 x^2\right )^{3/2}-\frac{34425687 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{840350 \sqrt{5}}-\frac{6 \sqrt{178175857354-53550689170 \sqrt{11}} \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 )}{184877}+\frac{6 \sqrt{\frac{2}{11} \left (8098902607+2434122235 \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 )}{16807}\\ \end{align*}

Mathematica [A]  time = 0.948347, size = 202, normalized size = 0.96 $\frac{-5 \left (77 \sqrt{5 x^2+2 x+3} \left (42875 x^3+344225 x^2+744870 x+1911108\right )+600 \sqrt{1572625-425459 \sqrt{11}} \left (61 \sqrt{11}-143\right ) \tanh ^{-1}\left (\frac{-5 \sqrt{11} x+17 x-\sqrt{11}+23}{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right )-600 \left (143+61 \sqrt{11}\right ) \sqrt{1572625+425459 \sqrt{11}} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{250+34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right )\right )-757365114 \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{92438500}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-757365114*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]] - 5*(77*Sqrt[3 + 2*x + 5*x^2]*(1911108 + 744870*x + 344225*x^2
+ 42875*x^3) + 600*Sqrt[1572625 - 425459*Sqrt[11]]*(-143 + 61*Sqrt[11])*ArcTanh[(23 - Sqrt[11] + 17*x - 5*Sqr
t[11]*x)/(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])] - 600*(143 + 61*Sqrt[11])*Sqrt[1572625 + 425459*Sqrt
[11]]*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[250 + 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])]))/9243850
0

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Maple [B]  time = 0.116, size = 730, normalized size = 3.5 \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)*(5*x^2+2*x+3)^(3/2)/(-7*x^2+4*x+1),x)

[Out]

-1/280*(10*x+2)*(5*x^2+2*x+3)^(3/2)-3/200*(10*x+2)*(5*x^2+2*x+3)^(1/2)-21/250*5^(1/2)*arcsinh(5/14*14^(1/2)*(x
+1/5))-3/154*(61+13*11^(1/2))*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+3
4/49*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))))-3/154*(-61+13*11^(1/2))*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*1
1^(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 [B]  time = 1.87847, size = 722, normalized size = 3.44 \begin{align*} \text{result too large to display} \end{align*}

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),x, algorithm="maxima")

[Out]

1/92438500*sqrt(11)*(19500*sqrt(11)*sqrt(2)*(17*sqrt(11) + 125)^(3/2)*arcsinh(5/7*sqrt(11)*sqrt(7)*sqrt(2)*x/a
bs(14*x - 2*sqrt(11) - 4) + 17/7*sqrt(7)*sqrt(2)*x/abs(14*x - 2*sqrt(11) - 4) + 1/7*sqrt(11)*sqrt(7)*sqrt(2)/a
bs(14*x - 2*sqrt(11) - 4) + 23/7*sqrt(7)*sqrt(2)/abs(14*x - 2*sqrt(11) - 4)) - 300125*sqrt(11)*(5*x^2 + 2*x +
3)^(3/2)*x - 3344250*sqrt(11)*(-34/49*sqrt(11) + 250/49)^(3/2)*arcsinh(5/7*sqrt(11)*sqrt(7)*sqrt(2)*x/abs(14*x
+ 2*sqrt(11) - 4) - 17/7*sqrt(7)*sqrt(2)*x/abs(14*x + 2*sqrt(11) - 4) + 1/7*sqrt(11)*sqrt(7)*sqrt(2)/abs(14*x
+ 2*sqrt(11) - 4) - 23/7*sqrt(7)*sqrt(2)/abs(14*x + 2*sqrt(11) - 4)) + 91500*sqrt(2)*(17*sqrt(11) + 125)^(3/2
)*arcsinh(5/7*sqrt(11)*sqrt(7)*sqrt(2)*x/abs(14*x - 2*sqrt(11) - 4) + 17/7*sqrt(7)*sqrt(2)*x/abs(14*x - 2*sqrt
(11) - 4) + 1/7*sqrt(11)*sqrt(7)*sqrt(2)/abs(14*x - 2*sqrt(11) - 4) + 23/7*sqrt(7)*sqrt(2)/abs(14*x - 2*sqrt(1
1) - 4)) + 15692250*(-34/49*sqrt(11) + 250/49)^(3/2)*arcsinh(5/7*sqrt(11)*sqrt(7)*sqrt(2)*x/abs(14*x + 2*sqrt(
11) - 4) - 17/7*sqrt(7)*sqrt(2)*x/abs(14*x + 2*sqrt(11) - 4) + 1/7*sqrt(11)*sqrt(7)*sqrt(2)/abs(14*x + 2*sqrt(
11) - 4) - 23/7*sqrt(7)*sqrt(2)/abs(14*x + 2*sqrt(11) - 4)) - 2289525*sqrt(11)*(5*x^2 + 2*x + 3)^(3/2) - 20591
025*sqrt(11)*sqrt(5*x^2 + 2*x + 3)*x - 68851374*sqrt(11)*sqrt(5)*arcsinh(5/14*sqrt(7)*sqrt(2)*x + 1/14*sqrt(7)
*sqrt(2)) - 60020205*sqrt(11)*sqrt(5*x^2 + 2*x + 3))

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Fricas [B]  time = 1.49094, size = 1442, normalized size = 6.87 \begin{align*} \frac{3}{184877} \, \sqrt{11} \sqrt{2} \sqrt{2434122235 \, \sqrt{11} + 8098902607} \log \left (\frac{12 \,{\left (\sqrt{2} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{2434122235 \, \sqrt{11} + 8098902607}{\left (7690 \, \sqrt{11} - 24697\right )} + 40555291 \, \sqrt{11}{\left (x + 3\right )} + 121665873 \, x - 202776455\right )}}{x}\right ) - \frac{3}{184877} \, \sqrt{11} \sqrt{2} \sqrt{2434122235 \, \sqrt{11} + 8098902607} \log \left (-\frac{12 \,{\left (\sqrt{2} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{2434122235 \, \sqrt{11} + 8098902607}{\left (7690 \, \sqrt{11} - 24697\right )} - 40555291 \, \sqrt{11}{\left (x + 3\right )} - 121665873 \, x + 202776455\right )}}{x}\right ) - \frac{1}{739508} \, \sqrt{11} \sqrt{-701027203680 \, \sqrt{11} + 2332483950816} \log \left (-\frac{\sqrt{5 \, x^{2} + 2 \, x + 3}{\left (7690 \, \sqrt{11} + 24697\right )} \sqrt{-701027203680 \, \sqrt{11} + 2332483950816} + 486663492 \, \sqrt{11}{\left (x + 3\right )} - 1459990476 \, x + 2433317460}{x}\right ) + \frac{1}{739508} \, \sqrt{11} \sqrt{-701027203680 \, \sqrt{11} + 2332483950816} \log \left (\frac{\sqrt{5 \, x^{2} + 2 \, x + 3}{\left (7690 \, \sqrt{11} + 24697\right )} \sqrt{-701027203680 \, \sqrt{11} + 2332483950816} - 486663492 \, \sqrt{11}{\left (x + 3\right )} + 1459990476 \, x - 2433317460}{x}\right ) - \frac{1}{240100} \,{\left (42875 \, x^{3} + 344225 \, x^{2} + 744870 \, x + 1911108\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{34425687}{8403500} \, \sqrt{5} \log \left (\sqrt{5} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \end{align*}

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),x, algorithm="fricas")

[Out]

3/184877*sqrt(11)*sqrt(2)*sqrt(2434122235*sqrt(11) + 8098902607)*log(12*(sqrt(2)*sqrt(5*x^2 + 2*x + 3)*sqrt(24
34122235*sqrt(11) + 8098902607)*(7690*sqrt(11) - 24697) + 40555291*sqrt(11)*(x + 3) + 121665873*x - 202776455)
/x) - 3/184877*sqrt(11)*sqrt(2)*sqrt(2434122235*sqrt(11) + 8098902607)*log(-12*(sqrt(2)*sqrt(5*x^2 + 2*x + 3)*
sqrt(2434122235*sqrt(11) + 8098902607)*(7690*sqrt(11) - 24697) - 40555291*sqrt(11)*(x + 3) - 121665873*x + 202
776455)/x) - 1/739508*sqrt(11)*sqrt(-701027203680*sqrt(11) + 2332483950816)*log(-(sqrt(5*x^2 + 2*x + 3)*(7690*
sqrt(11) + 24697)*sqrt(-701027203680*sqrt(11) + 2332483950816) + 486663492*sqrt(11)*(x + 3) - 1459990476*x + 2
433317460)/x) + 1/739508*sqrt(11)*sqrt(-701027203680*sqrt(11) + 2332483950816)*log((sqrt(5*x^2 + 2*x + 3)*(769
0*sqrt(11) + 24697)*sqrt(-701027203680*sqrt(11) + 2332483950816) - 486663492*sqrt(11)*(x + 3) + 1459990476*x -
2433317460)/x) - 1/240100*(42875*x^3 + 344225*x^2 + 744870*x + 1911108)*sqrt(5*x^2 + 2*x + 3) + 34425687/8403
500*sqrt(5)*log(sqrt(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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),x)

[Out]

Timed out

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

[Out]

Exception raised: TypeError