3.384 \(\int \frac{(2+5 x+x^2) (3+2 x+5 x^2)^{3/2}}{(1+4 x-7 x^2)^2} \, dx\)
Optimal. Leaf size=222 \[ \frac{3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{154 \left (-7 x^2+4 x+1\right )}+\frac{(3395 x+5826) \sqrt{5 x^2+2 x+3}}{3773}-\frac{\sqrt{\frac{1}{22} \left (52175400311-13155376531 \sqrt{11}\right )} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{26411}-\frac{\sqrt{\frac{1}{22} \left (52175400311+13155376531 \sqrt{11}\right )} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{26411}+\frac{16691 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{2401 \sqrt{5}} \]
[Out]
((5826 + 3395*x)*Sqrt[3 + 2*x + 5*x^2])/3773 + (3*(3 + 61*x)*(3 + 2*x + 5*x^2)^(3/2))/(154*(1 + 4*x - 7*x^2))
+ (16691*ArcSinh[(1 + 5*x)/Sqrt[14]])/(2401*Sqrt[5]) - (Sqrt[(52175400311 - 13155376531*Sqrt[11])/22]*ArcTanh[
(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/26411 - (Sqrt[(521
75400311 + 13155376531*Sqrt[11])/22]*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])
]*Sqrt[3 + 2*x + 5*x^2])])/26411
________________________________________________________________________________________
Rubi [A] time = 0.3216, antiderivative size = 222, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used
= {1054, 1066, 1076, 619, 215, 1032, 724, 206} \[ \frac{3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{154 \left (-7 x^2+4 x+1\right )}+\frac{(3395 x+5826) \sqrt{5 x^2+2 x+3}}{3773}-\frac{\sqrt{\frac{1}{22} \left (52175400311-13155376531 \sqrt{11}\right )} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{26411}-\frac{\sqrt{\frac{1}{22} \left (52175400311+13155376531 \sqrt{11}\right )} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{26411}+\frac{16691 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{2401 \sqrt{5}} \]
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)^2,x]
[Out]
((5826 + 3395*x)*Sqrt[3 + 2*x + 5*x^2])/3773 + (3*(3 + 61*x)*(3 + 2*x + 5*x^2)^(3/2))/(154*(1 + 4*x - 7*x^2))
+ (16691*ArcSinh[(1 + 5*x)/Sqrt[14]])/(2401*Sqrt[5]) - (Sqrt[(52175400311 - 13155376531*Sqrt[11])/22]*ArcTanh[
(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/26411 - (Sqrt[(521
75400311 + 13155376531*Sqrt[11])/22]*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])
]*Sqrt[3 + 2*x + 5*x^2])])/26411
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 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}}{\left (1+4 x-7 x^2\right )^2} \, dx &=\frac{3 (3+61 x) \left (3+2 x+5 x^2\right )^{3/2}}{154 \left (1+4 x-7 x^2\right )}-\frac{1}{308} \int \frac{\sqrt{3+2 x+5 x^2} \left (-912+724 x+3880 x^2\right )}{1+4 x-7 x^2} \, dx\\ &=\frac{(5826+3395 x) \sqrt{3+2 x+5 x^2}}{3773}+\frac{3 (3+61 x) \left (3+2 x+5 x^2\right )^{3/2}}{154 \left (1+4 x-7 x^2\right )}+\frac{\int \frac{700200-4304880 x-7344040 x^2}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx}{150920}\\ &=\frac{(5826+3395 x) \sqrt{3+2 x+5 x^2}}{3773}+\frac{3 (3+61 x) \left (3+2 x+5 x^2\right )^{3/2}}{154 \left (1+4 x-7 x^2\right )}-\frac{\int \frac{2442640+59510320 x}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx}{1056440}+\frac{16691 \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx}{2401}\\ &=\frac{(5826+3395 x) \sqrt{3+2 x+5 x^2}}{3773}+\frac{3 (3+61 x) \left (3+2 x+5 x^2\right )^{3/2}}{154 \left (1+4 x-7 x^2\right )}+\frac{16691 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )}{4802 \sqrt{70}}-\frac{\left (2 \left (8182669-1701489 \sqrt{11}\right )\right ) \int \frac{1}{\left (4-2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{290521}-\frac{\left (2 \left (8182669+1701489 \sqrt{11}\right )\right ) \int \frac{1}{\left (4+2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{290521}\\ &=\frac{(5826+3395 x) \sqrt{3+2 x+5 x^2}}{3773}+\frac{3 (3+61 x) \left (3+2 x+5 x^2\right )^{3/2}}{154 \left (1+4 x-7 x^2\right )}+\frac{16691 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{2401 \sqrt{5}}+\frac{\left (4 \left (8182669-1701489 \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 )}{290521}+\frac{\left (4 \left (8182669+1701489 \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 )}{290521}\\ &=\frac{(5826+3395 x) \sqrt{3+2 x+5 x^2}}{3773}+\frac{3 (3+61 x) \left (3+2 x+5 x^2\right )^{3/2}}{154 \left (1+4 x-7 x^2\right )}+\frac{16691 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{2401 \sqrt{5}}-\frac{\sqrt{\frac{1}{22} \left (52175400311-13155376531 \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 )}{26411}-\frac{\sqrt{\frac{1}{22} \left (52175400311+13155376531 \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 )}{26411}\\ \end{align*}
Mathematica [A] time = 1.8726, size = 354, normalized size = 1.59 \[ \frac{\frac{770 \sqrt{5 x^2+2 x+3} \left (2695 x^3+34265 x^2-81181 x-12975\right )}{7 x^2-4 x-1}+5 \sqrt{\frac{22}{125-17 \sqrt{11}}} \left (743879 \sqrt{11}-1701489\right ) \log \left (49 x^2+14 \left (\sqrt{11}-2\right ) x-4 \sqrt{11}+15\right )-10 \sqrt{\frac{22}{125+17 \sqrt{11}}} \left (1701489+743879 \sqrt{11}\right ) \log \left (\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left (55+17 \sqrt{11}\right ) x+23 \sqrt{11}+11\right )+10 \sqrt{\frac{22}{125-17 \sqrt{11}}} \left (743879 \sqrt{11}-1701489\right ) \tanh ^{-1}\left (\frac{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}{\left (5 \sqrt{11}-17\right ) x+\sqrt{11}-23}\right )+10 \sqrt{\frac{22}{125+17 \sqrt{11}}} \left (1701489+743879 \sqrt{11}\right ) \log \left (-7 x+\sqrt{11}+2\right )-5 \sqrt{\frac{22}{125-17 \sqrt{11}}} \left (743879 \sqrt{11}-1701489\right ) \log \left (\left (7 x+\sqrt{11}-2\right )^2\right )+8078444 \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{5810420} \]
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)^2,x]
[Out]
((770*Sqrt[3 + 2*x + 5*x^2]*(-12975 - 81181*x + 34265*x^2 + 2695*x^3))/(-1 - 4*x + 7*x^2) + 8078444*Sqrt[5]*Ar
cSinh[(1 + 5*x)/Sqrt[14]] + 10*Sqrt[22/(125 - 17*Sqrt[11])]*(-1701489 + 743879*Sqrt[11])*ArcTanh[(Sqrt[250 - 3
4*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])/(-23 + Sqrt[11] + (-17 + 5*Sqrt[11])*x)] + 10*Sqrt[22/(125 + 17*Sqrt[11])]*
(1701489 + 743879*Sqrt[11])*Log[2 + Sqrt[11] - 7*x] - 5*Sqrt[22/(125 - 17*Sqrt[11])]*(-1701489 + 743879*Sqrt[1
1])*Log[(-2 + Sqrt[11] + 7*x)^2] + 5*Sqrt[22/(125 - 17*Sqrt[11])]*(-1701489 + 743879*Sqrt[11])*Log[15 - 4*Sqrt
[11] + 14*(-2 + Sqrt[11])*x + 49*x^2] - 10*Sqrt[22/(125 + 17*Sqrt[11])]*(1701489 + 743879*Sqrt[11])*Log[11 + 2
3*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]])/5810420
________________________________________________________________________________________
Maple [B] time = 0.118, size = 1828, normalized size = 8.2 \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)^2,x)
[Out]
-161/484*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*(250/4
9+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))+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))^(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+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))))+20/49/(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*1
1^(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)-1/2
0*(34/7+10/7*11^(1/2))^2)^(1/2)*(x+1/5)))))+161/484*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*1
1^(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*(500/49-68/4
9*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/(250/49-34/4
9*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*(5000/49-680/49
*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*(500/49-6
8/49*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+4
9*(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)+3/80*(50
00/49-680/49*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)))))
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 )}^{2}}\,{d x} \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)^2,x, algorithm="maxima")
[Out]
integrate((5*x^2 + 2*x + 3)^(3/2)*(x^2 + 5*x + 2)/(7*x^2 - 4*x - 1)^2, x)
________________________________________________________________________________________
Fricas [B] time = 1.54977, size = 1515, normalized size = 6.82 \begin{align*} \frac{5 \, \sqrt{11}{\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt{26310753062 \, \sqrt{11} + 104350800622} \log \left (\frac{\sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{26310753062 \, \sqrt{11} + 104350800622}{\left (16206 \, \sqrt{11} - 68441\right )} + 1795191685 \, \sqrt{11}{\left (x + 3\right )} + 5385575055 \, x - 8975958425}{x}\right ) - 5 \, \sqrt{11}{\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt{26310753062 \, \sqrt{11} + 104350800622} \log \left (-\frac{\sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{26310753062 \, \sqrt{11} + 104350800622}{\left (16206 \, \sqrt{11} - 68441\right )} - 1795191685 \, \sqrt{11}{\left (x + 3\right )} - 5385575055 \, x + 8975958425}{x}\right ) - 5 \, \sqrt{11}{\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt{-26310753062 \, \sqrt{11} + 104350800622} \log \left (-\frac{\sqrt{5 \, x^{2} + 2 \, x + 3}{\left (16206 \, \sqrt{11} + 68441\right )} \sqrt{-26310753062 \, \sqrt{11} + 104350800622} + 1795191685 \, \sqrt{11}{\left (x + 3\right )} - 5385575055 \, x + 8975958425}{x}\right ) + 5 \, \sqrt{11}{\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt{-26310753062 \, \sqrt{11} + 104350800622} \log \left (\frac{\sqrt{5 \, x^{2} + 2 \, x + 3}{\left (16206 \, \sqrt{11} + 68441\right )} \sqrt{-26310753062 \, \sqrt{11} + 104350800622} - 1795191685 \, \sqrt{11}{\left (x + 3\right )} + 5385575055 \, x - 8975958425}{x}\right ) + 4039222 \, \sqrt{5}{\left (7 \, x^{2} - 4 \, 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 ) + 770 \,{\left (2695 \, x^{3} + 34265 \, x^{2} - 81181 \, x - 12975\right )} \sqrt{5 \, x^{2} + 2 \, x + 3}}{5810420 \,{\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)*(5*x^2+2*x+3)^(3/2)/(-7*x^2+4*x+1)^2,x, algorithm="fricas")
[Out]
1/5810420*(5*sqrt(11)*(7*x^2 - 4*x - 1)*sqrt(26310753062*sqrt(11) + 104350800622)*log((sqrt(5*x^2 + 2*x + 3)*s
qrt(26310753062*sqrt(11) + 104350800622)*(16206*sqrt(11) - 68441) + 1795191685*sqrt(11)*(x + 3) + 5385575055*x
- 8975958425)/x) - 5*sqrt(11)*(7*x^2 - 4*x - 1)*sqrt(26310753062*sqrt(11) + 104350800622)*log(-(sqrt(5*x^2 +
2*x + 3)*sqrt(26310753062*sqrt(11) + 104350800622)*(16206*sqrt(11) - 68441) - 1795191685*sqrt(11)*(x + 3) - 53
85575055*x + 8975958425)/x) - 5*sqrt(11)*(7*x^2 - 4*x - 1)*sqrt(-26310753062*sqrt(11) + 104350800622)*log(-(sq
rt(5*x^2 + 2*x + 3)*(16206*sqrt(11) + 68441)*sqrt(-26310753062*sqrt(11) + 104350800622) + 1795191685*sqrt(11)*
(x + 3) - 5385575055*x + 8975958425)/x) + 5*sqrt(11)*(7*x^2 - 4*x - 1)*sqrt(-26310753062*sqrt(11) + 1043508006
22)*log((sqrt(5*x^2 + 2*x + 3)*(16206*sqrt(11) + 68441)*sqrt(-26310753062*sqrt(11) + 104350800622) - 179519168
5*sqrt(11)*(x + 3) + 5385575055*x - 8975958425)/x) + 4039222*sqrt(5)*(7*x^2 - 4*x - 1)*log(-sqrt(5)*sqrt(5*x^2
+ 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8) + 770*(2695*x^3 + 34265*x^2 - 81181*x - 12975)*sqrt(5*x^2 + 2*x + 3
))/(7*x^2 - 4*x - 1)
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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)**2,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)^2,x, algorithm="giac")
[Out]
Exception raised: TypeError