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

Optimal. Leaf size=166 \[ -\frac{131-605 x}{3556 \sqrt{5 x^2+2 x+3}}-\frac{3 \sqrt{\frac{281693-25015 \sqrt{11}}{1397}} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{1016}+\frac{3 \sqrt{\frac{281693+25015 \sqrt{11}}{1397}} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{1016} \]

[Out]

-(131 - 605*x)/(3556*Sqrt[3 + 2*x + 5*x^2]) - (3*Sqrt[(281693 - 25015*Sqrt[11])/1397]*ArcTanh[(23 - Sqrt[11] +
(17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/1016 + (3*Sqrt[(281693 + 25015*Sqr
t[11])/1397]*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])
])/1016

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Rubi [A]  time = 0.215755, antiderivative size = 166, 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} \[ -\frac{131-605 x}{3556 \sqrt{5 x^2+2 x+3}}-\frac{3 \sqrt{\frac{281693-25015 \sqrt{11}}{1397}} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{1016}+\frac{3 \sqrt{\frac{281693+25015 \sqrt{11}}{1397}} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{1016} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(131 - 605*x)/(3556*Sqrt[3 + 2*x + 5*x^2]) - (3*Sqrt[(281693 - 25015*Sqrt[11])/1397]*ArcTanh[(23 - Sqrt[11] +
(17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/1016 + (3*Sqrt[(281693 + 25015*Sqr
t[11])/1397]*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])
])/1016

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 ) \left (3+2 x+5 x^2\right )^{3/2}} \, dx &=-\frac{131-605 x}{3556 \sqrt{3+2 x+5 x^2}}+\frac{\int \frac{13776+14112 x}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx}{28448}\\ &=-\frac{131-605 x}{3556 \sqrt{3+2 x+5 x^2}}+\frac{\left (21 \left (66-53 \sqrt{11}\right )\right ) \int \frac{1}{\left (4-2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{2794}+\frac{\left (21 \left (66+53 \sqrt{11}\right )\right ) \int \frac{1}{\left (4+2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{2794}\\ &=-\frac{131-605 x}{3556 \sqrt{3+2 x+5 x^2}}-\frac{\left (21 \left (66-53 \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 )}{1397}-\frac{\left (21 \left (66+53 \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 )}{1397}\\ &=-\frac{131-605 x}{3556 \sqrt{3+2 x+5 x^2}}-\frac{3 \sqrt{\frac{281693-25015 \sqrt{11}}{1397}} \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 )}{1016}+\frac{3 \sqrt{\frac{281693+25015 \sqrt{11}}{1397}} \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 )}{1016}\\ \end{align*}

Mathematica [A]  time = 1.17675, size = 174, normalized size = 1.05 \[ \frac{\frac{2794 (605 x-131)}{\sqrt{5 x^2+2 x+3}}-21 \sqrt{127 \left (125+17 \sqrt{11}\right )} \left (53 \sqrt{11}-66\right ) \tanh ^{-1}\left (\frac{-5 \sqrt{11} x+17 x-\sqrt{11}+23}{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right )+21 \sqrt{127 \left (125-17 \sqrt{11}\right )} \left (66+53 \sqrt{11}\right ) \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{250+34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right )}{9935464} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2794*(-131 + 605*x))/Sqrt[3 + 2*x + 5*x^2] - 21*Sqrt[127*(125 + 17*Sqrt[11])]*(-66 + 53*Sqrt[11])*ArcTanh[(2
3 - Sqrt[11] + 17*x - 5*Sqrt[11]*x)/(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])] + 21*Sqrt[127*(125 - 17*S
qrt[11])]*(66 + 53*Sqrt[11])*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[250 + 34*Sqrt[11]]*Sqrt[3 + 2
*x + 5*x^2])])/9935464

________________________________________________________________________________________

Maple [B]  time = 0.104, size = 489, normalized size = 3. \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)/(5*x^2+2*x+3)^(3/2),x)

[Out]

-1/196*(10*x+2)/(5*x^2+2*x+3)^(1/2)-3/154*(61+13*11^(1/2))*11^(1/2)*(1/7/(250/49+34/49*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)-1/7*(34/7+10/7*11^(1/2))/(
250/49+34/49*11^(1/2))*(10*x+2)/(5000/49+680/49*11^(1/2)-(34/7+10/7*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))^(1/2)-1/(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)*(1/7/(250/49-34/49*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*1
1^(1/2))+250/49-34/49*11^(1/2))^(1/2)-1/7*(34/7-10/7*11^(1/2))/(250/49-34/49*11^(1/2))*(10*x+2)/(5000/49-680/4
9*11^(1/2)-(34/7-10/7*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))^(1/2)-1/(250/49-34/49*11^(1/2))/(250-34*11^(1/2))^(1/2)*arctanh(49/2*(500/49-68/49*11^(1/2)+(3
4/7-10/7*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 = 1.80126, size = 1049, normalized size = 6.32 \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)/(-7*x^2+4*x+1)/(5*x^2+2*x+3)^(3/2),x, algorithm="maxima")

[Out]

-1/4312*sqrt(11)*(20*sqrt(11)*x/sqrt(5*x^2 + 2*x + 3) - 7890*sqrt(11)*x/(17*sqrt(11)*sqrt(5*x^2 + 2*x + 3) + 1
25*sqrt(5*x^2 + 2*x + 3)) + 7890*sqrt(11)*x/(17*sqrt(11)*sqrt(5*x^2 + 2*x + 3) - 125*sqrt(5*x^2 + 2*x + 3)) -
13377*sqrt(11)*sqrt(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))/(17*sqrt(11) + 125)^(3/2) + 4*sqrt(11)/sqrt(5*x^2 + 2*x + 3) - 26280*x/(17*sqrt(
11)*sqrt(5*x^2 + 2*x + 3) + 125*sqrt(5*x^2 + 2*x + 3)) - 26280*x/(17*sqrt(11)*sqrt(5*x^2 + 2*x + 3) - 125*sqrt
(5*x^2 + 2*x + 3)) + 156*sqrt(11)*arcsinh(5/7*sqrt(11)*sqrt(7)*sqrt(2)*x/abs(14*x + 2*sqrt(11) - 4) - 17/7*sqr
t(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*sqr
t(7)*sqrt(2)/abs(14*x + 2*sqrt(11) - 4))/(-34/49*sqrt(11) + 250/49)^(3/2) - 62769*sqrt(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(1
1)*sqrt(7)*sqrt(2)/abs(14*x - 2*sqrt(11) - 4) + 23/7*sqrt(7)*sqrt(2)/abs(14*x - 2*sqrt(11) - 4))/(17*sqrt(11)
+ 125)^(3/2) + 2244*sqrt(11)/(17*sqrt(11)*sqrt(5*x^2 + 2*x + 3) + 125*sqrt(5*x^2 + 2*x + 3)) - 2244*sqrt(11)/(
17*sqrt(11)*sqrt(5*x^2 + 2*x + 3) - 125*sqrt(5*x^2 + 2*x + 3)) - 732*arcsinh(5/7*sqrt(11)*sqrt(7)*sqrt(2)*x/ab
s(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)/ab
s(14*x + 2*sqrt(11) - 4) - 23/7*sqrt(7)*sqrt(2)/abs(14*x + 2*sqrt(11) - 4))/(-34/49*sqrt(11) + 250/49)^(3/2) +
12678/(17*sqrt(11)*sqrt(5*x^2 + 2*x + 3) + 125*sqrt(5*x^2 + 2*x + 3)) + 12678/(17*sqrt(11)*sqrt(5*x^2 + 2*x +
3) - 125*sqrt(5*x^2 + 2*x + 3)))

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Fricas [B]  time = 1.40603, size = 1269, normalized size = 7.64 \begin{align*} -\frac{21 \, \sqrt{1397}{\left (5 \, x^{2} + 2 \, x + 3\right )} \sqrt{25015 \, \sqrt{11} + 281693} \log \left (\frac{3 \,{\left (\sqrt{1397} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{25015 \, \sqrt{11} + 281693}{\left (1335 \, \sqrt{11} - 8173\right )} + 23596727 \, \sqrt{11}{\left (x + 3\right )} + 70790181 \, x - 117983635\right )}}{x}\right ) - 21 \, \sqrt{1397}{\left (5 \, x^{2} + 2 \, x + 3\right )} \sqrt{25015 \, \sqrt{11} + 281693} \log \left (-\frac{3 \,{\left (\sqrt{1397} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{25015 \, \sqrt{11} + 281693}{\left (1335 \, \sqrt{11} - 8173\right )} - 23596727 \, \sqrt{11}{\left (x + 3\right )} - 70790181 \, x + 117983635\right )}}{x}\right ) + 7 \, \sqrt{1397}{\left (5 \, x^{2} + 2 \, x + 3\right )} \sqrt{-225135 \, \sqrt{11} + 2535237} \log \left (-\frac{\sqrt{1397} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (1335 \, \sqrt{11} + 8173\right )} \sqrt{-225135 \, \sqrt{11} + 2535237} + 70790181 \, \sqrt{11}{\left (x + 3\right )} - 212370543 \, x + 353950905}{x}\right ) - 7 \, \sqrt{1397}{\left (5 \, x^{2} + 2 \, x + 3\right )} \sqrt{-225135 \, \sqrt{11} + 2535237} \log \left (\frac{\sqrt{1397} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (1335 \, \sqrt{11} + 8173\right )} \sqrt{-225135 \, \sqrt{11} + 2535237} - 70790181 \, \sqrt{11}{\left (x + 3\right )} + 212370543 \, x - 353950905}{x}\right ) - 5588 \, \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (605 \, x - 131\right )}}{19870928 \,{\left (5 \, x^{2} + 2 \, x + 3\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)/(5*x^2+2*x+3)^(3/2),x, algorithm="fricas")

[Out]

-1/19870928*(21*sqrt(1397)*(5*x^2 + 2*x + 3)*sqrt(25015*sqrt(11) + 281693)*log(3*(sqrt(1397)*sqrt(5*x^2 + 2*x
+ 3)*sqrt(25015*sqrt(11) + 281693)*(1335*sqrt(11) - 8173) + 23596727*sqrt(11)*(x + 3) + 70790181*x - 117983635
)/x) - 21*sqrt(1397)*(5*x^2 + 2*x + 3)*sqrt(25015*sqrt(11) + 281693)*log(-3*(sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*
sqrt(25015*sqrt(11) + 281693)*(1335*sqrt(11) - 8173) - 23596727*sqrt(11)*(x + 3) - 70790181*x + 117983635)/x)
+ 7*sqrt(1397)*(5*x^2 + 2*x + 3)*sqrt(-225135*sqrt(11) + 2535237)*log(-(sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*(1335
*sqrt(11) + 8173)*sqrt(-225135*sqrt(11) + 2535237) + 70790181*sqrt(11)*(x + 3) - 212370543*x + 353950905)/x) -
7*sqrt(1397)*(5*x^2 + 2*x + 3)*sqrt(-225135*sqrt(11) + 2535237)*log((sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*(1335*s
qrt(11) + 8173)*sqrt(-225135*sqrt(11) + 2535237) - 70790181*sqrt(11)*(x + 3) + 212370543*x - 353950905)/x) - 5
588*sqrt(5*x^2 + 2*x + 3)*(605*x - 131))/(5*x^2 + 2*x + 3)

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

[Out]

Exception raised: TypeError