3.2294 \(\int \frac{\sqrt{d+e x}}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=634 \[ -\frac{\sqrt{c} \left (-8 c^2 d e \left (-3 d \sqrt{b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (-2 b \left (6 d \sqrt{b^2-4 a c}+13 a e\right )+10 a e \sqrt{b^2-4 a c}+23 b^2 d\right )+b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{c} \left (-8 c^2 d e \left (3 d \sqrt{b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (12 b d \sqrt{b^2-4 a c}-10 a e \sqrt{b^2-4 a c}-26 a b e+23 b^2 d\right )+b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac{(b+2 c x) \sqrt{d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\sqrt{d+e x} \left (-c x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-4 b c \left (2 a e^2+3 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+13 b^2 c d e\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

[Out]

-((b + 2*c*x)*Sqrt[d + e*x])/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (Sqrt[d + e
*x]*(13*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - 4*b*c*(3*c*d^2 + 2*a*e^2) - c*(24*c^
2*d^2 + b^2*e^2 - 4*c*e*(6*b*d - 5*a*e))*x))/(4*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e +
a*e^2)*(a + b*x + c*x^2)) - (Sqrt[c]*(96*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*
e^3 - 8*c^2*d*e*(18*b*d - 3*Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b^2*d +
10*a*Sqrt[b^2 - 4*a*c]*e - 2*b*(6*Sqrt[b^2 - 4*a*c]*d + 13*a*e)))*ArcTanh[(Sqrt[
2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*(
b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^
2)) + (Sqrt[c]*(96*c^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(18*b*d
+ 3*Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b^2*d + 12*b*Sqrt[b^2 - 4*a*c]*
d - 26*a*b*e - 10*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x]
)/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[
2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2))

_______________________________________________________________________________________

Rubi [A]  time = 11.8048, antiderivative size = 634, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\sqrt{c} \left (-8 c^2 d e \left (-3 d \sqrt{b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (-2 b \left (6 d \sqrt{b^2-4 a c}+13 a e\right )+10 a e \sqrt{b^2-4 a c}+23 b^2 d\right )+b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{c} \left (-8 c^2 d e \left (3 d \sqrt{b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (12 b d \sqrt{b^2-4 a c}-10 a e \sqrt{b^2-4 a c}-26 a b e+23 b^2 d\right )+b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac{(b+2 c x) \sqrt{d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\sqrt{d+e x} \left (-c x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-4 b c \left (2 a e^2+3 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+13 b^2 c d e\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x]/(a + b*x + c*x^2)^3,x]

[Out]

-((b + 2*c*x)*Sqrt[d + e*x])/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (Sqrt[d + e
*x]*(13*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - 4*b*c*(3*c*d^2 + 2*a*e^2) - c*(24*c^
2*d^2 + b^2*e^2 - 4*c*e*(6*b*d - 5*a*e))*x))/(4*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e +
a*e^2)*(a + b*x + c*x^2)) - (Sqrt[c]*(96*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*
e^3 - 8*c^2*d*e*(18*b*d - 3*Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b^2*d +
10*a*Sqrt[b^2 - 4*a*c]*e - 2*b*(6*Sqrt[b^2 - 4*a*c]*d + 13*a*e)))*ArcTanh[(Sqrt[
2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*(
b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^
2)) + (Sqrt[c]*(96*c^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(18*b*d
+ 3*Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b^2*d + 12*b*Sqrt[b^2 - 4*a*c]*
d - 26*a*b*e - 10*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x]
)/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[
2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2))

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(1/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 6.20616, size = 701, normalized size = 1.11 \[ \sqrt{d+e x} \left (\frac{-b-2 c x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{8 a b c e^2+4 a c^2 d e+20 a c^2 e^2 x+b^3 e^2-13 b^2 c d e+b^2 c e^2 x+12 b c^2 d^2-24 b c^2 d e x+24 c^3 d^2 x}{4 \left (4 a c-b^2\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\right )-\frac{\left (-24 c^3 d^2 e \sqrt{b^2-4 a c}+24 b c^2 d e^2 \sqrt{b^2-4 a c}-20 a c^2 e^3 \sqrt{b^2-4 a c}-b^2 c e^3 \sqrt{b^2-4 a c}-52 a b c^2 e^3+104 a c^3 d e^2+b^3 c e^3+46 b^2 c^2 d e^2-144 b c^3 d^2 e+96 c^4 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{-e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{4 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{5/2} \sqrt{-e \sqrt{b^2-4 a c}-b e+2 c d} \left (-a e^2+b d e-c d^2\right )}-\frac{\left (-24 c^3 d^2 e \sqrt{b^2-4 a c}+24 b c^2 d e^2 \sqrt{b^2-4 a c}-20 a c^2 e^3 \sqrt{b^2-4 a c}-b^2 c e^3 \sqrt{b^2-4 a c}+52 a b c^2 e^3-104 a c^3 d e^2-b^3 c e^3-46 b^2 c^2 d e^2+144 b c^3 d^2 e-96 c^4 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{4 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{5/2} \sqrt{e \sqrt{b^2-4 a c}-b e+2 c d} \left (-a e^2+b d e-c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d + e*x]/(a + b*x + c*x^2)^3,x]

[Out]

Sqrt[d + e*x]*((-b - 2*c*x)/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (12*b*c^2*d^
2 - 13*b^2*c*d*e + 4*a*c^2*d*e + b^3*e^2 + 8*a*b*c*e^2 + 24*c^3*d^2*x - 24*b*c^2
*d*e*x + b^2*c*e^2*x + 20*a*c^2*e^2*x)/(4*(-b^2 + 4*a*c)^2*(c*d^2 - b*d*e + a*e^
2)*(a + b*x + c*x^2))) - ((96*c^4*d^3 - 144*b*c^3*d^2*e - 24*c^3*Sqrt[b^2 - 4*a*
c]*d^2*e + 46*b^2*c^2*d*e^2 + 104*a*c^3*d*e^2 + 24*b*c^2*Sqrt[b^2 - 4*a*c]*d*e^2
+ b^3*c*e^3 - 52*a*b*c^2*e^3 - b^2*c*Sqrt[b^2 - 4*a*c]*e^3 - 20*a*c^2*Sqrt[b^2
- 4*a*c]*e^3)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e - Sqrt[b^
2 - 4*a*c]*e]])/(4*Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - b*e - Sqrt[b
^2 - 4*a*c]*e]*(-(c*d^2) + b*d*e - a*e^2)) - ((-96*c^4*d^3 + 144*b*c^3*d^2*e - 2
4*c^3*Sqrt[b^2 - 4*a*c]*d^2*e - 46*b^2*c^2*d*e^2 - 104*a*c^3*d*e^2 + 24*b*c^2*Sq
rt[b^2 - 4*a*c]*d*e^2 - b^3*c*e^3 + 52*a*b*c^2*e^3 - b^2*c*Sqrt[b^2 - 4*a*c]*e^3
- 20*a*c^2*Sqrt[b^2 - 4*a*c]*e^3)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[
2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(4*Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^(5/2)*Sqrt
[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*(-(c*d^2) + b*d*e - a*e^2))

_______________________________________________________________________________________

Maple [B]  time = 0.209, size = 3348, normalized size = 5.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(1/2)/(c*x^2+b*x+a)^3,x)

[Out]

6*e*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c
+b^2))^(1/2))^2*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2
))*(e*x+d)^(3/2)*d+10*e^3*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(-b*e+2*c*d
+(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(
1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(
1/2))*a-3*e^2*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2
*(-4*a*c+b^2))^(1/2))^2*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e
^2)^(1/2))*(e*x+d)^(3/2)*b-24*e*c^4/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(-b*e
+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2)
)*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2)
)*c)^(1/2))*d^2-3*e^2*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c+1/
2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-b*e+2*c*d-(-4*a*c*e
^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*b+6*e*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^
2/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-
b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*d-17/2*e^3*c^2/(-e^2*(4*a*c-
b^2))^(1/2)/(4*a*c-b^2)^2/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e
+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e
+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2+10*e^3*c^3/(-e^2*(4*a*c-b^2))^(1/
2)/(4*a*c-b^2)^2/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-e
^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2
*(4*a*c-b^2))^(1/2))*c)^(1/2))*a-17/2*e^3*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^
2)^2/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^
2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2)
)^(1/2))*c)^(1/2))*b^2-24*e*c^4/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(-b*e+2*c
*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^
(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1
/2))*d^2-7/4*e^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^
2*(-4*a*c+b^2))^(1/2))^2*(e*x+d)^(1/2)*b^2+7/4*e^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a
*c-b^2)^2/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(e*x+d)^(1/2)*b^2+7*e
^3*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^
2))^(1/2))^2*(e*x+d)^(1/2)*a-7*e^3*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x
+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(e*x+d)^(1/2)*a+3/2*e^2/(-e^2*(4*a*
c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(-4
*a*c*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)*b+3/2*e^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-
b^2)^2/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(-4*a*c*e^2+b^2*e^2)^(1/
2)*(e*x+d)^(1/2)*b+9*e^2*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(-b*e+2*c*d+
(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1
/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1
/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)*b-18*e*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^
2/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2)
)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2)
)^(1/2))*c)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)*d+24*e^2*c^3/(-e^2*(4*a*c-b^2))^(1
/2)/(4*a*c-b^2)^2/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-
e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^
2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d-9*e^2*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^
2)^2/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^
2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2)
)^(1/2))*c)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)*b+18*e*c^3/(-e^2*(4*a*c-b^2))^(1/2
)/(4*a*c-b^2)^2/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-e^
2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*
(4*a*c-b^2))^(1/2))*c)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)*d-3*e*c/(-e^2*(4*a*c-b^
2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(-4*a*c
*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)*d+24*e^2*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b
^2)^2/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-
b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-
b^2))^(1/2))*c)^(1/2))*b*d-5/2*e^3*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x
+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1
/2))*(e*x+d)^(3/2)*b^2-3*e*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e
/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(-4*a*c*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)*d-
10*e^3*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*
a*c+b^2))^(1/2))^2/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*a+5/2*e
^3*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^
2))^(1/2))^2/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*b^2+10*e^3*c^
2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))
^(1/2))^2/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*a

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{{\left (c x^{2} + b x + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

integrate(sqrt(e*x + d)/(c*x^2 + b*x + a)^3, x)

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(1/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

Timed out