3.1627 \(\int \frac{b+2 c x}{\sqrt{d+e x} \left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=673 \[ \frac{e \sqrt{d+e x} \left (c x \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )+(c d-3 b e) \left (2 a c e+b^2 (-e)+b c d\right )+5 a c e (2 c d-b e)\right )}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{\sqrt{c} e \left (-2 c^2 d e \left (-d \sqrt{b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (b d \sqrt{b^2-4 a c}+5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )^2}+\frac{\sqrt{c} e \left (-2 c^2 d e \left (d \sqrt{b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (-b d \sqrt{b^2-4 a c}-5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )^2}-\frac{\sqrt{d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )} \]

[Out]

-(Sqrt[d + e*x]*((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x))/(2*(b^2 - 4*a
*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) + (e*Sqrt[d + e*x]*(5*a*c*e*(2*
c*d - b*e) + (c*d - 3*b*e)*(b*c*d - b^2*e + 2*a*c*e) + c*(2*c^2*d^2 + 3*b^2*e^2
- 2*c*e*(b*d + 5*a*e))*x))/(4*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x +
c*x^2)) - (Sqrt[c]*e*(8*c^3*d^3 + 3*b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e
*(6*b*d - Sqrt[b^2 - 4*a*c]*d - 16*a*e) - 2*c*e^2*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d
+ 8*a*b*e + 5*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/S
qrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c
*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)^2) + (Sqrt[c]*e*(8*c^3*d
^3 + 3*b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*d
- 16*a*e) - 2*c*e^2*(b^2*d - b*Sqrt[b^2 - 4*a*c]*d + 8*a*b*e - 5*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)^(3/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e
]*(c*d^2 - b*d*e + a*e^2)^2)

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Rubi [A]  time = 11.1876, antiderivative size = 673, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{e \sqrt{d+e x} \left (c x \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )+(c d-3 b e) \left (2 a c e+b^2 (-e)+b c d\right )+5 a c e (2 c d-b e)\right )}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{\sqrt{c} e \left (-2 c^2 d e \left (-d \sqrt{b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (b d \sqrt{b^2-4 a c}+5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )^2}+\frac{\sqrt{c} e \left (-2 c^2 d e \left (d \sqrt{b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (-b d \sqrt{b^2-4 a c}-5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )^2}-\frac{\sqrt{d+e x} (-b e+c d-c e x)}{2 \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d + e*x]*(c*d - b*e - c*e*x))/(2*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2
)^2) + (e*Sqrt[d + e*x]*(5*a*c*e*(2*c*d - b*e) + (c*d - 3*b*e)*(b*c*d - b^2*e +
2*a*c*e) + c*(2*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(b*d + 5*a*e))*x))/(4*(b^2 - 4*a*c)*
(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2)) - (Sqrt[c]*e*(8*c^3*d^3 + 3*b^2*(b
+ Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d - Sqrt[b^2 - 4*a*c]*d - 16*a*e) - 2*
c*e^2*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d + 8*a*b*e + 5*a*Sqrt[b^2 - 4*a*c]*e))*ArcTa
nh[(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)^(3/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d
*e + a*e^2)^2) + (Sqrt[c]*e*(8*c^3*d^3 + 3*b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 2*c
^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*d - 16*a*e) - 2*c*e^2*(b^2*d - b*Sqrt[b^2 - 4*
a*c]*d + 8*a*b*e - 5*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)^(3/2)*Sq
rt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)^2)

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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((2*c*x+b)/(c*x**2+b*x+a)**3/(e*x+d)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 6.32364, size = 732, normalized size = 1.09 \[ \sqrt{d+e x} \left (\frac{-11 a b c e^3+12 a c^2 d e^2-10 a c^2 e^3 x+3 b^3 e^3-4 b^2 c d e^2+3 b^2 c e^3 x+b c^2 d^2 e-2 b c^2 d e^2 x+2 c^3 d^2 e x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (-a e^2+b d e-c d^2\right )^2}+\frac{b e-c d+c e x}{2 \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\right )-\frac{\left (2 c^3 d^2 e^2 \sqrt{b^2-4 a c}-2 b c^2 d e^3 \sqrt{b^2-4 a c}-10 a c^2 e^4 \sqrt{b^2-4 a c}+3 b^2 c e^4 \sqrt{b^2-4 a c}+16 a b c^2 e^4-32 a c^3 d e^3-3 b^3 c e^4+2 b^2 c^2 d e^3+12 b c^3 d^2 e^2-8 c^4 d^3 e\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{-e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{4 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{-e \sqrt{b^2-4 a c}-b e+2 c d} \left (-a e^2+b d e-c d^2\right )^2}-\frac{\left (2 c^3 d^2 e^2 \sqrt{b^2-4 a c}-2 b c^2 d e^3 \sqrt{b^2-4 a c}-10 a c^2 e^4 \sqrt{b^2-4 a c}+3 b^2 c e^4 \sqrt{b^2-4 a c}-16 a b c^2 e^4+32 a c^3 d e^3+3 b^3 c e^4-2 b^2 c^2 d e^3-12 b c^3 d^2 e^2+8 c^4 d^3 e\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{4 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{e \sqrt{b^2-4 a c}-b e+2 c d} \left (-a e^2+b d e-c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d + e*x]*((-(c*d) + b*e + c*e*x)/(2*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^
2)^2) + (b*c^2*d^2*e - 4*b^2*c*d*e^2 + 12*a*c^2*d*e^2 + 3*b^3*e^3 - 11*a*b*c*e^3
+ 2*c^3*d^2*e*x - 2*b*c^2*d*e^2*x + 3*b^2*c*e^3*x - 10*a*c^2*e^3*x)/(4*(b^2 - 4
*a*c)*(-(c*d^2) + b*d*e - a*e^2)^2*(a + b*x + c*x^2))) - ((-8*c^4*d^3*e + 12*b*c
^3*d^2*e^2 + 2*c^3*Sqrt[b^2 - 4*a*c]*d^2*e^2 + 2*b^2*c^2*d*e^3 - 32*a*c^3*d*e^3
- 2*b*c^2*Sqrt[b^2 - 4*a*c]*d*e^3 - 3*b^3*c*e^4 + 16*a*b*c^2*e^4 + 3*b^2*c*Sqrt[
b^2 - 4*a*c]*e^4 - 10*a*c^2*Sqrt[b^2 - 4*a*c]*e^4)*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)^(3/2)*Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]*(-(c*d^2) + b*d*e - a*e^2)^2
) - ((8*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 2*c^3*Sqrt[b^2 - 4*a*c]*d^2*e^2 - 2*b^2*c
^2*d*e^3 + 32*a*c^3*d*e^3 - 2*b*c^2*Sqrt[b^2 - 4*a*c]*d*e^3 + 3*b^3*c*e^4 - 16*a
*b*c^2*e^4 + 3*b^2*c*Sqrt[b^2 - 4*a*c]*e^4 - 10*a*c^2*Sqrt[b^2 - 4*a*c]*e^4)*Arc
Tanh[(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)^(3/2)*Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*(-
(c*d^2) + b*d*e - a*e^2)^2)

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Maple [B]  time = 0.181, size = 2734, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^3*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b
^2))^(1/2))^2/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2-(-4*a*c*e^2+b^2*e^2)^(1/2)
*b*e+2*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*(e*x+d)^(3/2)*b-2*e^2*c^3/(-e^2*(4*a*c-b^
2))^(1/2)/(4*a*c-b^2)/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-2*a*c*e
^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2-(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e+2*(-4*a*c*e^2+b^2*
e^2)^(1/2)*c*d)*(e*x+d)^(3/2)*d-5/2*e^2*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)
/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+
2*c^2*d^2-(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e+2*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*(e*x+
d)^(3/2)*(-4*a*c*e^2+b^2*e^2)^(1/2)-e^3*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^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)^(1/2)*b+2*e^2*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^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)^(1/2)*d+7/2*e^2*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^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)^
(1/2)*(-4*a*c*e^2+b^2*e^2)^(1/2)-e^3*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(-
2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2-(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e+2*(-4*a*c*e
^2+b^2*e^2)^(1/2)*c*d)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*a
rctanh(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*e^2*c^4/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2
*c^2*d^2-(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e+2*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*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+5/2*e^2*c^3/(-e^2*(4*a*c-b^
2))^(1/2)/(4*a*c-b^2)/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2-(-4*a*c*e^2+b^2*e^
2)^(1/2)*b*e+2*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*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)-e^3*c^2/(-e^2*(4*a*c-b^2))^(1
/2)/(4*a*c-b^2)/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-2*a*c*e^2+b^2
*e^2-2*b*c*d*e+2*c^2*d^2+(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e-2*(-4*a*c*e^2+b^2*e^2)^(
1/2)*c*d)*(e*x+d)^(3/2)*b+2*e^2*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(e*x+1/
2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^
2+(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e-2*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*(e*x+d)^(3/2)
*d-5/2*e^2*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(e*x+1/2*b*e/c+1/2/c*(e^2*(-
4*a*c+b^2))^(1/2))^2/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2+(-4*a*c*e^2+b^2*e^2
)^(1/2)*b*e-2*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*(e*x+d)^(3/2)*(-4*a*c*e^2+b^2*e^2)
^(1/2)+e^3*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^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)^(1/2)*b-2*e^2
*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^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)^(1/2)*d+7/2*e^2*c/(-e
^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^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)^(1/2)*(-4*a*c*e^2+b^2*e^2)^(1
/2)-e^3*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2
*c^2*d^2+(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e-2*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*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*e^2*c^4/(-e^2*(4*a*c-b^2))^(
1/2)/(4*a*c-b^2)/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2+(-4*a*c*e^2+b^2*e^2)^(1
/2)*b*e-2*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*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-5/2*e^2*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(-2*a*c*e^2+b
^2*e^2-2*b*c*d*e+2*c^2*d^2+(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e-2*(-4*a*c*e^2+b^2*e^2)
^(1/2)*c*d)*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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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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((2*c*x+b)/(c*x**2+b*x+a)**3/(e*x+d)**(1/2),x)

[Out]

Timed out

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

[Out]

Timed out