3.2295 \(\int \frac{1}{\sqrt{d+e x} \left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=835 \[ -\frac{\sqrt{d+e x} \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac{3 \sqrt{c} \left (32 c^4 d^4-8 c^3 e \left (8 b d-\sqrt{b^2-4 a c} d-9 a e\right ) d^2+b^3 \left (b+\sqrt{b^2-4 a c}\right ) e^4+2 b c e^3 \left (d b^2+\sqrt{b^2-4 a c} d b-5 a e b-4 a \sqrt{b^2-4 a c} e\right )+2 c^2 e^2 \left (15 b^2 d^2-6 b \left (\sqrt{b^2-4 a c} d+6 a e\right ) d+4 a e \left (2 \sqrt{b^2-4 a c} d+7 a e\right )\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e} \left (c d^2-b e d+a e^2\right )^2}+\frac{3 \sqrt{c} \left (32 c^4 d^4-8 c^3 e \left (8 b d+\sqrt{b^2-4 a c} d-9 a e\right ) d^2+b^3 \left (b-\sqrt{b^2-4 a c}\right ) e^4+2 c^2 e^2 \left (15 b^2 d^2+6 b \left (\sqrt{b^2-4 a c} d-6 a e\right ) d-4 a e \left (2 \sqrt{b^2-4 a c} d-7 a e\right )\right )+2 b c e^3 \left (d b^2-\left (\sqrt{b^2-4 a c} d+5 a e\right ) b+4 a \sqrt{b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} \left (c d^2-e (b d-a e)\right )^2}-\frac{\sqrt{d+e x} \left (5 a c e (2 c d-b e)^2-3 c \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (12 c^2 d^2-3 b^2 e^2-7 c e (b d-2 a e)\right )\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )} \]

[Out]

-(Sqrt[d + e*x]*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*
(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (Sqrt[d + e*x]*(5*a*c*e*(2*c*d -
b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(12*c^2*d^2 - 3*b^2*e^2 - 7*c*e*(b*d - 2*a*e)
) - 3*c*(2*c*d - b*e)*(4*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - 2*a*e))*x))/(4*(b^2 -
4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2)) - (3*Sqrt[c]*(32*c^4*d^4 +
b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(8*b*d - Sqrt[b^2 - 4*a*c]*d - 9*
a*e) + 2*b*c*e^3*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 5*a*b*e - 4*a*Sqrt[b^2 - 4*a*c
]*e) + 2*c^2*e^2*(15*b^2*d^2 - 6*b*d*(Sqrt[b^2 - 4*a*c]*d + 6*a*e) + 4*a*e*(2*Sq
rt[b^2 - 4*a*c]*d + 7*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)^2) + (3*Sqrt[c]*(32*c^4*d^4 + b^3*
(b - Sqrt[b^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(8*b*d + Sqrt[b^2 - 4*a*c]*d - 9*a*e)
+ 2*c^2*e^2*(15*b^2*d^2 - 4*a*e*(2*Sqrt[b^2 - 4*a*c]*d - 7*a*e) + 6*b*d*(Sqrt[b^
2 - 4*a*c]*d - 6*a*e)) + 2*b*c*e^3*(b^2*d + 4*a*Sqrt[b^2 - 4*a*c]*e - b*(Sqrt[b^
2 - 4*a*c]*d + 5*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 - e*(b*d - a*e))^2)

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Rubi [A]  time = 22.2387, antiderivative size = 836, 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 \[ -\frac{\sqrt{d+e x} \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac{3 \sqrt{c} \left (32 c^4 d^4-8 c^3 e \left (8 b d-\sqrt{b^2-4 a c} d-9 a e\right ) d^2+b^3 \left (b+\sqrt{b^2-4 a c}\right ) e^4+2 b c e^3 \left (d b^2+\sqrt{b^2-4 a c} d b-5 a e b-4 a \sqrt{b^2-4 a c} e\right )+2 c^2 e^2 \left (15 b^2 d^2-6 b \left (\sqrt{b^2-4 a c} d+6 a e\right ) d+4 a e \left (2 \sqrt{b^2-4 a c} d+7 a e\right )\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e} \left (c d^2-e (b d-a e)\right )^2}+\frac{3 \sqrt{c} \left (32 c^4 d^4-8 c^3 e \left (8 b d+\sqrt{b^2-4 a c} d-9 a e\right ) d^2+b^3 \left (b-\sqrt{b^2-4 a c}\right ) e^4+2 c^2 e^2 \left (15 b^2 d^2+6 b \left (\sqrt{b^2-4 a c} d-6 a e\right ) d-4 a e \left (2 \sqrt{b^2-4 a c} d-7 a e\right )\right )+2 b c e^3 \left (d b^2-\left (\sqrt{b^2-4 a c} d+5 a e\right ) b+4 a \sqrt{b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} \left (c d^2-e (b d-a e)\right )^2}-\frac{\sqrt{d+e x} \left (5 a c e (2 c d-b e)^2-3 c \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (12 c^2 d^2-3 b^2 e^2-7 c e (b d-2 a e)\right )\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d + e*x]*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*
(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (Sqrt[d + e*x]*(5*a*c*e*(2*c*d -
b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(12*c^2*d^2 - 3*b^2*e^2 - 7*c*e*(b*d - 2*a*e)
) - 3*c*(2*c*d - b*e)*(4*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - 2*a*e))*x))/(4*(b^2 -
4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2)) - (3*Sqrt[c]*(32*c^4*d^4 +
b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(8*b*d - Sqrt[b^2 - 4*a*c]*d - 9*
a*e) + 2*b*c*e^3*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 5*a*b*e - 4*a*Sqrt[b^2 - 4*a*c
]*e) + 2*c^2*e^2*(15*b^2*d^2 - 6*b*d*(Sqrt[b^2 - 4*a*c]*d + 6*a*e) + 4*a*e*(2*Sq
rt[b^2 - 4*a*c]*d + 7*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 - e*(b*d - a*e))^2) + (3*Sqrt[c]*(32*c^4*d^4 + b^3*
(b - Sqrt[b^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(8*b*d + Sqrt[b^2 - 4*a*c]*d - 9*a*e)
+ 2*c^2*e^2*(15*b^2*d^2 - 4*a*e*(2*Sqrt[b^2 - 4*a*c]*d - 7*a*e) + 6*b*d*(Sqrt[b^
2 - 4*a*c]*d - 6*a*e)) + 2*b*c*e^3*(b^2*d + 4*a*Sqrt[b^2 - 4*a*c]*e - b*(Sqrt[b^
2 - 4*a*c]*d + 5*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 - e*(b*d - a*e))^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(1/(e*x+d)**(1/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 6.3671, size = 987, normalized size = 1.18 \[ \sqrt{d+e x} \left (\frac{-e b^2+c d b-c e x b+2 a c e+2 c^2 d x}{2 \left (4 a c-b^2\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}+\frac{3 e^3 b^4+4 c d e^2 b^3+3 c e^3 x b^3-25 a c e^3 b^2-19 c^2 d^2 e b^2+6 c^2 d e^2 x b^2+12 c^3 d^3 b+20 a c^2 d e^2 b-24 a c^2 e^3 x b-36 c^3 d^2 e x b+28 a^2 c^2 e^3+4 a c^3 d^2 e+24 c^4 d^3 x+48 a c^3 d e^2 x}{4 \left (4 a c-b^2\right )^2 \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )}\right )-\frac{3 \left (-32 d^4 c^5-72 a d^2 e^2 c^4+64 b d^3 e c^4+8 \sqrt{b^2-4 a c} d^3 e c^4-56 a^2 e^4 c^3+72 a b d e^3 c^3+16 a \sqrt{b^2-4 a c} d e^3 c^3-30 b^2 d^2 e^2 c^3-12 b \sqrt{b^2-4 a c} d^2 e^2 c^3+10 a b^2 e^4 c^2-8 a b \sqrt{b^2-4 a c} e^4 c^2-2 b^3 d e^3 c^2+2 b^2 \sqrt{b^2-4 a c} d e^3 c^2-b^4 e^4 c+b^3 \sqrt{b^2-4 a c} e^4 c\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-b e-\sqrt{b^2-4 a c} e}}\right )}{4 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-b e-\sqrt{b^2-4 a c} e} \left (-c d^2+b e d-a e^2\right )^2}-\frac{3 \left (32 d^4 c^5+72 a d^2 e^2 c^4-64 b d^3 e c^4+8 \sqrt{b^2-4 a c} d^3 e c^4+56 a^2 e^4 c^3-72 a b d e^3 c^3+16 a \sqrt{b^2-4 a c} d e^3 c^3+30 b^2 d^2 e^2 c^3-12 b \sqrt{b^2-4 a c} d^2 e^2 c^3-10 a b^2 e^4 c^2-8 a b \sqrt{b^2-4 a c} e^4 c^2+2 b^3 d e^3 c^2+2 b^2 \sqrt{b^2-4 a c} d e^3 c^2+b^4 e^4 c+b^3 \sqrt{b^2-4 a c} e^4 c\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-b e+\sqrt{b^2-4 a c} e}}\right )}{4 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-b e+\sqrt{b^2-4 a c} e} \left (-c d^2+b e d-a e^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d + e*x]*((b*c*d - b^2*e + 2*a*c*e + 2*c^2*d*x - b*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) + (12*b*c^3*d^3 - 19*b^2*c^2*d^2*e
+ 4*a*c^3*d^2*e + 4*b^3*c*d*e^2 + 20*a*b*c^2*d*e^2 + 3*b^4*e^3 - 25*a*b^2*c*e^3
+ 28*a^2*c^2*e^3 + 24*c^4*d^3*x - 36*b*c^3*d^2*e*x + 6*b^2*c^2*d*e^2*x + 48*a*c
^3*d*e^2*x + 3*b^3*c*e^3*x - 24*a*b*c^2*e^3*x)/(4*(-b^2 + 4*a*c)^2*(c*d^2 - b*d*
e + a*e^2)^2*(a + b*x + c*x^2))) - (3*(-32*c^5*d^4 + 64*b*c^4*d^3*e + 8*c^4*Sqrt
[b^2 - 4*a*c]*d^3*e - 30*b^2*c^3*d^2*e^2 - 72*a*c^4*d^2*e^2 - 12*b*c^3*Sqrt[b^2
- 4*a*c]*d^2*e^2 - 2*b^3*c^2*d*e^3 + 72*a*b*c^3*d*e^3 + 2*b^2*c^2*Sqrt[b^2 - 4*a
*c]*d*e^3 + 16*a*c^3*Sqrt[b^2 - 4*a*c]*d*e^3 - b^4*c*e^4 + 10*a*b^2*c^2*e^4 - 56
*a^2*c^3*e^4 + b^3*c*Sqrt[b^2 - 4*a*c]*e^4 - 8*a*b*c^2*Sqrt[b^2 - 4*a*c]*e^4)*Ar
cTanh[(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)^2) - (3*(32*c^5*d^4 - 64*b*c^4*d^3*e + 8*c^4*Sqrt[b^2
- 4*a*c]*d^3*e + 30*b^2*c^3*d^2*e^2 + 72*a*c^4*d^2*e^2 - 12*b*c^3*Sqrt[b^2 - 4*a
*c]*d^2*e^2 + 2*b^3*c^2*d*e^3 - 72*a*b*c^3*d*e^3 + 2*b^2*c^2*Sqrt[b^2 - 4*a*c]*d
*e^3 + 16*a*c^3*Sqrt[b^2 - 4*a*c]*d*e^3 + b^4*c*e^4 - 10*a*b^2*c^2*e^4 + 56*a^2*
c^3*e^4 + b^3*c*Sqrt[b^2 - 4*a*c]*e^4 - 8*a*b*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*Sq
rt[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)^2)

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Maple [B]  time = 0.19, size = 4470, normalized size = 5.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

42*e^3*c^4/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^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)*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+22*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)^(1/2)*a-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)^(1/2)*d+42*e^3*c^4/(
-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^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)*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-24*e*c^5/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-
b^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)*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^2-9/2*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/(-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^2-22*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)^(1/2)*a+11/2*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)^(1/2)*b^2-18*e^3*c^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/(-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)*a+9/2*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/(-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^2-24*e*c^5/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^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)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arc
tanh(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+6*e*c^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*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-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-33/2*e^3*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^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)*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+6*e*c^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*
(-4*a*c*e^2+b^2*e^2)^(1/2)/(-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-33/2*e^3*c^3
/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^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)*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^2-3*e^2*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)/(-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+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)^(1/2)*b-3*e^2*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)/(-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+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)^(1/2)*b+18*e^3*c^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/(-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)*a+15*e^2*
c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^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)*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-30*e*c
^4/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^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)*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^4/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^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)*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+30*e*c^4/(-e^2*(4*a*c-b^2))^(1/2)/
(4*a*c-b^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)*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+24*e^2*c^4/(-e^2*(4*a*c-b^2))^(1/2)/(4
*a*c-b^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)*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-15*e^2*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^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)*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-11/2*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)^(1/2)*b^2-6*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)^(1/2)*d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{3} \sqrt{e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^3*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

integrate(1/((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(1/((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(1/(e*x+d)**(1/2)/(c*x**2+b*x+a)**3,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(1/((c*x^2 + b*x + a)^3*sqrt(e*x + d)),x, algorithm="giac")

[Out]

Timed out