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

Optimal. Leaf size=224 \[ -\frac{\left (16 c^2 \left (3 a^2-8 a d+8 d^2\right )+8 b^2 c (a-4 d)+3 b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a-d} (b+2 c x)}{\sqrt{b^2-4 c d} \sqrt{a+b x+c x^2}}\right )}{4 (a-d)^{5/2} \left (b^2-4 c d\right )^{5/2}}+\frac{3 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt{a+b x+c x^2}}{4 (a-d)^2 \left (b^2-4 c d\right )^2 \left (b x+c x^2+d\right )}-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.971241, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{\left (16 c^2 \left (3 a^2-8 a d+8 d^2\right )+8 b^2 c (a-4 d)+3 b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a-d} (b+2 c x)}{\sqrt{b^2-4 c d} \sqrt{a+b x+c x^2}}\right )}{4 (a-d)^{5/2} \left (b^2-4 c d\right )^{5/2}}+\frac{3 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt{a+b x+c x^2}}{4 (a-d)^2 \left (b^2-4 c d\right )^2 \left (b x+c x^2+d\right )}-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

[Out]

Timed out

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Mathematica [B]  time = 1.11885, size = 486, normalized size = 2.17 \[ \frac{\left (16 c^2 \left (3 a^2-8 a d+8 d^2\right )+8 b^2 c (a-4 d)+3 b^4\right ) (x (b+c x)+d)^2 \log \left (-\sqrt{b^2-4 c d}+b+2 c x\right )-\left (16 c^2 \left (3 a^2-8 a d+8 d^2\right )+8 b^2 c (a-4 d)+3 b^4\right ) (x (b+c x)+d)^2 \log \left (\sqrt{b^2-4 c d}+b+2 c x\right )+\left (16 c^2 \left (3 a^2-8 a d+8 d^2\right )+8 b^2 c (a-4 d)+3 b^4\right ) (x (b+c x)+d)^2 \log \left (2 c \left (-2 \sqrt{a-d} \sqrt{a+x (b+c x)}-2 a+x \sqrt{b^2-4 c d}\right )+b \sqrt{b^2-4 c d}+b^2\right )-\left (16 c^2 \left (3 a^2-8 a d+8 d^2\right )+8 b^2 c (a-4 d)+3 b^4\right ) (x (b+c x)+d)^2 \log \left (2 c \left (2 \sqrt{a-d} \sqrt{a+x (b+c x)}+2 a+x \sqrt{b^2-4 c d}\right )+b \sqrt{b^2-4 c d}-b^2\right )-2 \sqrt{a-d} \sqrt{b^2-4 c d} (b+2 c x) \sqrt{a+x (b+c x)} \left (2 (a-d) \left (b^2-4 c d\right )-3 \left (4 c (a-2 d)+b^2\right ) (x (b+c x)+d)\right )}{8 (a-d)^{5/2} \left (b^2-4 c d\right )^{5/2} (x (b+c x)+d)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a - d]*Sqrt[b^2 - 4*c*d]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(2*(a - d)*(
b^2 - 4*c*d) - 3*(b^2 + 4*c*(a - 2*d))*(d + x*(b + c*x))) + (3*b^4 + 8*b^2*c*(a
- 4*d) + 16*c^2*(3*a^2 - 8*a*d + 8*d^2))*(d + x*(b + c*x))^2*Log[b - Sqrt[b^2 -
4*c*d] + 2*c*x] - (3*b^4 + 8*b^2*c*(a - 4*d) + 16*c^2*(3*a^2 - 8*a*d + 8*d^2))*(
d + x*(b + c*x))^2*Log[b + Sqrt[b^2 - 4*c*d] + 2*c*x] + (3*b^4 + 8*b^2*c*(a - 4*
d) + 16*c^2*(3*a^2 - 8*a*d + 8*d^2))*(d + x*(b + c*x))^2*Log[b^2 + b*Sqrt[b^2 -
4*c*d] + 2*c*(-2*a + Sqrt[b^2 - 4*c*d]*x - 2*Sqrt[a - d]*Sqrt[a + x*(b + c*x)])]
- (3*b^4 + 8*b^2*c*(a - 4*d) + 16*c^2*(3*a^2 - 8*a*d + 8*d^2))*(d + x*(b + c*x)
)^2*Log[-b^2 + b*Sqrt[b^2 - 4*c*d] + 2*c*(2*a + Sqrt[b^2 - 4*c*d]*x + 2*Sqrt[a -
d]*Sqrt[a + x*(b + c*x)])])/(8*(a - d)^(5/2)*(b^2 - 4*c*d)^(5/2)*(d + x*(b + c*
x))^2)

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Maple [B]  time = 0.037, size = 1884, normalized size = 8.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(b^2-4*c*d)^(3/2)/(a-d)/(x+1/2*b/c-1/2/c*(b^2-4*c*d)^(1/2))^2*((x-1/2*(-b+(
b^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)
^(1/2)+3/4/(b^2-4*c*d)/(a-d)^2/(x+1/2*b/c-1/2/c*(b^2-4*c*d)^(1/2))*((x-1/2*(-b+(
b^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)
^(1/2)-3/8/(b^2-4*c*d)^(3/2)/(a-d)^(5/2)*ln((2*a-2*d+(b^2-4*c*d)^(1/2)*(x-1/2*(-
b+(b^2-4*c*d)^(1/2))/c)+2*(a-d)^(1/2)*((x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c+(b^2
-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2))/(x-1/2*(-b+(b^2-4*c*d
)^(1/2))/c))*b^2+3/2/(b^2-4*c*d)^(3/2)/(a-d)^(5/2)*ln((2*a-2*d+(b^2-4*c*d)^(1/2)
*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+2*(a-d)^(1/2)*((x-1/2*(-b+(b^2-4*c*d)^(1/2))/c
)^2*c+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2))/(x-1/2*(-b+
(b^2-4*c*d)^(1/2))/c))*c*d-1/(b^2-4*c*d)^(3/2)*c/(a-d)^(3/2)*ln((2*a-2*d+(b^2-4*
c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+2*(a-d)^(1/2)*((x-1/2*(-b+(b^2-4*c*d
)^(1/2))/c)^2*c+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2))/(
x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))+3/(b^2-4*c*d)^2*c/(a-d)/(x+1/2*b/c-1/2/c*(b^2-4
*c*d)^(1/2))*((x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+
(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2)-6*c^2/(b^2-4*c*d)^(5/2)/(a-d)^(1/2)*ln((2*a-2*d
+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+2*(a-d)^(1/2)*((x-1/2*(-b+(b
^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^
(1/2))/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))+1/2/(b^2-4*c*d)^(3/2)/(a-d)/(x+1/2*b/c+
1/2/c*(b^2-4*c*d)^(1/2))^2*((x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-(b^2-4*c*d)^(1/2
)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2)+3/4/(b^2-4*c*d)/(a-d)^2/(x+1/2*b/c+
1/2/c*(b^2-4*c*d)^(1/2))*((x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-(b^2-4*c*d)^(1/2)*
(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2)+3/8/(b^2-4*c*d)^(3/2)/(a-d)^(5/2)*ln(
(2*a-2*d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)+2*(a-d)^(1/2)*((x+1/2
*(b+(b^2-4*c*d)^(1/2))/c)^2*c-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)+
a-d)^(1/2))/(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))*b^2-3/2/(b^2-4*c*d)^(3/2)/(a-d)^(5/
2)*ln((2*a-2*d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)+2*(a-d)^(1/2)*(
(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2
))/c)+a-d)^(1/2))/(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))*c*d+1/(b^2-4*c*d)^(3/2)*c/(a-
d)^(3/2)*ln((2*a-2*d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)+2*(a-d)^(
1/2)*((x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d
)^(1/2))/c)+a-d)^(1/2))/(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))+3/(b^2-4*c*d)^2*c/(a-d)
/(x+1/2*b/c+1/2/c*(b^2-4*c*d)^(1/2))*((x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-(b^2-4
*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2)+6*c^2/(b^2-4*c*d)^(5/2)/(
a-d)^(1/2)*ln((2*a-2*d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)+2*(a-d)
^(1/2)*((x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c
*d)^(1/2))/c)+a-d)^(1/2))/(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(c*x^2 + b*x + a)*(c*x^2 + b*x + d)^3), x)

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Fricas [A]  time = 1.79157, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(4*(2*a*b^3 + 32*b*c*d^2 - 6*(b^2*c^2 + 4*a*c^3 - 8*c^3*d)*x^3 - 9*(b^3*c
+ 4*a*b*c^2 - 8*b*c^2*d)*x^2 - 5*(b^3 + 4*a*b*c)*d - (3*b^4 + 8*a*b^2*c - 64*c^
2*d^2 - 2*(7*b^2*c - 20*a*c^2)*d)*x)*sqrt(a*b^2 + 4*c*d^2 - (b^2 + 4*a*c)*d)*sqr
t(c*x^2 + b*x + a) - (128*c^2*d^4 + (3*b^4*c^2 + 8*a*b^2*c^3 + 48*a^2*c^4 + 128*
c^4*d^2 - 32*(b^2*c^3 + 4*a*c^4)*d)*x^4 - 32*(b^2*c + 4*a*c^2)*d^3 + 2*(3*b^5*c
+ 8*a*b^3*c^2 + 48*a^2*b*c^3 + 128*b*c^3*d^2 - 32*(b^3*c^2 + 4*a*b*c^3)*d)*x^3 +
(3*b^4 + 8*a*b^2*c + 48*a^2*c^2)*d^2 + (3*b^6 + 8*a*b^4*c + 48*a^2*b^2*c^2 + 25
6*c^3*d^3 + 64*(b^2*c^2 - 4*a*c^3)*d^2 - 2*(13*b^4*c + 56*a*b^2*c^2 - 48*a^2*c^3
)*d)*x^2 + 2*(128*b*c^2*d^3 - 32*(b^3*c + 4*a*b*c^2)*d^2 + (3*b^5 + 8*a*b^3*c +
48*a^2*b*c^2)*d)*x)*log(((8*a^2*b^4 + (b^4*c^2 + 24*a*b^2*c^3 + 16*a^2*c^4 + 128
*c^4*d^2 - 32*(b^2*c^3 + 4*a*c^4)*d)*x^4 + 2*(b^5*c + 24*a*b^3*c^2 + 16*a^2*b*c^
3 + 128*b*c^3*d^2 - 32*(b^3*c^2 + 4*a*b*c^3)*d)*x^3 + (b^4 + 24*a*b^2*c + 16*a^2
*c^2)*d^2 + (b^6 + 32*a*b^4*c + 48*a^2*b^2*c^2 + 32*(5*b^2*c^2 + 4*a*c^3)*d^2 -
2*(19*b^4*c + 104*a*b^2*c^2 + 48*a^2*c^3)*d)*x^2 - 8*(a*b^4 + 4*a^2*b^2*c)*d + 2
*(4*a*b^5 + 16*a^2*b^3*c + 16*(b^3*c + 4*a*b*c^2)*d^2 - (3*b^5 + 40*a*b^3*c + 48
*a^2*b*c^2)*d)*x)*sqrt(a*b^2 + 4*c*d^2 - (b^2 + 4*a*c)*d) - 4*(2*a^2*b^5 - 4*(b^
3*c + 4*a*b*c^2)*d^3 + 2*(a*b^4*c^2 + 4*a^2*b^2*c^3 - 32*c^4*d^3 + 12*(b^2*c^3 +
4*a*c^4)*d^2 - (b^4*c^2 + 16*a*b^2*c^3 + 16*a^2*c^4)*d)*x^3 + (b^5 + 16*a*b^3*c
+ 16*a^2*b*c^2)*d^2 + 3*(a*b^5*c + 4*a^2*b^3*c^2 - 32*b*c^3*d^3 + 12*(b^3*c^2 +
4*a*b*c^3)*d^2 - (b^5*c + 16*a*b^3*c^2 + 16*a^2*b*c^3)*d)*x^2 - 3*(a*b^5 + 4*a^
2*b^3*c)*d + (a*b^6 + 8*a^2*b^4*c - 8*(5*b^2*c^2 + 4*a*c^3)*d^3 + 2*(7*b^4*c + 4
0*a*b^2*c^2 + 16*a^2*c^3)*d^2 - (b^6 + 22*a*b^4*c + 40*a^2*b^2*c^2)*d)*x)*sqrt(c
*x^2 + b*x + a))/(c^2*x^4 + 2*b*c*x^3 + 2*b*d*x + (b^2 + 2*c*d)*x^2 + d^2)))/((a
^2*b^4*d^2 + 16*c^2*d^6 - 8*(b^2*c + 4*a*c^2)*d^5 + (b^4 + 16*a*b^2*c + 16*a^2*c
^2)*d^4 + (a^2*b^4*c^2 + 16*c^4*d^4 - 8*(b^2*c^3 + 4*a*c^4)*d^3 + (b^4*c^2 + 16*
a*b^2*c^3 + 16*a^2*c^4)*d^2 - 2*(a*b^4*c^2 + 4*a^2*b^2*c^3)*d)*x^4 - 2*(a*b^4 +
4*a^2*b^2*c)*d^3 + 2*(a^2*b^5*c + 16*b*c^3*d^4 - 8*(b^3*c^2 + 4*a*b*c^3)*d^3 + (
b^5*c + 16*a*b^3*c^2 + 16*a^2*b*c^3)*d^2 - 2*(a*b^5*c + 4*a^2*b^3*c^2)*d)*x^3 +
(a^2*b^6 - 64*a*c^3*d^4 + 32*c^3*d^5 - 2*(3*b^4*c - 16*a^2*c^3)*d^3 + (b^6 + 12*
a*b^4*c)*d^2 - 2*(a*b^6 + 3*a^2*b^4*c)*d)*x^2 + 2*(a^2*b^5*d + 16*b*c^2*d^5 - 8*
(b^3*c + 4*a*b*c^2)*d^4 + (b^5 + 16*a*b^3*c + 16*a^2*b*c^2)*d^3 - 2*(a*b^5 + 4*a
^2*b^3*c)*d^2)*x)*sqrt(a*b^2 + 4*c*d^2 - (b^2 + 4*a*c)*d)), -1/8*(2*(2*a*b^3 + 3
2*b*c*d^2 - 6*(b^2*c^2 + 4*a*c^3 - 8*c^3*d)*x^3 - 9*(b^3*c + 4*a*b*c^2 - 8*b*c^2
*d)*x^2 - 5*(b^3 + 4*a*b*c)*d - (3*b^4 + 8*a*b^2*c - 64*c^2*d^2 - 2*(7*b^2*c - 2
0*a*c^2)*d)*x)*sqrt(-a*b^2 - 4*c*d^2 + (b^2 + 4*a*c)*d)*sqrt(c*x^2 + b*x + a) -
(128*c^2*d^4 + (3*b^4*c^2 + 8*a*b^2*c^3 + 48*a^2*c^4 + 128*c^4*d^2 - 32*(b^2*c^3
+ 4*a*c^4)*d)*x^4 - 32*(b^2*c + 4*a*c^2)*d^3 + 2*(3*b^5*c + 8*a*b^3*c^2 + 48*a^
2*b*c^3 + 128*b*c^3*d^2 - 32*(b^3*c^2 + 4*a*b*c^3)*d)*x^3 + (3*b^4 + 8*a*b^2*c +
48*a^2*c^2)*d^2 + (3*b^6 + 8*a*b^4*c + 48*a^2*b^2*c^2 + 256*c^3*d^3 + 64*(b^2*c
^2 - 4*a*c^3)*d^2 - 2*(13*b^4*c + 56*a*b^2*c^2 - 48*a^2*c^3)*d)*x^2 + 2*(128*b*c
^2*d^3 - 32*(b^3*c + 4*a*b*c^2)*d^2 + (3*b^5 + 8*a*b^3*c + 48*a^2*b*c^2)*d)*x)*a
rctan(-1/2*(2*a*b^2 + (b^2*c + 4*a*c^2 - 8*c^2*d)*x^2 - (b^2 + 4*a*c)*d + (b^3 +
4*a*b*c - 8*b*c*d)*x)*sqrt(-a*b^2 - 4*c*d^2 + (b^2 + 4*a*c)*d)/((a*b^3 + 4*b*c*
d^2 - (b^3 + 4*a*b*c)*d + 2*(a*b^2*c + 4*c^2*d^2 - (b^2*c + 4*a*c^2)*d)*x)*sqrt(
c*x^2 + b*x + a))))/((a^2*b^4*d^2 + 16*c^2*d^6 - 8*(b^2*c + 4*a*c^2)*d^5 + (b^4
+ 16*a*b^2*c + 16*a^2*c^2)*d^4 + (a^2*b^4*c^2 + 16*c^4*d^4 - 8*(b^2*c^3 + 4*a*c^
4)*d^3 + (b^4*c^2 + 16*a*b^2*c^3 + 16*a^2*c^4)*d^2 - 2*(a*b^4*c^2 + 4*a^2*b^2*c^
3)*d)*x^4 - 2*(a*b^4 + 4*a^2*b^2*c)*d^3 + 2*(a^2*b^5*c + 16*b*c^3*d^4 - 8*(b^3*c
^2 + 4*a*b*c^3)*d^3 + (b^5*c + 16*a*b^3*c^2 + 16*a^2*b*c^3)*d^2 - 2*(a*b^5*c + 4
*a^2*b^3*c^2)*d)*x^3 + (a^2*b^6 - 64*a*c^3*d^4 + 32*c^3*d^5 - 2*(3*b^4*c - 16*a^
2*c^3)*d^3 + (b^6 + 12*a*b^4*c)*d^2 - 2*(a*b^6 + 3*a^2*b^4*c)*d)*x^2 + 2*(a^2*b^
5*d + 16*b*c^2*d^5 - 8*(b^3*c + 4*a*b*c^2)*d^4 + (b^5 + 16*a*b^3*c + 16*a^2*b*c^
2)*d^3 - 2*(a*b^5 + 4*a^2*b^3*c)*d^2)*x)*sqrt(-a*b^2 - 4*c*d^2 + (b^2 + 4*a*c)*d
))]

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

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError