3.2290 \(\int \frac{1}{x^{5/2} \left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=360 \[ \frac{b \left (5 b^2-19 a c\right )}{a^3 \sqrt{x} \left (b^2-4 a c\right )}-\frac{5 b^2-14 a c}{3 a^2 x^{3/2} \left (b^2-4 a c\right )}+\frac{\sqrt{c} \left (28 a^2 c^2-29 a b^2 c+b \left (5 b^2-19 a c\right ) \sqrt{b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (28 a^2 c^2-29 a b^2 c-b \left (5 b^2-19 a c\right ) \sqrt{b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{-2 a c+b^2+b c x}{a x^{3/2} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[Out]

-(5*b^2 - 14*a*c)/(3*a^2*(b^2 - 4*a*c)*x^(3/2)) + (b*(5*b^2 - 19*a*c))/(a^3*(b^2
- 4*a*c)*Sqrt[x]) + (b^2 - 2*a*c + b*c*x)/(a*(b^2 - 4*a*c)*x^(3/2)*(a + b*x + c
*x^2)) + (Sqrt[c]*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + b*(5*b^2 - 19*a*c)*Sqrt[b^2
- 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[
2]*a^3*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[c]*(5*b^4 - 29*a
*b^2*c + 28*a^2*c^2 - b*(5*b^2 - 19*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt
[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a^3*(b^2 - 4*a*c)^(3/2)*Sqrt
[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 7.27618, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{b \left (5 b^2-19 a c\right )}{a^3 \sqrt{x} \left (b^2-4 a c\right )}-\frac{5 b^2-14 a c}{3 a^2 x^{3/2} \left (b^2-4 a c\right )}+\frac{\sqrt{c} \left (28 a^2 c^2-29 a b^2 c+b \left (5 b^2-19 a c\right ) \sqrt{b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (28 a^2 c^2-29 a b^2 c-b \left (5 b^2-19 a c\right ) \sqrt{b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{-2 a c+b^2+b c x}{a x^{3/2} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(a + b*x + c*x^2)^2),x]

[Out]

-(5*b^2 - 14*a*c)/(3*a^2*(b^2 - 4*a*c)*x^(3/2)) + (b*(5*b^2 - 19*a*c))/(a^3*(b^2
- 4*a*c)*Sqrt[x]) + (b^2 - 2*a*c + b*c*x)/(a*(b^2 - 4*a*c)*x^(3/2)*(a + b*x + c
*x^2)) + (Sqrt[c]*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + b*(5*b^2 - 19*a*c)*Sqrt[b^2
- 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[
2]*a^3*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[c]*(5*b^4 - 29*a
*b^2*c + 28*a^2*c^2 - b*(5*b^2 - 19*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt
[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a^3*(b^2 - 4*a*c)^(3/2)*Sqrt
[b + Sqrt[b^2 - 4*a*c]])

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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/x**(5/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 1.38687, size = 353, normalized size = 0.98 \[ \frac{\frac{6 \sqrt{x} \left (2 a^2 c^2-4 a b^2 c-3 a b c^2 x+b^4+b^3 c x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac{3 \sqrt{2} \sqrt{c} \left (28 a^2 c^2-29 a b^2 c-19 a b c \sqrt{b^2-4 a c}+5 b^3 \sqrt{b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \left (-28 a^2 c^2+29 a b^2 c-19 a b c \sqrt{b^2-4 a c}+5 b^3 \sqrt{b^2-4 a c}-5 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 a}{x^{3/2}}+\frac{24 b}{\sqrt{x}}}{6 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*(a + b*x + c*x^2)^2),x]

[Out]

((-4*a)/x^(3/2) + (24*b)/Sqrt[x] + (6*Sqrt[x]*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3
*c*x - 3*a*b*c^2*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (3*Sqrt[2]*Sqrt[c]*(5*b
^4 - 29*a*b^2*c + 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*c*Sqrt[b^2 - 4*a
*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*
c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(-5*b^4 + 29*a*b^2*c
- 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqr
t[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b
+ Sqrt[b^2 - 4*a*c]]))/(6*a^3)

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Maple [B]  time = 0.127, size = 2377, normalized size = 6.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(c*x^2+b*x+a)^2,x)

[Out]

-2/3/a^2/x^(3/2)+4/a^3*b/x^(1/2)+3/a^2/(c*x^2+b*x+a)*b*c^2/(4*a*c-b^2)*x^(3/2)-1
/a^3/(c*x^2+b*x+a)*b^3*c/(4*a*c-b^2)*x^(3/2)-2/a/(c*x^2+b*x+a)/(4*a*c-b^2)*x^(1/
2)*c^2+4/a^2/(c*x^2+b*x+a)/(4*a*c-b^2)*x^(1/2)*c*b^2-1/a^3/(c*x^2+b*x+a)/(4*a*c-
b^2)*x^(1/2)*b^4+224*a/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c-b^2)
*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x^(1/2
)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*c^5-344/(-
(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^
2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x^(1/2)*2^(1/2)/(c*(4*a*c-b^2)*
(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*c^4*b^2+170/a/(-(4*a*c-b^2)^3)^(1/2
)/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)
*arctan(1/2*(8*a*c^2-2*b^2*c)*x^(1/2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a
*c-b^2)^3)^(1/2)))^(1/2))*b^4*c^3-69/2/a^2/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^
(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a
*c^2-2*b^2*c)*x^(1/2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)
))^(1/2))*b^6*c^2+5/2/a^3*c/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c
-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x
^(1/2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*b^8+3
8/a/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/
2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x^(1/2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4
*a*c-b^2)^3)^(1/2)))^(1/2))*b*c^3-39/2/a^2/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c-b^2)*(4
*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x^(1/2)*2
^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*b^3*c^2+5/2/a
^3*c/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1
/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x^(1/2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(
4*a*c-b^2)^3)^(1/2)))^(1/2))*b^5-224*a/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2
)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*
c^2+2*b^2*c)*x^(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2
))^(1/2))*c^5+344/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4
*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x^(1/2)*
2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2))*c^4*b^2-170
/a/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1
/2))*c*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x^(1/2)*2^(1/2)/((-4*a*
b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2))*b^4*c^3+69/2/a^2/(-(4*a*c-
b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*
c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x^(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4
*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2))*b^6*c^2-5/2/a^3*c/(-(4*a*c-b^2)^3)^(1/
2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/
2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x^(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3
)^(1/2))*c*(4*a*c-b^2))^(1/2))*b^8+38/a/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*
a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x^(1/2)*2
^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2))*b*c^3-39/2/a
^2/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/
2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x^(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3
)^(1/2))*c*(4*a*c-b^2))^(1/2))*b^3*c^2+5/2/a^3*c/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+
b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*
x^(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2))*b^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 \,{\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 14 \, a^{2} c^{3}\right )} x^{\frac{5}{2}} + 3 \,{\left (5 \, b^{5} - 19 \, a b^{3} c - 5 \, a^{2} b c^{2}\right )} x^{\frac{3}{2}} + 2 \,{\left (15 \, a b^{4} - 67 \, a^{2} b^{2} c + 28 \, a^{3} c^{2}\right )} \sqrt{x} + \frac{10 \,{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )}}{\sqrt{x}} - \frac{2 \,{\left (a^{3} b^{2} - 4 \, a^{4} c\right )}}{x^{\frac{3}{2}}}}{3 \,{\left (a^{5} b^{2} - 4 \, a^{6} c +{\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{2} +{\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x\right )}} + \int -\frac{{\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 14 \, a^{2} c^{3}\right )} x^{\frac{3}{2}} +{\left (5 \, b^{5} - 29 \, a b^{3} c + 33 \, a^{2} b c^{2}\right )} \sqrt{x}}{2 \,{\left (a^{5} b^{2} - 4 \, a^{6} c +{\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{2} +{\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^2*x^(5/2)),x, algorithm="maxima")

[Out]

1/3*(3*(5*b^4*c - 24*a*b^2*c^2 + 14*a^2*c^3)*x^(5/2) + 3*(5*b^5 - 19*a*b^3*c - 5
*a^2*b*c^2)*x^(3/2) + 2*(15*a*b^4 - 67*a^2*b^2*c + 28*a^3*c^2)*sqrt(x) + 10*(a^2
*b^3 - 4*a^3*b*c)/sqrt(x) - 2*(a^3*b^2 - 4*a^4*c)/x^(3/2))/(a^5*b^2 - 4*a^6*c +
(a^4*b^2*c - 4*a^5*c^2)*x^2 + (a^4*b^3 - 4*a^5*b*c)*x) + integrate(-1/2*((5*b^4*
c - 24*a*b^2*c^2 + 14*a^2*c^3)*x^(3/2) + (5*b^5 - 29*a*b^3*c + 33*a^2*b*c^2)*sqr
t(x))/(a^5*b^2 - 4*a^6*c + (a^4*b^2*c - 4*a^5*c^2)*x^2 + (a^4*b^3 - 4*a^5*b*c)*x
), x)

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Fricas [A]  time = 1.01452, size = 4655, normalized size = 12.93 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^2*x^(5/2)),x, algorithm="fricas")

[Out]

-1/6*(4*a^2*b^2 - 16*a^3*c - 6*(5*b^3*c - 19*a*b*c^2)*x^3 - 3*sqrt(1/2)*((a^3*b^
2*c - 4*a^4*c^2)*x^3 + (a^3*b^3 - 4*a^4*b*c)*x^2 + (a^4*b^2 - 4*a^5*c)*x)*sqrt(x
)*sqrt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b
*c^4 + (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 -
8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 2410
8*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a
^17*c^3)))/(a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))*log(sqrt(1/2
)*(125*b^14 - 2425*a*b^12*c + 18940*a^2*b^10*c^2 - 75579*a^3*b^8*c^3 + 160932*a^
4*b^6*c^4 - 172990*a^5*b^4*c^5 + 79408*a^6*b^2*c^6 - 10976*a^7*c^7 - (5*a^7*b^11
- 94*a^8*b^9*c + 700*a^9*b^7*c^2 - 2576*a^10*b^5*c^3 + 4672*a^11*b^3*c^4 - 3328
*a^12*b*c^5)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*
c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*
b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))*sqrt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*
b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4 + (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b
^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630
*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 -
12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^8*b^4*c + 48*a
^9*b^2*c^2 - 64*a^10*c^3)) + 2*(1125*b^8*c^4 - 12325*a*b^6*c^5 + 43410*a^2*b^4*c
^6 - 50421*a^3*b^2*c^7 + 9604*a^4*c^8)*sqrt(x)) + 3*sqrt(1/2)*((a^3*b^2*c - 4*a^
4*c^2)*x^3 + (a^3*b^3 - 4*a^4*b*c)*x^2 + (a^4*b^2 - 4*a^5*c)*x)*sqrt(x)*sqrt(-(2
5*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4 + (a^
7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^1
0*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*
c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))
/(a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))*log(-sqrt(1/2)*(125*b^
14 - 2425*a*b^12*c + 18940*a^2*b^10*c^2 - 75579*a^3*b^8*c^3 + 160932*a^4*b^6*c^4
- 172990*a^5*b^4*c^5 + 79408*a^6*b^2*c^6 - 10976*a^7*c^7 - (5*a^7*b^11 - 94*a^8
*b^9*c + 700*a^9*b^7*c^2 - 2576*a^10*b^5*c^3 + 4672*a^11*b^3*c^4 - 3328*a^12*b*c
^5)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 766
86*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 4
8*a^16*b^2*c^2 - 64*a^17*c^3)))*sqrt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 -
2415*a^3*b^3*c^3 + 1260*a^4*b*c^4 + (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 -
64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*
c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*
b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^
2 - 64*a^10*c^3)) + 2*(1125*b^8*c^4 - 12325*a*b^6*c^5 + 43410*a^2*b^4*c^6 - 5042
1*a^3*b^2*c^7 + 9604*a^4*c^8)*sqrt(x)) - 3*sqrt(1/2)*((a^3*b^2*c - 4*a^4*c^2)*x^
3 + (a^3*b^3 - 4*a^4*b*c)*x^2 + (a^4*b^2 - 4*a^5*c)*x)*sqrt(x)*sqrt(-(25*b^9 - 3
15*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4 - (a^7*b^6 - 1
2*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 395
25*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 240
1*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6
- 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))*log(sqrt(1/2)*(125*b^14 - 2425*
a*b^12*c + 18940*a^2*b^10*c^2 - 75579*a^3*b^8*c^3 + 160932*a^4*b^6*c^4 - 172990*
a^5*b^4*c^5 + 79408*a^6*b^2*c^6 - 10976*a^7*c^7 + (5*a^7*b^11 - 94*a^8*b^9*c + 7
00*a^9*b^7*c^2 - 2576*a^10*b^5*c^3 + 4672*a^11*b^3*c^4 - 3328*a^12*b*c^5)*sqrt((
625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4
*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2
*c^2 - 64*a^17*c^3)))*sqrt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*
b^3*c^3 + 1260*a^4*b*c^4 - (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^
3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 7668
6*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48
*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^1
0*c^3)) + 2*(1125*b^8*c^4 - 12325*a*b^6*c^5 + 43410*a^2*b^4*c^6 - 50421*a^3*b^2*
c^7 + 9604*a^4*c^8)*sqrt(x)) + 3*sqrt(1/2)*((a^3*b^2*c - 4*a^4*c^2)*x^3 + (a^3*b
^3 - 4*a^4*b*c)*x^2 + (a^4*b^2 - 4*a^5*c)*x)*sqrt(x)*sqrt(-(25*b^9 - 315*a*b^7*c
+ 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4 - (a^7*b^6 - 12*a^8*b^4*
c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8
*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)
/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^8*
b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))*log(-sqrt(1/2)*(125*b^14 - 2425*a*b^12*c
+ 18940*a^2*b^10*c^2 - 75579*a^3*b^8*c^3 + 160932*a^4*b^6*c^4 - 172990*a^5*b^4*c
^5 + 79408*a^6*b^2*c^6 - 10976*a^7*c^7 + (5*a^7*b^11 - 94*a^8*b^9*c + 700*a^9*b^
7*c^2 - 2576*a^10*b^5*c^3 + 4672*a^11*b^3*c^4 - 3328*a^12*b*c^5)*sqrt((625*b^12
- 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24
108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64
*a^17*c^3)))*sqrt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 +
1260*a^4*b*c^4 - (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((
625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4
*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2
*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)) +
2*(1125*b^8*c^4 - 12325*a*b^6*c^5 + 43410*a^2*b^4*c^6 - 50421*a^3*b^2*c^7 + 960
4*a^4*c^8)*sqrt(x)) - 2*(15*b^4 - 62*a*b^2*c + 14*a^2*c^2)*x^2 - 20*(a*b^3 - 4*a
^2*b*c)*x)/(((a^3*b^2*c - 4*a^4*c^2)*x^3 + (a^3*b^3 - 4*a^4*b*c)*x^2 + (a^4*b^2
- 4*a^5*c)*x)*sqrt(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{\frac{5}{2}} \left (a + b x + c x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(5/2)/(c*x**2+b*x+a)**2,x)

[Out]

Integral(1/(x**(5/2)*(a + b*x + c*x**2)**2), x)

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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)^2*x^(5/2)),x, algorithm="giac")

[Out]

Timed out