\(\int \frac {1}{x^{5/2} (a+b x+c x^2)^2} \, dx\) [279]

Optimal result

Integrand size = 18, antiderivative size = 360 \[ \int \frac {1}{x^{5/2} \left (a+b x+c x^2\right )^2} \, dx=-\frac {5 b^2-14 a c}{3 a^2 \left (b^2-4 a c\right ) x^{3/2}}+\frac {b \left (5 b^2-19 a c\right )}{a^3 \left (b^2-4 a c\right ) \sqrt {x}}+\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) x^{3/2} \left (a+b x+c x^2\right )}+\frac {\sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2+b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2-b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

-1/3*(-14*a*c+5*b^2)/a^2/(-4*a*c+b^2)/x^(3/2)+b*(-19*a*c+5*b^2)/a^3/(-4*a* 
c+b^2)/x^(1/2)+(b*c*x-2*a*c+b^2)/a/(-4*a*c+b^2)/x^(3/2)/(c*x^2+b*x+a)+1/2* 
c^(1/2)*(5*b^4-29*a*b^2*c+28*a^2*c^2+b*(-19*a*c+5*b^2)*(-4*a*c+b^2)^(1/2)) 
*arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a^3/ 
(-4*a*c+b^2)^(3/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-1/2*c^(1/2)*(5*b^4-29*a*b^ 
2*c+28*a^2*c^2-b*(-19*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2 
)*x^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a^3/(-4*a*c+b^2)^(3/2)/(b+ 
(-4*a*c+b^2)^(1/2))^(1/2)
 

Mathematica [A] (verified)

Time = 2.28 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^{5/2} \left (a+b x+c x^2\right )^2} \, dx=\frac {-\frac {2 \left (8 a^3 c+15 b^3 x^2 (b+c x)+a b x \left (10 b^2-62 b c x-57 c^2 x^2\right )-2 a^2 \left (b^2+20 b c x-7 c^2 x^2\right )\right )}{x^{3/2} (a+x (b+c x))}-\frac {3 \sqrt {2} \sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2+5 b^3 \sqrt {b^2-4 a c}-19 a b c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {2} \sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2-5 b^3 \sqrt {b^2-4 a c}+19 a b c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{6 a^3 \left (-b^2+4 a c\right )} \] Input:

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

Output:

((-2*(8*a^3*c + 15*b^3*x^2*(b + c*x) + a*b*x*(10*b^2 - 62*b*c*x - 57*c^2*x 
^2) - 2*a^2*(b^2 + 20*b*c*x - 7*c^2*x^2)))/(x^(3/2)*(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]]])/(Sqrt[b^2 - 4*a*c]*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[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + 
Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(6*a 
^3*(-b^2 + 4*a*c))
 

Rubi [A] (verified)

Time = 1.97 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1165, 27, 1198, 25, 1198, 25, 1197, 1480, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1165

\(\displaystyle \frac {-2 a c+b^2+b c x}{a x^{3/2} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int -\frac {5 b^2+5 c x b-14 a c}{2 x^{5/2} \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {5 b^2+5 c x b-14 a c}{x^{5/2} \left (c x^2+b x+a\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{a x^{3/2} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1198

\(\displaystyle \frac {\frac {\int -\frac {b \left (5 b^2-19 a c\right )+c \left (5 b^2-14 a c\right ) x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 \left (5 b^2-14 a c\right )}{3 a x^{3/2}}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{a x^{3/2} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {\int \frac {b \left (5 b^2-19 a c\right )+c \left (5 b^2-14 a c\right ) x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 \left (5 b^2-14 a c\right )}{3 a x^{3/2}}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{a x^{3/2} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1198

\(\displaystyle \frac {-\frac {\frac {\int -\frac {5 b^4-24 a c b^2+c \left (5 b^2-19 a c\right ) x b+14 a^2 c^2}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 b \left (5 b^2-19 a c\right )}{a \sqrt {x}}}{a}-\frac {2 \left (5 b^2-14 a c\right )}{3 a x^{3/2}}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{a x^{3/2} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {-\frac {\int \frac {5 b^4-24 a c b^2+c \left (5 b^2-19 a c\right ) x b+14 a^2 c^2}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 b \left (5 b^2-19 a c\right )}{a \sqrt {x}}}{a}-\frac {2 \left (5 b^2-14 a c\right )}{3 a x^{3/2}}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{a x^{3/2} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1197

\(\displaystyle \frac {-\frac {-\frac {2 \int \frac {5 b^4-24 a c b^2+c \left (5 b^2-19 a c\right ) x b+14 a^2 c^2}{c x^2+b x+a}d\sqrt {x}}{a}-\frac {2 b \left (5 b^2-19 a c\right )}{a \sqrt {x}}}{a}-\frac {2 \left (5 b^2-14 a c\right )}{3 a x^{3/2}}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{a x^{3/2} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {-\frac {-\frac {2 \left (\frac {c \left (28 a^2 c^2-29 a b^2 c+b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \int \frac {1}{\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}}{2 \sqrt {b^2-4 a c}}-\frac {c \left (28 a^2 c^2-29 a b^2 c-b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \int \frac {1}{\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}}{2 \sqrt {b^2-4 a c}}\right )}{a}-\frac {2 b \left (5 b^2-19 a c\right )}{a \sqrt {x}}}{a}-\frac {2 \left (5 b^2-14 a c\right )}{3 a x^{3/2}}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{a x^{3/2} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {-\frac {-\frac {2 \left (\frac {\sqrt {c} \left (28 a^2 c^2-29 a b^2 c+b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (28 a^2 c^2-29 a b^2 c-b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {2 b \left (5 b^2-19 a c\right )}{a \sqrt {x}}}{a}-\frac {2 \left (5 b^2-14 a c\right )}{3 a x^{3/2}}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{a x^{3/2} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 1165

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) 
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 
2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d 
+ e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p 
+ 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + 
 b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] 
 && IntQuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1197

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)), x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - 
b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr 
eeQ[{a, b, c, d, e, f, g}, x]
 

rule 1198

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + 
(c_.)*(x_)^2), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c 
*d^2 - b*d*e + a*e^2))), x] + Simp[1/(c*d^2 - b*d*e + a*e^2)   Int[(d + e*x 
)^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^ 
2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1 
]
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.93

method	result	size
risch	\(-\frac {2 \left (-6 b x +a \right )}{3 a^{3} x^{\frac {3}{2}}}+\frac {\frac {\frac {2 b c \left (3 a c -b^{2}\right ) x^{\frac {3}{2}}}{8 a c -2 b^{2}}-\frac {\left (2 a^{2} c^{2}-4 c a \,b^{2}+b^{4}\right ) \sqrt {x}}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {4 c \left (-\frac {\left (19 a b c \sqrt {-4 a c +b^{2}}-5 b^{3} \sqrt {-4 a c +b^{2}}-28 a^{2} c^{2}+29 c a \,b^{2}-5 b^{4}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (19 a b c \sqrt {-4 a c +b^{2}}-5 b^{3} \sqrt {-4 a c +b^{2}}+28 a^{2} c^{2}-29 c a \,b^{2}+5 b^{4}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}}{a^{3}}\)	\(335\)
derivativedivides	\(-\frac {2 \left (\frac {-\frac {b c \left (3 a c -b^{2}\right ) x^{\frac {3}{2}}}{2 \left (4 a c -b^{2}\right )}+\frac {\left (2 a^{2} c^{2}-4 c a \,b^{2}+b^{4}\right ) \sqrt {x}}{8 a c -2 b^{2}}}{c \,x^{2}+b x +a}+\frac {2 c \left (-\frac {\left (-19 a b c \sqrt {-4 a c +b^{2}}+5 b^{3} \sqrt {-4 a c +b^{2}}+28 a^{2} c^{2}-29 c a \,b^{2}+5 b^{4}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-19 a b c \sqrt {-4 a c +b^{2}}+5 b^{3} \sqrt {-4 a c +b^{2}}-28 a^{2} c^{2}+29 c a \,b^{2}-5 b^{4}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}\right )}{a^{3}}-\frac {2}{3 a^{2} x^{\frac {3}{2}}}+\frac {4 b}{a^{3} \sqrt {x}}\)	\(338\)
default	\(-\frac {2 \left (\frac {-\frac {b c \left (3 a c -b^{2}\right ) x^{\frac {3}{2}}}{2 \left (4 a c -b^{2}\right )}+\frac {\left (2 a^{2} c^{2}-4 c a \,b^{2}+b^{4}\right ) \sqrt {x}}{8 a c -2 b^{2}}}{c \,x^{2}+b x +a}+\frac {2 c \left (-\frac {\left (-19 a b c \sqrt {-4 a c +b^{2}}+5 b^{3} \sqrt {-4 a c +b^{2}}+28 a^{2} c^{2}-29 c a \,b^{2}+5 b^{4}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-19 a b c \sqrt {-4 a c +b^{2}}+5 b^{3} \sqrt {-4 a c +b^{2}}-28 a^{2} c^{2}+29 c a \,b^{2}-5 b^{4}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}\right )}{a^{3}}-\frac {2}{3 a^{2} x^{\frac {3}{2}}}+\frac {4 b}{a^{3} \sqrt {x}}\)	\(338\)

Input:

int(1/x^(5/2)/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-2/3*(-6*b*x+a)/a^3/x^(3/2)+1/a^3*(2*(1/2*b*c*(3*a*c-b^2)/(4*a*c-b^2)*x^(3 
/2)-1/2*(2*a^2*c^2-4*a*b^2*c+b^4)/(4*a*c-b^2)*x^(1/2))/(c*x^2+b*x+a)+4/(4* 
a*c-b^2)*c*(-1/8*(19*a*b*c*(-4*a*c+b^2)^(1/2)-5*b^3*(-4*a*c+b^2)^(1/2)-28* 
a^2*c^2+29*c*a*b^2-5*b^4)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/ 
2))*c)^(1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))+ 
1/8*(19*a*b*c*(-4*a*c+b^2)^(1/2)-5*b^3*(-4*a*c+b^2)^(1/2)+28*a^2*c^2-29*c* 
a*b^2+5*b^4)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*a 
rctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3449 vs. \(2 (309) = 618\).

Time = 1.31 (sec) , antiderivative size = 3449, normalized size of antiderivative = 9.58 \[ \int \frac {1}{x^{5/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^{5/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{x^{5/2} \left (a+b x+c x^2\right )^2} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{2} x^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

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)*sqrt(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)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3656 vs. \(2 (309) = 618\).

Time = 0.50 (sec) , antiderivative size = 3656, normalized size of antiderivative = 10.16 \[ \int \frac {1}{x^{5/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/x^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="giac")
 

Output:

(b^3*c*x^(3/2) - 3*a*b*c^2*x^(3/2) + b^4*sqrt(x) - 4*a*b^2*c*sqrt(x) + 2*a 
^2*c^2*sqrt(x))/((a^3*b^2 - 4*a^4*c)*(c*x^2 + b*x + a)) - 1/8*(10*a^6*b^9* 
c^2 - 138*a^7*b^7*c^3 + 680*a^8*b^5*c^4 - 1376*a^9*b^3*c^5 + 896*a^10*b*c^ 
6 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^9 + 
69*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^7*c + 1 
0*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^8*c - 34 
0*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^5*c^2 - 
98*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^6*c^2 - 
 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^7*c^2 + 
 688*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^3*c^3 
 + 288*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^4*c 
^3 + 49*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^5* 
c^3 - 448*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^10*b 
*c^4 - 224*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b 
^2*c^4 - 144*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8 
*b^3*c^4 + 112*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a 
^9*b*c^5 - 10*(b^2 - 4*a*c)*a^6*b^7*c^2 + 98*(b^2 - 4*a*c)*a^7*b^5*c^3 - 2 
88*(b^2 - 4*a*c)*a^8*b^3*c^4 + 224*(b^2 - 4*a*c)*a^9*b*c^5 + (10*b^5*c^2 - 
 78*a*b^3*c^3 + 152*a^2*b*c^4 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqr 
t(b^2 - 4*a*c)*c)*b^5 + 39*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^...
 

Mupad [B] (verification not implemented)

Time = 11.77 (sec) , antiderivative size = 8768, normalized size of antiderivative = 24.36 \[ \int \frac {1}{x^{5/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((((-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 63 
66*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5* 
c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^1 
3*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^ 
2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^ 
2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*(x^ 
(1/2)*(-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 636 
6*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c 
^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13 
*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2 
)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 
 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*(327 
68*a^21*b*c^8 + 8*a^15*b^13*c^2 - 192*a^16*b^11*c^3 + 1920*a^17*b^9*c^4 - 
10240*a^18*b^7*c^5 + 30720*a^19*b^5*c^6 - 49152*a^20*b^3*c^7) - 57344*a^19 
*c^9 + 20*a^12*b^14*c^2 - 496*a^13*b^12*c^3 + 5176*a^14*b^10*c^4 - 29280*a 
^15*b^8*c^5 + 96000*a^16*b^6*c^6 - 179200*a^17*b^4*c^7 + 169984*a^18*b^2*c 
^8) - x^(1/2)*(50176*a^16*c^10 - 50*a^9*b^14*c^3 + 1180*a^10*b^12*c^4 - 11 
602*a^11*b^10*c^5 + 61012*a^12*b^8*c^6 - 182336*a^13*b^6*c^7 + 300160*a^14 
*b^4*c^8 - 233984*a^15*b^2*c^9))*(-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1 
/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*...
 

Reduce [B] (verification not implemented)

Time = 1.64 (sec) , antiderivative size = 3914, normalized size of antiderivative = 10.87 \[ \int \frac {1}{x^{5/2} \left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 312*sqrt(x)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*s 
qrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b*c**2* 
x + 204*sqrt(x)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*s 
qrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b**3*c* 
x - 312*sqrt(x)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*s 
qrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b**2*c* 
*2*x**2 - 312*sqrt(x)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqr 
t(c)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b 
*c**3*x**3 - 30*sqrt(x)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*s 
qrt(c)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a*b* 
*5*x + 204*sqrt(x)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c 
)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a*b**4*c* 
x**2 + 204*sqrt(x)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c 
)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a*b**3*c* 
*2*x**3 - 30*sqrt(x)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt 
(c)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*b**6*x* 
*2 - 30*sqrt(x)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*s 
qrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*b**5*c*x**3 
+ 168*sqrt(x)*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqr 
t(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a**4*c**2*x...