\(\int \frac {A+B x}{x^{5/2} (a+b x+c x^2)} \, dx\) [89]

Optimal result

Integrand size = 23, antiderivative size = 284 \[ \int \frac {A+B x}{x^{5/2} \left (a+b x+c x^2\right )} \, dx=-\frac {2 A}{3 a x^{3/2}}+\frac {2 (A b-a B)}{a^2 \sqrt {x}}-\frac {\sqrt {2} \sqrt {c} \left (a B \left (b+\sqrt {b^2-4 a c}\right )-A \left (b^2-2 a c+b \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a^2 \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (a B \left (b-\sqrt {b^2-4 a c}\right )-A \left (b^2-2 a c-b \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a^2 \sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

-2/3*A/a/x^(3/2)+2*(A*b-B*a)/a^2/x^(1/2)-2^(1/2)*c^(1/2)*(a*B*(b+(-4*a*c+b 
^2)^(1/2))-A*(b^2-2*a*c+b*(-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))/a^2/(-4*a*c+b^2)^(1/2)/(b-(-4*a*c+b^2)^( 
1/2))^(1/2)+2^(1/2)*c^(1/2)*(a*B*(b-(-4*a*c+b^2)^(1/2))-A*(b^2-2*a*c-b*(-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))/a^2/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
 

Mathematica [A] (verified)

Time = 1.28 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x}{x^{5/2} \left (a+b x+c x^2\right )} \, dx=\frac {\frac {6 A b x-2 a (A+3 B x)}{x^{3/2}}-\frac {3 \sqrt {2} \sqrt {c} \left (a B \left (b+\sqrt {b^2-4 a c}\right )-A \left (b^2-2 a c+b \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {2} \sqrt {c} \left (a B \left (b-\sqrt {b^2-4 a c}\right )+A \left (-b^2+2 a c+b \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{3 a^2} \] Input:

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

Output:

((6*A*b*x - 2*a*(A + 3*B*x))/x^(3/2) - (3*Sqrt[2]*Sqrt[c]*(a*B*(b + Sqrt[b 
^2 - 4*a*c]) - A*(b^2 - 2*a*c + b*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 - Sqr 
t[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(a*B*(b - Sqrt[b^2 - 4*a*c]) + A*(-b 
^2 + 2*a*c + b*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*a^2)
 

Rubi [F]

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.

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

\(\Big \downarrow \) 1198

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1198

\(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{a \sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\)

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 25

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

Maple [A] (verified)

Time = 1.53 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.82

method	result	size
risch	\(-\frac {2 \left (-3 A b x +3 B a x +A a \right )}{3 a^{2} x^{\frac {3}{2}}}+\frac {8 c \left (-\frac {\left (A b \sqrt {-4 a c +b^{2}}-2 A a c +b^{2} A -a B \sqrt {-4 a c +b^{2}}-a b B \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (A b \sqrt {-4 a c +b^{2}}+2 A a c -b^{2} A -a B \sqrt {-4 a c +b^{2}}+a b B \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {x}\, c \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a^{2}}\)	\(233\)
derivativedivides	\(\frac {8 c \left (-\frac {\left (A b \sqrt {-4 a c +b^{2}}-2 A a c +b^{2} A -a B \sqrt {-4 a c +b^{2}}-a b B \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (A b \sqrt {-4 a c +b^{2}}+2 A a c -b^{2} A -a B \sqrt {-4 a c +b^{2}}+a b B \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {x}\, c \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a^{2}}-\frac {2 A}{3 a \,x^{\frac {3}{2}}}-\frac {2 \left (-A b +B a \right )}{a^{2} \sqrt {x}}\)	\(236\)
default	\(\frac {8 c \left (-\frac {\left (A b \sqrt {-4 a c +b^{2}}-2 A a c +b^{2} A -a B \sqrt {-4 a c +b^{2}}-a b B \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (A b \sqrt {-4 a c +b^{2}}+2 A a c -b^{2} A -a B \sqrt {-4 a c +b^{2}}+a b B \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {x}\, c \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a^{2}}-\frac {2 A}{3 a \,x^{\frac {3}{2}}}-\frac {2 \left (-A b +B a \right )}{a^{2} \sqrt {x}}\)	\(236\)

Input:

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

Output:

-2/3*(-3*A*b*x+3*B*a*x+A*a)/a^2/x^(3/2)+8/a^2*c*(-1/8*(A*b*(-4*a*c+b^2)^(1 
/2)-2*A*a*c+b^2*A-a*B*(-4*a*c+b^2)^(1/2)-a*b*B)/(-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*(A*b*(-4*a*c+b^2)^(1/2)+2*A*a*c-b^2*A-a*B*(-4*a* 
c+b^2)^(1/2)+a*b*B)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^ 
(1/2)*arctan(x^(1/2)*c*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. 5453 vs. \(2 (230) = 460\).

Time = 1.42 (sec) , antiderivative size = 5453, normalized size of antiderivative = 19.20 \[ \int \frac {A+B x}{x^{5/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-2/3*(A*a^2/x^(3/2) + 3*(B*a*b - (b^2 - a*c)*A)*sqrt(x) + 3*(B*a^2 - A*a*b 
)/sqrt(x))/a^3 + integrate(((B*a*b*c - (b^2*c - a*c^2)*A)*x^(3/2) - ((b^3 
- 2*a*b*c)*A - (a*b^2 - a^2*c)*B)*sqrt(x))/(a^3*c*x^2 + a^3*b*x + a^4), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2870 vs. \(2 (230) = 460\).

Time = 0.83 (sec) , antiderivative size = 2870, normalized size of antiderivative = 10.11 \[ \int \frac {A+B x}{x^{5/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*((sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6 - 9*sqrt(2)*sqrt(b*c + s 
qrt(b^2 - 4*a*c)*c)*a*b^4*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^ 
5*c - 2*b^6*c + 24*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 + 1 
0*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 + sqrt(2)*sqrt(b*c + s 
qrt(b^2 - 4*a*c)*c)*b^4*c^2 + 18*a*b^4*c^2 - 2*b^5*c^2 - 16*sqrt(2)*sqrt(b 
*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^3 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c) 
*c)*a^2*b*c^3 - 5*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 - 48*a 
^2*b^2*c^3 + 14*a*b^3*c^3 + 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2* 
c^4 + 32*a^3*c^4 - 24*a^2*b*c^4 + sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqr 
t(b^2 - 4*a*c)*c)*b^5 - 7*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 
4*a*c)*c)*a*b^3*c - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a* 
c)*c)*b^4*c + 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c) 
*a^2*b*c^2 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a 
*b^2*c^2 + sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c 
^2 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 + 
 2*(b^2 - 4*a*c)*b^4*c - 10*(b^2 - 4*a*c)*a*b^2*c^2 + 2*(b^2 - 4*a*c)*b^3* 
c^2 + 8*(b^2 - 4*a*c)*a^2*c^3 - 6*(b^2 - 4*a*c)*a*b*c^3)*A - (sqrt(2)*sqrt 
(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c) 
*c)*a^2*b^3*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c - 2*a*b^ 
5*c + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^2 + 8*sqrt(2)*...
 

Mupad [B] (verification not implemented)

Time = 2.14 (sec) , antiderivative size = 10133, normalized size of antiderivative = 35.68 \[ \int \frac {A+B x}{x^{5/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((x^(1/2)*(16*A^2*a^8*c^5 - 16*B^2*a^9*c^4 + 8*A^2*a^6*b^4*c^3 - 32*A 
^2*a^7*b^2*c^4 + 8*B^2*a^8*b^2*c^3 - 16*A*B*a^7*b^3*c^3 + 48*A*B*a^8*b*c^4 
) + (-(A^2*b^7 + B^2*a^2*b^5 + A^2*b^4*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a* 
b^6 + 25*A^2*a^2*b^3*c^2 + A^2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2* 
b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^4*c^3 - 9*A^2*a*b^5*c - 20*A^2*a^3 
*b*c^3 - 7*B^2*a^3*b^3*c + 12*B^2*a^4*b*c^2 - B^2*a^3*c*(-(4*a*c - b^2)^3) 
^(1/2) - 36*A*B*a^3*b^2*c^2 - 3*A^2*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A 
*B*a*b^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^2*b^4*c + 4*A*B*a^2*b*c*(-(4* 
a*c - b^2)^3)^(1/2))/(2*(a^5*b^4 + 16*a^7*c^2 - 8*a^6*b^2*c)))^(1/2)*(32*A 
*a^10*c^4 - x^(1/2)*(32*a^11*b*c^3 - 8*a^10*b^3*c^2)*(-(A^2*b^7 + B^2*a^2* 
b^5 + A^2*b^4*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^6 + 25*A^2*a^2*b^3*c^2 
+ A^2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^3)^(1 
/2) + 16*A*B*a^4*c^3 - 9*A^2*a*b^5*c - 20*A^2*a^3*b*c^3 - 7*B^2*a^3*b^3*c 
+ 12*B^2*a^4*b*c^2 - B^2*a^3*c*(-(4*a*c - b^2)^3)^(1/2) - 36*A*B*a^3*b^2*c 
^2 - 3*A^2*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^3*(-(4*a*c - b^2)^ 
3)^(1/2) + 16*A*B*a^2*b^4*c + 4*A*B*a^2*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*( 
a^5*b^4 + 16*a^7*c^2 - 8*a^6*b^2*c)))^(1/2) + 32*B*a^10*b*c^3 + 8*A*a^8*b^ 
4*c^2 - 40*A*a^9*b^2*c^3 - 8*B*a^9*b^3*c^2))*(-(A^2*b^7 + B^2*a^2*b^5 + A^ 
2*b^4*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^6 + 25*A^2*a^2*b^3*c^2 + A^2*a^ 
2*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^3)^(1/2) +...
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.42 \[ \int \frac {A+B x}{x^{5/2} \left (a+b x+c x^2\right )} \, dx=\frac {-6 \sqrt {x}\, \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}-2 \sqrt {x}\, \sqrt {c}}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) b c x +12 \sqrt {x}\, \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}-2 \sqrt {x}\, \sqrt {c}}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) a c x +6 \sqrt {x}\, \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}+2 \sqrt {x}\, \sqrt {c}}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) b c x -12 \sqrt {x}\, \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}+2 \sqrt {x}\, \sqrt {c}}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) a c x +3 \sqrt {x}\, \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathrm {log}\left (-\sqrt {x}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}+\sqrt {a}+\sqrt {c}\, x \right ) b c x -3 \sqrt {x}\, \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathrm {log}\left (\sqrt {x}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}+\sqrt {a}+\sqrt {c}\, x \right ) b c x +6 \sqrt {x}\, \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathrm {log}\left (-\sqrt {x}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}+\sqrt {a}+\sqrt {c}\, x \right ) a c x -6 \sqrt {x}\, \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathrm {log}\left (\sqrt {x}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}+\sqrt {a}+\sqrt {c}\, x \right ) a c x -16 a^{2} c +4 a \,b^{2}}{6 \sqrt {x}\, a x \left (4 a c -b^{2}\right )} \] Input:

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

Output:

( - 6*sqrt(x)*sqrt(a)*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))*b*c*x + 12*sqr 
t(x)*sqrt(c)*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*c*x + 6*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*c*x - 12*sqrt(x)*sqrt(c)*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*c*x + 3*sqrt(x)*sqrt(a)*sqrt(2*sqrt(c)*s 
qrt(a) - b)*log( - sqrt(x)*sqrt(2*sqrt(c)*sqrt(a) - b) + sqrt(a) + sqrt(c) 
*x)*b*c*x - 3*sqrt(x)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - b)*log(sqrt(x)*sqrt 
(2*sqrt(c)*sqrt(a) - b) + sqrt(a) + sqrt(c)*x)*b*c*x + 6*sqrt(x)*sqrt(c)*s 
qrt(2*sqrt(c)*sqrt(a) - b)*log( - sqrt(x)*sqrt(2*sqrt(c)*sqrt(a) - b) + sq 
rt(a) + sqrt(c)*x)*a*c*x - 6*sqrt(x)*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) - b)*l 
og(sqrt(x)*sqrt(2*sqrt(c)*sqrt(a) - b) + sqrt(a) + sqrt(c)*x)*a*c*x - 16*a 
**2*c + 4*a*b**2)/(6*sqrt(x)*a*x*(4*a*c - b**2))