\(\int \frac {A+B x+C x^2}{x^5 (c+d x) \sqrt {a+b x^2}} \, dx\) [123]

Optimal result

Integrand size = 32, antiderivative size = 330 \[ \int \frac {A+B x+C x^2}{x^5 (c+d x) \sqrt {a+b x^2}} \, dx=-\frac {A \sqrt {a+b x^2}}{4 a c x^4}-\frac {(B c-A d) \sqrt {a+b x^2}}{3 a c^2 x^3}-\frac {\left (4 a c (c C-B d)-A \left (3 b c^2-4 a d^2\right )\right ) \sqrt {a+b x^2}}{8 a^2 c^3 x^2}+\frac {\left (2 b c^2 (B c-A d)+3 a d \left (c^2 C-B c d+A d^2\right )\right ) \sqrt {a+b x^2}}{3 a^2 c^4 x}+\frac {d^3 \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^5 \sqrt {b c^2+a d^2}}+\frac {\left (4 a c (c C-B d) \left (b c^2-2 a d^2\right )-A \left (3 b^2 c^4-4 a b c^2 d^2+8 a^2 d^4\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{5/2} c^5} \] Output:

-1/4*A*(b*x^2+a)^(1/2)/a/c/x^4-1/3*(-A*d+B*c)*(b*x^2+a)^(1/2)/a/c^2/x^3-1/ 
8*(4*a*c*(-B*d+C*c)-A*(-4*a*d^2+3*b*c^2))*(b*x^2+a)^(1/2)/a^2/c^3/x^2+1/3* 
(2*b*c^2*(-A*d+B*c)+3*a*d*(A*d^2-B*c*d+C*c^2))*(b*x^2+a)^(1/2)/a^2/c^4/x+d 
^3*(A*d^2-B*c*d+C*c^2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^ 
(1/2))/c^5/(a*d^2+b*c^2)^(1/2)+1/8*(4*a*c*(-B*d+C*c)*(-2*a*d^2+b*c^2)-A*(8 
*a^2*d^4-4*a*b*c^2*d^2+3*b^2*c^4))*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(5/2 
)/c^5
 

Mathematica [A] (verified)

Time = 2.72 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x+C x^2}{x^5 (c+d x) \sqrt {a+b x^2}} \, dx=\frac {\frac {c \sqrt {a+b x^2} \left (b c^2 x^2 (9 A c+16 B c x-16 A d x)-2 a \left (A \left (3 c^3-4 c^2 d x+6 c d^2 x^2-12 d^3 x^3\right )+2 c x \left (3 c C x (c-2 d x)+B \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )\right )\right )}{a^2 x^4}+\frac {48 d^3 \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}+\frac {48 A d^4 \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {6 c \left (A b c \left (3 b c^2-4 a d^2\right )+4 a (c C-B d) \left (-b c^2+2 a d^2\right )\right ) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}}}{24 c^5} \] Input:

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

Output:

((c*Sqrt[a + b*x^2]*(b*c^2*x^2*(9*A*c + 16*B*c*x - 16*A*d*x) - 2*a*(A*(3*c 
^3 - 4*c^2*d*x + 6*c*d^2*x^2 - 12*d^3*x^3) + 2*c*x*(3*c*C*x*(c - 2*d*x) + 
B*(2*c^2 - 3*c*d*x + 6*d^2*x^2)))))/(a^2*x^4) + (48*d^3*(c^2*C - B*c*d + A 
*d^2)*ArcTan[(Sqrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2 
]])/Sqrt[-(b*c^2) - a*d^2] + (48*A*d^4*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2 
])/Sqrt[a]])/Sqrt[a] - (6*c*(A*b*c*(3*b*c^2 - 4*a*d^2) + 4*a*(c*C - B*d)*( 
-(b*c^2) + 2*a*d^2))*ArcTanh[(-(Sqrt[b]*x) + Sqrt[a + b*x^2])/Sqrt[a]])/a^ 
(5/2))/(24*c^5)
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2353, 2009}

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.

\(\displaystyle \int \frac {A+B x+C x^2}{x^5 \sqrt {a+b x^2} (c+d x)} \, dx\)

\(\Big \downarrow \) 2353

\(\displaystyle \int \left (\frac {B c-A d}{c^2 x^4 \sqrt {a+b x^2}}+\frac {d^2 \left (A d^2-B c d+c^2 C\right )}{c^5 x \sqrt {a+b x^2}}-\frac {d^3 \left (A d^2-B c d+c^2 C\right )}{c^5 \sqrt {a+b x^2} (c+d x)}-\frac {d \left (A d^2-B c d+c^2 C\right )}{c^4 x^2 \sqrt {a+b x^2}}+\frac {A d^2-B c d+c^2 C}{c^3 x^3 \sqrt {a+b x^2}}+\frac {A}{c x^5 \sqrt {a+b x^2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 A b^2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{5/2} c}+\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \left (A d^2-B c d+c^2 C\right )}{2 a^{3/2} c^3}+\frac {2 b \sqrt {a+b x^2} (B c-A d)}{3 a^2 c^2 x}+\frac {3 A b \sqrt {a+b x^2}}{8 a^2 c x^2}-\frac {d^2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \left (A d^2-B c d+c^2 C\right )}{\sqrt {a} c^5}+\frac {d^3 \left (A d^2-B c d+c^2 C\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^5 \sqrt {a d^2+b c^2}}-\frac {\sqrt {a+b x^2} (B c-A d)}{3 a c^2 x^3}+\frac {d \sqrt {a+b x^2} \left (A d^2-B c d+c^2 C\right )}{a c^4 x}-\frac {\sqrt {a+b x^2} \left (A d^2-B c d+c^2 C\right )}{2 a c^3 x^2}-\frac {A \sqrt {a+b x^2}}{4 a c x^4}\)

Input:

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

Output:

-1/4*(A*Sqrt[a + b*x^2])/(a*c*x^4) - ((B*c - A*d)*Sqrt[a + b*x^2])/(3*a*c^ 
2*x^3) + (3*A*b*Sqrt[a + b*x^2])/(8*a^2*c*x^2) - ((c^2*C - B*c*d + A*d^2)* 
Sqrt[a + b*x^2])/(2*a*c^3*x^2) + (2*b*(B*c - A*d)*Sqrt[a + b*x^2])/(3*a^2* 
c^2*x) + (d*(c^2*C - B*c*d + A*d^2)*Sqrt[a + b*x^2])/(a*c^4*x) + (d^3*(c^2 
*C - B*c*d + A*d^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b* 
x^2])])/(c^5*Sqrt[b*c^2 + a*d^2]) - (3*A*b^2*ArcTanh[Sqrt[a + b*x^2]/Sqrt[ 
a]])/(8*a^(5/2)*c) + (b*(c^2*C - B*c*d + A*d^2)*ArcTanh[Sqrt[a + b*x^2]/Sq 
rt[a]])/(2*a^(3/2)*c^3) - (d^2*(c^2*C - B*c*d + A*d^2)*ArcTanh[Sqrt[a + b* 
x^2]/Sqrt[a]])/(Sqrt[a]*c^5)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2353

Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) 
^(p_), x_Symbol] :> Int[ExpandIntegrand[Px*(e*x)^m*(c + d*x)^n*(a + b*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && PolyQ[Px, x] && (Integer 
Q[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0]))
 

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.21

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-24 A a \,d^{3} x^{3}+16 A b \,c^{2} d \,x^{3}+24 B a c \,d^{2} x^{3}-16 B b \,c^{3} x^{3}-24 C a \,c^{2} d \,x^{3}+12 A a c \,d^{2} x^{2}-9 A b \,c^{3} x^{2}-12 B a \,c^{2} d \,x^{2}+12 C a \,c^{3} x^{2}-8 A a \,c^{2} d x +8 B a \,c^{3} x +6 A a \,c^{3}\right )}{24 a^{2} c^{4} x^{4}}-\frac {\frac {\left (8 A \,a^{2} d^{4}-4 A a b \,c^{2} d^{2}+3 A \,b^{2} c^{4}-8 B \,a^{2} c \,d^{3}+4 B a b \,c^{3} d +8 C \,a^{2} c^{2} d^{2}-4 C a b \,c^{4}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c \sqrt {a}}-\frac {8 a^{2} d^{2} \left (A \,d^{2}-B c d +C \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{8 c^{4} a^{2}}\)	\(398\)
default	\(\frac {A \left (-\frac {\sqrt {b \,x^{2}+a}}{4 a \,x^{4}}-\frac {3 b \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )}{c}-\frac {\left (A d -B c \right ) \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )}{c^{2}}+\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{c^{3}}-\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) d^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c^{5} \sqrt {a}}+\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) d \sqrt {b \,x^{2}+a}}{c^{4} a x}+\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c^{5} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\)	\(424\)

Input:

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

Output:

-1/24*(b*x^2+a)^(1/2)*(-24*A*a*d^3*x^3+16*A*b*c^2*d*x^3+24*B*a*c*d^2*x^3-1 
6*B*b*c^3*x^3-24*C*a*c^2*d*x^3+12*A*a*c*d^2*x^2-9*A*b*c^3*x^2-12*B*a*c^2*d 
*x^2+12*C*a*c^3*x^2-8*A*a*c^2*d*x+8*B*a*c^3*x+6*A*a*c^3)/a^2/c^4/x^4-1/8/c 
^4/a^2*((8*A*a^2*d^4-4*A*a*b*c^2*d^2+3*A*b^2*c^4-8*B*a^2*c*d^3+4*B*a*b*c^3 
*d+8*C*a^2*c^2*d^2-4*C*a*b*c^4)/c/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2 
))/x)-8*a^2*d^2*(A*d^2-B*c*d+C*c^2)/c/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d 
^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b 
*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))
 

Fricas [A] (verification not implemented)

Time = 19.91 (sec) , antiderivative size = 2207, normalized size of antiderivative = 6.69 \[ \int \frac {A+B x+C x^2}{x^5 (c+d x) \sqrt {a+b x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/48*(24*(C*a^3*c^2*d^3 - B*a^3*c*d^4 + A*a^3*d^5)*sqrt(b*c^2 + a*d^2)*x^ 
4*log((2*a*b*c*d*x - a*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 + 2*s 
qrt(b*c^2 + a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2 
)) + 3*(4*B*a*b^2*c^5*d - 4*B*a^2*b*c^3*d^3 - 8*B*a^3*c*d^5 + 8*A*a^3*d^6 
- (4*C*a*b^2 - 3*A*b^3)*c^6 + (4*C*a^2*b - A*a*b^2)*c^4*d^2 + 4*(2*C*a^3 + 
 A*a^2*b)*c^2*d^4)*sqrt(a)*x^4*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2 
*a)/x^2) - 2*(6*A*a^2*b*c^6 + 6*A*a^3*c^4*d^2 - 8*(2*B*a*b^2*c^6 - B*a^2*b 
*c^4*d^2 - 3*B*a^3*c^2*d^4 + 3*A*a^3*c*d^5 + (3*C*a^2*b - 2*A*a*b^2)*c^5*d 
 + (3*C*a^3 + A*a^2*b)*c^3*d^3)*x^3 - 3*(4*B*a^2*b*c^5*d + 4*B*a^3*c^3*d^3 
 - 4*A*a^3*c^2*d^4 - (4*C*a^2*b - 3*A*a*b^2)*c^6 - (4*C*a^3 + A*a^2*b)*c^4 
*d^2)*x^2 + 8*(B*a^2*b*c^6 - A*a^2*b*c^5*d + B*a^3*c^4*d^2 - A*a^3*c^3*d^3 
)*x)*sqrt(b*x^2 + a))/((a^3*b*c^7 + a^4*c^5*d^2)*x^4), 1/48*(48*(C*a^3*c^2 
*d^3 - B*a^3*c*d^4 + A*a^3*d^5)*sqrt(-b*c^2 - a*d^2)*x^4*arctan(sqrt(-b*c^ 
2 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + (b^2*c^2 + a 
*b*d^2)*x^2)) + 3*(4*B*a*b^2*c^5*d - 4*B*a^2*b*c^3*d^3 - 8*B*a^3*c*d^5 + 8 
*A*a^3*d^6 - (4*C*a*b^2 - 3*A*b^3)*c^6 + (4*C*a^2*b - A*a*b^2)*c^4*d^2 + 4 
*(2*C*a^3 + A*a^2*b)*c^2*d^4)*sqrt(a)*x^4*log(-(b*x^2 - 2*sqrt(b*x^2 + a)* 
sqrt(a) + 2*a)/x^2) - 2*(6*A*a^2*b*c^6 + 6*A*a^3*c^4*d^2 - 8*(2*B*a*b^2*c^ 
6 - B*a^2*b*c^4*d^2 - 3*B*a^3*c^2*d^4 + 3*A*a^3*c*d^5 + (3*C*a^2*b - 2*A*a 
*b^2)*c^5*d + (3*C*a^3 + A*a^2*b)*c^3*d^3)*x^3 - 3*(4*B*a^2*b*c^5*d + 4...
 

Sympy [F]

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

integrate((C*x**2+B*x+A)/x**5/(d*x+c)/(b*x**2+a)**(1/2),x)
 

Output:

Integral((A + B*x + C*x**2)/(x**5*sqrt(a + b*x**2)*(c + d*x)), x)
 

Maxima [F]

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

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

Output:

integrate((C*x^2 + B*x + A)/(sqrt(b*x^2 + a)*(d*x + c)*x^5), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1161 vs. \(2 (301) = 602\).

Time = 0.21 (sec) , antiderivative size = 1161, normalized size of antiderivative = 3.52 \[ \int \frac {A+B x+C x^2}{x^5 (c+d x) \sqrt {a+b x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-2*(C*c^2*d^3 - B*c*d^4 + A*d^5)*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d 
+ sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/(sqrt(-b*c^2 - a*d^2)*c^5) - 1/4*(4*C*a 
*b*c^4 - 3*A*b^2*c^4 - 4*B*a*b*c^3*d - 8*C*a^2*c^2*d^2 + 4*A*a*b*c^2*d^2 + 
 8*B*a^2*c*d^3 - 8*A*a^2*d^4)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(- 
a))/(sqrt(-a)*a^2*c^5) + 1/12*(12*(sqrt(b)*x - sqrt(b*x^2 + a))^7*C*a*b*c^ 
3 - 9*(sqrt(b)*x - sqrt(b*x^2 + a))^7*A*b^2*c^3 - 12*(sqrt(b)*x - sqrt(b*x 
^2 + a))^7*B*a*b*c^2*d + 12*(sqrt(b)*x - sqrt(b*x^2 + a))^7*A*a*b*c*d^2 - 
24*(sqrt(b)*x - sqrt(b*x^2 + a))^6*C*a^2*sqrt(b)*c^2*d + 24*(sqrt(b)*x - s 
qrt(b*x^2 + a))^6*B*a^2*sqrt(b)*c*d^2 - 24*(sqrt(b)*x - sqrt(b*x^2 + a))^6 
*A*a^2*sqrt(b)*d^3 - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^5*C*a^2*b*c^3 + 33*( 
sqrt(b)*x - sqrt(b*x^2 + a))^5*A*a*b^2*c^3 + 12*(sqrt(b)*x - sqrt(b*x^2 + 
a))^5*B*a^2*b*c^2*d - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^5*A*a^2*b*c*d^2 + 4 
8*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^2*b^(3/2)*c^3 + 72*(sqrt(b)*x - sqrt 
(b*x^2 + a))^4*C*a^3*sqrt(b)*c^2*d - 48*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A* 
a^2*b^(3/2)*c^2*d - 72*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^3*sqrt(b)*c*d^2 
 + 72*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^3*sqrt(b)*d^3 - 12*(sqrt(b)*x - 
sqrt(b*x^2 + a))^3*C*a^3*b*c^3 + 33*(sqrt(b)*x - sqrt(b*x^2 + a))^3*A*a^2* 
b^2*c^3 + 12*(sqrt(b)*x - sqrt(b*x^2 + a))^3*B*a^3*b*c^2*d - 12*(sqrt(b)*x 
 - sqrt(b*x^2 + a))^3*A*a^3*b*c*d^2 - 64*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B 
*a^3*b^(3/2)*c^3 - 72*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a^4*sqrt(b)*c^2...
 

Mupad [F(-1)]

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

int((A + B*x + C*x^2)/(x^5*(a + b*x^2)^(1/2)*(c + d*x)),x)
 

Output:

int((A + B*x + C*x^2)/(x^5*(a + b*x^2)^(1/2)*(c + d*x)), x)
 

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 1280, normalized size of antiderivative = 3.88 \[ \int \frac {A+B x+C x^2}{x^5 (c+d x) \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

int((C*x^2+B*x+A)/x^5/(d*x+c)/(b*x^2+a)^(1/2),x)
 

Output:

(48*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - 
a*d + b*c*x)*a**3*d**5*x**4 - 48*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x 
**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b*c*d**4*x**4 + 48*sqrt(a*d 
**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x) 
*a**2*c**3*d**3*x**4 - 48*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**3*d**5*x** 
4 + 48*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*c*d**4*x**4 - 48*sqrt(a*d 
**2 + b*c**2)*log(c + d*x)*a**2*c**3*d**3*x**4 - 12*sqrt(a + b*x**2)*a**3* 
c**4*d**2 + 16*sqrt(a + b*x**2)*a**3*c**3*d**3*x - 24*sqrt(a + b*x**2)*a** 
3*c**2*d**4*x**2 + 48*sqrt(a + b*x**2)*a**3*c*d**5*x**3 - 12*sqrt(a + b*x* 
*2)*a**2*b*c**6 + 16*sqrt(a + b*x**2)*a**2*b*c**5*d*x - 6*sqrt(a + b*x**2) 
*a**2*b*c**4*d**2*x**2 - 16*sqrt(a + b*x**2)*a**2*b*c**4*d**2*x + 16*sqrt( 
a + b*x**2)*a**2*b*c**3*d**3*x**3 + 24*sqrt(a + b*x**2)*a**2*b*c**3*d**3*x 
**2 - 48*sqrt(a + b*x**2)*a**2*b*c**2*d**4*x**3 - 24*sqrt(a + b*x**2)*a**2 
*c**5*d**2*x**2 + 48*sqrt(a + b*x**2)*a**2*c**4*d**3*x**3 + 18*sqrt(a + b* 
x**2)*a*b**2*c**6*x**2 - 16*sqrt(a + b*x**2)*a*b**2*c**6*x - 32*sqrt(a + b 
*x**2)*a*b**2*c**5*d*x**3 + 24*sqrt(a + b*x**2)*a*b**2*c**5*d*x**2 - 16*sq 
rt(a + b*x**2)*a*b**2*c**4*d**2*x**3 - 24*sqrt(a + b*x**2)*a*b*c**7*x**2 + 
 48*sqrt(a + b*x**2)*a*b*c**6*d*x**3 + 32*sqrt(a + b*x**2)*b**3*c**6*x**3 
+ 24*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a**3*d**6*x**4 + 12*sqrt(a)*l 
og(sqrt(a + b*x**2) - sqrt(a))*a**2*b*c**2*d**4*x**4 - 24*sqrt(a)*log(s...