\(\int \frac {A+B x+C x^2}{x^3 (c+d x)^2 \sqrt {a+b x^2}} \, dx\) [130]

Optimal result

Integrand size = 32, antiderivative size = 270 \[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^2 \sqrt {a+b x^2}} \, dx=-\frac {A \sqrt {a+b x^2}}{2 a c^2 x^2}-\frac {(B c-2 A d) \sqrt {a+b x^2}}{a c^3 x}+\frac {d^2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b x^2}}{c^3 \left (b c^2+a d^2\right ) (c+d x)}+\frac {d \left (a d^2 \left (c^2 C-2 B c d+3 A d^2\right )+b c^2 \left (2 c^2 C-3 B c d+4 A d^2\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^4 \left (b c^2+a d^2\right )^{3/2}}+\frac {\left (A b c^2-2 a c^2 C+4 a B c d-6 a A d^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2} c^4} \] Output:

-1/2*A*(b*x^2+a)^(1/2)/a/c^2/x^2-(-2*A*d+B*c)*(b*x^2+a)^(1/2)/a/c^3/x+d^2* 
(A*d^2-B*c*d+C*c^2)*(b*x^2+a)^(1/2)/c^3/(a*d^2+b*c^2)/(d*x+c)+d*(a*d^2*(3* 
A*d^2-2*B*c*d+C*c^2)+b*c^2*(4*A*d^2-3*B*c*d+2*C*c^2))*arctanh((-b*c*x+a*d) 
/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/c^4/(a*d^2+b*c^2)^(3/2)+1/2*(-6*A*a* 
d^2+A*b*c^2+4*B*a*c*d-2*C*a*c^2)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(3/2)/ 
c^4
 

Mathematica [A] (verified)

Time = 2.55 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.24 \[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {\frac {c \sqrt {a+b x^2} \left (-2 c x \left (b B c^2 (c+d x)+a d^2 (B c-c C x+2 B d x)\right )+A \left (b c^2 \left (-c^2+3 c d x+4 d^2 x^2\right )+a d^2 \left (-c^2+3 c d x+6 d^2 x^2\right )\right )\right )}{a \left (b c^2+a d^2\right ) x^2 (c+d x)}-\frac {4 d \left (a d^2 \left (c^2 C-2 B c d+3 A d^2\right )+b c^2 \left (2 c^2 C-3 B c d+4 A d^2\right )\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}+\frac {12 A d^2 \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {2 c (A b c-2 a c C+4 a B d) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}}{2 c^4} \] Input:

Integrate[(A + B*x + C*x^2)/(x^3*(c + d*x)^2*Sqrt[a + b*x^2]),x]
 

Output:

((c*Sqrt[a + b*x^2]*(-2*c*x*(b*B*c^2*(c + d*x) + a*d^2*(B*c - c*C*x + 2*B* 
d*x)) + A*(b*c^2*(-c^2 + 3*c*d*x + 4*d^2*x^2) + a*d^2*(-c^2 + 3*c*d*x + 6* 
d^2*x^2))))/(a*(b*c^2 + a*d^2)*x^2*(c + d*x)) - (4*d*(a*d^2*(c^2*C - 2*B*c 
*d + 3*A*d^2) + b*c^2*(2*c^2*C - 3*B*c*d + 4*A*d^2))*ArcTan[(Sqrt[b]*(c + 
d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]])/(-(b*c^2) - a*d^2)^(3/2 
) + (12*A*d^2*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/Sqrt[a] + (2 
*c*(A*b*c - 2*a*c*C + 4*a*B*d)*ArcTanh[(-(Sqrt[b]*x) + Sqrt[a + b*x^2])/Sq 
rt[a]])/a^(3/2))/(2*c^4)
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.24, 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.

\(\displaystyle \int \frac {A+B x+C x^2}{x^3 \sqrt {a+b x^2} (c+d x)^2} \, dx\)

\(\Big \downarrow \) 2353

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

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*x + C*x^2)/(x^3*(c + d*x)^2*Sqrt[a + b*x^2]),x]
 

Output:

-1/2*(A*Sqrt[a + b*x^2])/(a*c^2*x^2) - ((B*c - 2*A*d)*Sqrt[a + b*x^2])/(a* 
c^3*x) + (d^2*(c^2*C - B*c*d + A*d^2)*Sqrt[a + b*x^2])/(c^3*(b*c^2 + a*d^2 
)*(c + d*x)) + (b*d*(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^2*(b*c^2 + a*d^2)^(3/2)) + (d*(c^2*C - 
2*B*c*d + 3*A*d^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x 
^2])])/(c^4*Sqrt[b*c^2 + a*d^2]) + (A*b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/ 
(2*a^(3/2)*c^2) - ((c^2*C - 2*B*c*d + 3*A*d^2)*ArcTanh[Sqrt[a + b*x^2]/Sqr 
t[a]])/(Sqrt[a]*c^4)
 

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 = 479, normalized size of antiderivative = 1.77

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-4 A d x +2 B c x +A c \right )}{2 a \,c^{3} x^{2}}-\frac {\frac {\left (6 A a \,d^{2}-b A \,c^{2}-4 B a c d +2 C a \,c^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c \sqrt {a}}+\frac {2 a \left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d}-\frac {2 a \left (3 A \,d^{2}-2 B c d +C \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{2 c^{3} a}\)	\(479\)
default	\(\frac {A \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{c^{2}}+\frac {\left (2 A d -B c \right ) \sqrt {b \,x^{2}+a}}{c^{3} a x}-\frac {\left (3 A \,d^{2}-2 B c d +C \,c^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c^{4} \sqrt {a}}+\frac {\left (3 A \,d^{2}-2 B c d +C \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c^{4} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d \,c^{3}}\)	\(506\)

Input:

int((C*x^2+B*x+A)/x^3/(d*x+c)^2/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2*(b*x^2+a)^(1/2)*(-4*A*d*x+2*B*c*x+A*c)/a/c^3/x^2-1/2/c^3/a*(1/c*(6*A* 
a*d^2-A*b*c^2-4*B*a*c*d+2*C*a*c^2)/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/ 
2))/x)+2*a/d*(A*d^2-B*c*d+C*c^2)*(-1/(a*d^2+b*c^2)*d^2/(x+c/d)*(b*(x+c/d)^ 
2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-b*c*d/(a*d^2+b*c^2)/((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)))- 
2*a/c*(3*A*d^2-2*B*c*d+C*c^2)/((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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 775 vs. \(2 (250) = 500\).

Time = 31.18 (sec) , antiderivative size = 3165, normalized size of antiderivative = 11.72 \[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^2 \sqrt {a+b x^2}} \, dx=\text {Too large to display} \] Input:

integrate((C*x^2+B*x+A)/x^3/(d*x+c)^2/(b*x^2+a)^(1/2),x, algorithm="fricas 
")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^2 \sqrt {a+b x^2}} \, dx=\int \frac {A + B x + C x^{2}}{x^{3} \sqrt {a + b x^{2}} \left (c + d x\right )^{2}}\, dx \] Input:

integrate((C*x**2+B*x+A)/x**3/(d*x+c)**2/(b*x**2+a)**(1/2),x)
 

Output:

Integral((A + B*x + C*x**2)/(x**3*sqrt(a + b*x**2)*(c + d*x)**2), x)
 

Maxima [F]

\[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^2 \sqrt {a+b x^2}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {b x^{2} + a} {\left (d x + c\right )}^{2} x^{3}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/x^3/(d*x+c)^2/(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)^2*x^3), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^2 \sqrt {a+b x^2}} \, dx=\text {Timed out} \] Input:

integrate((C*x^2+B*x+A)/x^3/(d*x+c)^2/(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^2 \sqrt {a+b x^2}} \, dx=\int \frac {C\,x^2+B\,x+A}{x^3\,\sqrt {b\,x^2+a}\,{\left (c+d\,x\right )}^2} \,d x \] Input:

int((A + B*x + C*x^2)/(x^3*(a + b*x^2)^(1/2)*(c + d*x)^2),x)
 

Output:

int((A + B*x + C*x^2)/(x^3*(a + b*x^2)^(1/2)*(c + d*x)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 2620, normalized size of antiderivative = 9.70 \[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^2 \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

int((C*x^2+B*x+A)/x^3/(d*x+c)^2/(b*x^2+a)^(1/2),x)
 

Output:

(12*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*c*d**5*x**2 + 12*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**6*x**3 + 16*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**3*d**3*x**2 + 16*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sq 
rt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b*c**2*d**4*x**3 - 8*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**2*d**4*x**2 - 8*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**5*x**3 + 4*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**4*d 
**3*x**2 + 4*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**4*x**3 - 12*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*b**2*c**4*d**2*x 
**2 - 12*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*b**2*c**3*d**3*x**3 + 8*sqrt(a*d**2 + b*c**2)*log( - s 
qrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b*c**6*d*x**2 + 8*s 
qrt(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*b*c**5*d**2*x**3 - 12*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**3*c* 
d**5*x**2 - 12*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**3*d**6*x**3 - 16*sqrt 
(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*c**3*d**3*x**2 - 16*sqrt(a*d**2 +...