\(\int \frac {(c+d x) (A+B x+C x^2)}{x^2 (a+b x^2)^{3/2}} \, dx\) [137]

Optimal result

Integrand size = 30, antiderivative size = 120 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {a C d-b (B c+A d)}{a b \sqrt {a+b x^2}}-\frac {(A b c-a c C-a B d) x}{a^2 \sqrt {a+b x^2}}-\frac {A c \sqrt {a+b x^2}}{a^2 x}-\frac {(B c+A d) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.98 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {-a^2 C d x-2 A b^2 c x^2+a b (A (-c+d x)+x (B c+c C x+B d x))}{b x \sqrt {a+b x^2}}-\sqrt {a} (B c+A d) \log (x)+\sqrt {a} (B c+A d) \log \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{a^2} \] Input:

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

Output:

((-(a^2*C*d*x) - 2*A*b^2*c*x^2 + a*b*(A*(-c + d*x) + x*(B*c + c*C*x + B*d* 
x)))/(b*x*Sqrt[a + b*x^2]) - Sqrt[a]*(B*c + A*d)*Log[x] + Sqrt[a]*(B*c + A 
*d)*Log[-Sqrt[a] + Sqrt[a + b*x^2]])/a^2
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2336, 25, 534, 243, 73, 221}

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.

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

\(\Big \downarrow \) 2336

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 534

\(\displaystyle \frac {(A d+B c) \int \frac {1}{x \sqrt {b x^2+a}}dx-\frac {A c \sqrt {a+b x^2}}{a x}}{a}-\frac {b x \left (\frac {A b c}{a}-B d-c C\right )+a C d-b (A d+B c)}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {1}{2} (A d+B c) \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {A c \sqrt {a+b x^2}}{a x}}{a}-\frac {b x \left (\frac {A b c}{a}-B d-c C\right )+a C d-b (A d+B c)}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {(A d+B c) \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}-\frac {A c \sqrt {a+b x^2}}{a x}}{a}-\frac {b x \left (\frac {A b c}{a}-B d-c C\right )+a C d-b (A d+B c)}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 534

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d   Int[ 
x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 
0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
 

rule 2336

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema 
inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) 
^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* 
b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex 
pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F 
reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.14

method	result	size
default	\(\frac {B d x}{a \sqrt {b \,x^{2}+a}}+\frac {C c x}{a \sqrt {b \,x^{2}+a}}+\left (A d +B c \right ) \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )+A c \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )-\frac {d C}{b \sqrt {b \,x^{2}+a}}\)	\(137\)
risch	\(-\frac {A c \sqrt {b \,x^{2}+a}}{a^{2} x}-\frac {\frac {A b c x}{a \sqrt {b \,x^{2}+a}}-\frac {B d x}{\sqrt {b \,x^{2}+a}}-\frac {C c x}{\sqrt {b \,x^{2}+a}}-a \left (A d +B c \right ) \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )+\frac {a C d}{b \sqrt {b \,x^{2}+a}}}{a}\)	\(140\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.43 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {{\left ({\left (B b^{2} c + A b^{2} d\right )} x^{3} + {\left (B a b c + A a b d\right )} x\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (A a b c - {\left (B a b d + {\left (C a b - 2 \, A b^{2}\right )} c\right )} x^{2} - {\left (B a b c - {\left (C a^{2} - A a b\right )} d\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a^{2} b^{2} x^{3} + a^{3} b x\right )}}, \frac {{\left ({\left (B b^{2} c + A b^{2} d\right )} x^{3} + {\left (B a b c + A a b d\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) - {\left (A a b c - {\left (B a b d + {\left (C a b - 2 \, A b^{2}\right )} c\right )} x^{2} - {\left (B a b c - {\left (C a^{2} - A a b\right )} d\right )} x\right )} \sqrt {b x^{2} + a}}{a^{2} b^{2} x^{3} + a^{3} b x}\right ] \] Input:

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

Output:

[1/2*(((B*b^2*c + A*b^2*d)*x^3 + (B*a*b*c + A*a*b*d)*x)*sqrt(a)*log(-(b*x^ 
2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(A*a*b*c - (B*a*b*d + (C*a*b 
 - 2*A*b^2)*c)*x^2 - (B*a*b*c - (C*a^2 - A*a*b)*d)*x)*sqrt(b*x^2 + a))/(a^ 
2*b^2*x^3 + a^3*b*x), (((B*b^2*c + A*b^2*d)*x^3 + (B*a*b*c + A*a*b*d)*x)*s 
qrt(-a)*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) - (A*a*b*c - (B*a*b*d + (C*a*b 
- 2*A*b^2)*c)*x^2 - (B*a*b*c - (C*a^2 - A*a*b)*d)*x)*sqrt(b*x^2 + a))/(a^2 
*b^2*x^3 + a^3*b*x)]
 

Sympy [A] (verification not implemented)

Time = 10.84 (sec) , antiderivative size = 500, normalized size of antiderivative = 4.17 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=A c \left (- \frac {1}{a \sqrt {b} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2 \sqrt {b}}{a^{2} \sqrt {\frac {a}{b x^{2}} + 1}}\right ) + A d \left (\frac {2 a^{3} \sqrt {1 + \frac {b x^{2}}{a}}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} + \frac {a^{3} \log {\left (\frac {b x^{2}}{a} \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} + \frac {a^{2} b x^{2} \log {\left (\frac {b x^{2}}{a} \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} - \frac {2 a^{2} b x^{2} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}}\right ) + B c \left (\frac {2 a^{3} \sqrt {1 + \frac {b x^{2}}{a}}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} + \frac {a^{3} \log {\left (\frac {b x^{2}}{a} \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} + \frac {a^{2} b x^{2} \log {\left (\frac {b x^{2}}{a} \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} - \frac {2 a^{2} b x^{2} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}}\right ) + \frac {B d x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} + C d \left (\begin {cases} - \frac {1}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {C c x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} \] Input:

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

Output:

A*c*(-1/(a*sqrt(b)*x**2*sqrt(a/(b*x**2) + 1)) - 2*sqrt(b)/(a**2*sqrt(a/(b* 
x**2) + 1))) + A*d*(2*a**3*sqrt(1 + b*x**2/a)/(2*a**(9/2) + 2*a**(7/2)*b*x 
**2) + a**3*log(b*x**2/a)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) - 2*a**3*log(sq 
rt(1 + b*x**2/a) + 1)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) + a**2*b*x**2*log(b 
*x**2/a)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) - 2*a**2*b*x**2*log(sqrt(1 + b*x 
**2/a) + 1)/(2*a**(9/2) + 2*a**(7/2)*b*x**2)) + B*c*(2*a**3*sqrt(1 + b*x** 
2/a)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) + a**3*log(b*x**2/a)/(2*a**(9/2) + 2 
*a**(7/2)*b*x**2) - 2*a**3*log(sqrt(1 + b*x**2/a) + 1)/(2*a**(9/2) + 2*a** 
(7/2)*b*x**2) + a**2*b*x**2*log(b*x**2/a)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) 
 - 2*a**2*b*x**2*log(sqrt(1 + b*x**2/a) + 1)/(2*a**(9/2) + 2*a**(7/2)*b*x* 
*2)) + B*d*x/(a**(3/2)*sqrt(1 + b*x**2/a)) + C*d*Piecewise((-1/(b*sqrt(a + 
 b*x**2)), Ne(b, 0)), (x**2/(2*a**(3/2)), True)) + C*c*x/(a**(3/2)*sqrt(1 
+ b*x**2/a))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.08 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {C c x}{\sqrt {b x^{2} + a} a} - \frac {2 \, A b c x}{\sqrt {b x^{2} + a} a^{2}} + \frac {B d x}{\sqrt {b x^{2} + a} a} - \frac {C d}{\sqrt {b x^{2} + a} b} - \frac {{\left (B c + A d\right )} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {3}{2}}} + \frac {B c + A d}{\sqrt {b x^{2} + a} a} - \frac {A c}{\sqrt {b x^{2} + a} a x} \] Input:

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

Output:

C*c*x/(sqrt(b*x^2 + a)*a) - 2*A*b*c*x/(sqrt(b*x^2 + a)*a^2) + B*d*x/(sqrt( 
b*x^2 + a)*a) - C*d/(sqrt(b*x^2 + a)*b) - (B*c + A*d)*arcsinh(a/(sqrt(a*b) 
*abs(x)))/a^(3/2) + (B*c + A*d)/(sqrt(b*x^2 + a)*a) - A*c/(sqrt(b*x^2 + a) 
*a*x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.24 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {2 \, A \sqrt {b} c}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a} + \frac {\frac {{\left (C a^{2} b c - A a b^{2} c + B a^{2} b d\right )} x}{a^{3} b} + \frac {B a^{2} b c - C a^{3} d + A a^{2} b d}{a^{3} b}}{\sqrt {b x^{2} + a}} + \frac {2 \, {\left (B c + A d\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} \] Input:

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

Output:

2*A*sqrt(b)*c/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)*a) + ((C*a^2*b*c - A* 
a*b^2*c + B*a^2*b*d)*x/(a^3*b) + (B*a^2*b*c - C*a^3*d + A*a^2*b*d)/(a^3*b) 
)/sqrt(b*x^2 + a) + 2*(B*c + A*d)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sq 
rt(-a))/(sqrt(-a)*a)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 454, normalized size of antiderivative = 3.78 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-\sqrt {b \,x^{2}+a}\, a^{2} b c +\sqrt {b \,x^{2}+a}\, a^{2} b d x -\sqrt {b \,x^{2}+a}\, a^{2} c d x -2 \sqrt {b \,x^{2}+a}\, a \,b^{2} c \,x^{2}+\sqrt {b \,x^{2}+a}\, a \,b^{2} c x +\sqrt {b \,x^{2}+a}\, a \,b^{2} d \,x^{2}+\sqrt {b \,x^{2}+a}\, a b \,c^{2} x^{2}+\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b d x +\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} c x +\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d \,x^{3}+\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c \,x^{3}-\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b d x -\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} c x -\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d \,x^{3}-\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c \,x^{3}-2 \sqrt {b}\, a^{2} b c x +\sqrt {b}\, a^{2} b d x +\sqrt {b}\, a^{2} c^{2} x -2 \sqrt {b}\, a \,b^{2} c \,x^{3}+\sqrt {b}\, a \,b^{2} d \,x^{3}+\sqrt {b}\, a b \,c^{2} x^{3}}{a^{2} b x \left (b \,x^{2}+a \right )} \] Input:

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

Output:

( - sqrt(a + b*x**2)*a**2*b*c + sqrt(a + b*x**2)*a**2*b*d*x - sqrt(a + b*x 
**2)*a**2*c*d*x - 2*sqrt(a + b*x**2)*a*b**2*c*x**2 + sqrt(a + b*x**2)*a*b* 
*2*c*x + sqrt(a + b*x**2)*a*b**2*d*x**2 + sqrt(a + b*x**2)*a*b*c**2*x**2 + 
 sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a**2*b*d*x 
+ sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**2*c*x 
 + sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**2*d* 
x**3 + sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**3* 
c*x**3 - sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a** 
2*b*d*x - sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a* 
b**2*c*x - sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a 
*b**2*d*x**3 - sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a 
))*b**3*c*x**3 - 2*sqrt(b)*a**2*b*c*x + sqrt(b)*a**2*b*d*x + sqrt(b)*a**2* 
c**2*x - 2*sqrt(b)*a*b**2*c*x**3 + sqrt(b)*a*b**2*d*x**3 + sqrt(b)*a*b*c** 
2*x**3)/(a**2*b*x*(a + b*x**2))