\(\int \frac {A+B x^2+C x^4+D x^6}{x^3 (a+b x^2)^{3/2}} \, dx\) [228]

Optimal result

Integrand size = 32, antiderivative size = 119 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {A b^3-a \left (b^2 B-a b C+a^2 D\right )}{a^2 b^2 \sqrt {a+b x^2}}+\frac {D \sqrt {a+b x^2}}{b^2}-\frac {A \sqrt {a+b x^2}}{2 a^2 x^2}+\frac {(3 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}} \] Output:

-(A*b^3-a*(B*b^2-C*a*b+D*a^2))/a^2/b^2/(b*x^2+a)^(1/2)+D*(b*x^2+a)^(1/2)/b 
^2-1/2*A*(b*x^2+a)^(1/2)/a^2/x^2+1/2*(3*A*b-2*B*a)*arctanh((b*x^2+a)^(1/2) 
/a^(1/2))/a^(5/2)
 

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {a} \left (-3 A b^3 x^2+4 a^3 D x^2-a b^2 \left (A-2 B x^2\right )+2 a^2 b x^2 \left (-C+D x^2\right )\right )}{b^2 x^2 \sqrt {a+b x^2}}+(3 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}} \] Input:

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^3*(a + b*x^2)^(3/2)),x]
 

Output:

((Sqrt[a]*(-3*A*b^3*x^2 + 4*a^3*D*x^2 - a*b^2*(A - 2*B*x^2) + 2*a^2*b*x^2* 
(-C + D*x^2)))/(b^2*x^2*Sqrt[a + b*x^2]) + (3*A*b - 2*a*B)*ArcTanh[Sqrt[a 
+ b*x^2]/Sqrt[a]])/(2*a^(5/2))
 

Rubi [A] (warning: unable to verify)

Time = 0.52 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2331, 2124, 27, 1192, 25, 1584, 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.

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^3 \left (a+b x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2331

\(\displaystyle \frac {1}{2} \int \frac {D x^6+C x^4+B x^2+A}{x^4 \left (b x^2+a\right )^{3/2}}dx^2\)

\(\Big \downarrow \) 2124

\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {-2 a D x^4-2 a C x^2+3 A b-2 a B}{2 x^2 \left (b x^2+a\right )^{3/2}}dx^2}{a}-\frac {A}{a x^2 \sqrt {a+b x^2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {-2 a D x^4-2 a C x^2+3 A b-2 a B}{x^2 \left (b x^2+a\right )^{3/2}}dx^2}{2 a}-\frac {A}{a x^2 \sqrt {a+b x^2}}\right )\)

\(\Big \downarrow \) 1192

\(\displaystyle \frac {1}{2} \left (-\frac {\int -\frac {-2 a D x^8-2 a (b C-2 a D) x^4+3 A b^3-2 a \left (D a^2-b C a+b^2 B\right )}{x^4 \left (a-x^4\right )}d\sqrt {b x^2+a}}{a b^2}-\frac {A}{a x^2 \sqrt {a+b x^2}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {-2 a D x^8-2 a (b C-2 a D) x^4+3 A b^3-2 a \left (D a^2-b C a+b^2 B\right )}{x^4 \left (a-x^4\right )}d\sqrt {b x^2+a}}{a b^2}-\frac {A}{a x^2 \sqrt {a+b x^2}}\right )\)

\(\Big \downarrow \) 1584

\(\displaystyle \frac {1}{2} \left (\frac {\int \left (-\frac {(2 a B-3 A b) b^2}{a \left (a-x^4\right )}+2 a D+\frac {3 A b^3-2 a \left (D a^2-b C a+b^2 B\right )}{a x^4}\right )d\sqrt {b x^2+a}}{a b^2}-\frac {A}{a x^2 \sqrt {a+b x^2}}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (-\frac {-\frac {b^2 (3 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {3 A b^3-2 a \left (a^2 D-a b C+b^2 B\right )}{a x^2}-2 a D \sqrt {a+b x^2}}{a b^2}-\frac {A}{a x^2 \sqrt {a+b x^2}}\right )\)

Input:

Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^3*(a + b*x^2)^(3/2)),x]
 

Output:

(-(A/(a*x^2*Sqrt[a + b*x^2])) - ((3*A*b^3 - 2*a*(b^2*B - a*b*C + a^2*D))/( 
a*x^2) - 2*a*D*Sqrt[a + b*x^2] - (b^2*(3*A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x 
^2]/Sqrt[a]])/a^(3/2))/(a*b^2))/2
 

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 1192

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1)   Subst[Int[x^( 
2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + 
 c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && 
IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
 

rule 1584

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* 
(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ 
b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
 

rule 2009

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

rule 2124

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px 
, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - 
 a*d))), x] + Simp[1/((m + 1)*(b*c - a*d))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] ||  ! 
ILtQ[n, -1])
 

rule 2331

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2   S 
ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; 
 FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
 

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.96

method	result	size
pseudoelliptic	\(\frac {\frac {3 \sqrt {b \,x^{2}+a}\, x^{2} b^{2} \left (A b -\frac {2 B a}{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{2}+2 \left (a^{3} x^{2} D-\frac {b \,x^{2} \left (-D x^{2}+C \right ) a^{2}}{2}-\frac {b^{2} \left (-2 x^{2} B +A \right ) a}{4}-\frac {3 A \,x^{2} b^{3}}{4}\right ) \sqrt {a}}{b^{2} a^{\frac {5}{2}} x^{2} \sqrt {b \,x^{2}+a}}\)	\(114\)
default	\(A \left (-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )+B \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )-\frac {C}{b \sqrt {b \,x^{2}+a}}+D \left (\frac {x^{2}}{b \sqrt {b \,x^{2}+a}}+\frac {2 a}{b^{2} \sqrt {b \,x^{2}+a}}\right )\)	\(163\)

Input:

int((D*x^6+C*x^4+B*x^2+A)/x^3/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.69 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\left [-\frac {{\left ({\left (2 \, B a b^{3} - 3 \, A b^{4}\right )} x^{4} + {\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{2}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (2 \, D a^{3} b x^{4} - A a^{2} b^{2} + {\left (4 \, D a^{4} - 2 \, C a^{3} b + 2 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{4 \, {\left (a^{3} b^{3} x^{4} + a^{4} b^{2} x^{2}\right )}}, \frac {{\left ({\left (2 \, B a b^{3} - 3 \, A b^{4}\right )} x^{4} + {\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (2 \, D a^{3} b x^{4} - A a^{2} b^{2} + {\left (4 \, D a^{4} - 2 \, C a^{3} b + 2 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a^{3} b^{3} x^{4} + a^{4} b^{2} x^{2}\right )}}\right ] \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^3/(b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

[-1/4*(((2*B*a*b^3 - 3*A*b^4)*x^4 + (2*B*a^2*b^2 - 3*A*a*b^3)*x^2)*sqrt(a) 
*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(2*D*a^3*b*x^4 - 
A*a^2*b^2 + (4*D*a^4 - 2*C*a^3*b + 2*B*a^2*b^2 - 3*A*a*b^3)*x^2)*sqrt(b*x^ 
2 + a))/(a^3*b^3*x^4 + a^4*b^2*x^2), 1/2*(((2*B*a*b^3 - 3*A*b^4)*x^4 + (2* 
B*a^2*b^2 - 3*A*a*b^3)*x^2)*sqrt(-a)*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + 
(2*D*a^3*b*x^4 - A*a^2*b^2 + (4*D*a^4 - 2*C*a^3*b + 2*B*a^2*b^2 - 3*A*a*b^ 
3)*x^2)*sqrt(b*x^2 + a))/(a^3*b^3*x^4 + a^4*b^2*x^2)]
 

Sympy [A] (verification not implemented)

Time = 40.67 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.80 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=A \left (- \frac {1}{2 a \sqrt {b} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 \sqrt {b}}{2 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {5}{2}}}\right ) + B \left (\frac {2 a^{3} \sqrt {1 + \frac {b x^{2}}{a}}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} + \frac {a^{3} \log {\left (\frac {b x^{2}}{a} \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} + \frac {a^{2} b x^{2} \log {\left (\frac {b x^{2}}{a} \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} - \frac {2 a^{2} b x^{2} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}}\right ) + C \left (\begin {cases} - \frac {1}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + D \left (\begin {cases} \frac {2 a}{b^{2} \sqrt {a + b x^{2}}} + \frac {x^{2}}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) \] Input:

integrate((D*x**6+C*x**4+B*x**2+A)/x**3/(b*x**2+a)**(3/2),x)
 

Output:

A*(-1/(2*a*sqrt(b)*x**3*sqrt(a/(b*x**2) + 1)) - 3*sqrt(b)/(2*a**2*x*sqrt(a 
/(b*x**2) + 1)) + 3*b*asinh(sqrt(a)/(sqrt(b)*x))/(2*a**(5/2))) + B*(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)) + C*Piecewise((-1/(b*sqrt(a + b*x**2)), Ne(b, 0)), (x* 
*2/(2*a**(3/2)), True)) + D*Piecewise((2*a/(b**2*sqrt(a + b*x**2)) + x**2/ 
(b*sqrt(a + b*x**2)), Ne(b, 0)), (x**4/(4*a**(3/2)), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.13 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {D x^{2}}{\sqrt {b x^{2} + a} b} - \frac {B \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {3}{2}}} + \frac {3 \, A b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {5}{2}}} + \frac {B}{\sqrt {b x^{2} + a} a} + \frac {2 \, D a}{\sqrt {b x^{2} + a} b^{2}} - \frac {C}{\sqrt {b x^{2} + a} b} - \frac {3 \, A b}{2 \, \sqrt {b x^{2} + a} a^{2}} - \frac {A}{2 \, \sqrt {b x^{2} + a} a x^{2}} \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^3/(b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

D*x^2/(sqrt(b*x^2 + a)*b) - B*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) + 3/2* 
A*b*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) + B/(sqrt(b*x^2 + a)*a) + 2*D*a/ 
(sqrt(b*x^2 + a)*b^2) - C/(sqrt(b*x^2 + a)*b) - 3/2*A*b/(sqrt(b*x^2 + a)*a 
^2) - 1/2*A/(sqrt(b*x^2 + a)*a*x^2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.39 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {b x^{2} + a} D}{b^{2}} + \frac {{\left (2 \, B a - 3 \, A b\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a} a^{2}} + \frac {2 \, {\left (b x^{2} + a\right )} D a^{3} - 2 \, D a^{4} - 2 \, {\left (b x^{2} + a\right )} C a^{2} b + 2 \, C a^{3} b + 2 \, {\left (b x^{2} + a\right )} B a b^{2} - 2 \, B a^{2} b^{2} - 3 \, {\left (b x^{2} + a\right )} A b^{3} + 2 \, A a b^{3}}{2 \, {\left ({\left (b x^{2} + a\right )}^{\frac {3}{2}} - \sqrt {b x^{2} + a} a\right )} a^{2} b^{2}} \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^3/(b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

sqrt(b*x^2 + a)*D/b^2 + 1/2*(2*B*a - 3*A*b)*arctan(sqrt(b*x^2 + a)/sqrt(-a 
))/(sqrt(-a)*a^2) + 1/2*(2*(b*x^2 + a)*D*a^3 - 2*D*a^4 - 2*(b*x^2 + a)*C*a 
^2*b + 2*C*a^3*b + 2*(b*x^2 + a)*B*a*b^2 - 2*B*a^2*b^2 - 3*(b*x^2 + a)*A*b 
^3 + 2*A*a*b^3)/(((b*x^2 + a)^(3/2) - sqrt(b*x^2 + a)*a)*a^2*b^2)
 

Mupad [B] (verification not implemented)

Time = 2.01 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {B}{a\,\sqrt {b\,x^2+a}}-\frac {C}{b\,\sqrt {b\,x^2+a}}-\frac {B\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {\left (b\,x^2+2\,a\right )\,D}{b^2\,\sqrt {b\,x^2+a}}-\frac {3\,A\,b}{2\,a^2\,\sqrt {b\,x^2+a}}-\frac {A}{2\,a\,x^2\,\sqrt {b\,x^2+a}}+\frac {3\,A\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{5/2}} \] Input:

int((A + B*x^2 + C*x^4 + x^6*D)/(x^3*(a + b*x^2)^(3/2)),x)
                                                                                    
                                                                                    
 

Output:

B/(a*(a + b*x^2)^(1/2)) - C/(b*(a + b*x^2)^(1/2)) - (B*atanh((a + b*x^2)^( 
1/2)/a^(1/2)))/a^(3/2) + ((2*a + b*x^2)*D)/(b^2*(a + b*x^2)^(1/2)) - (3*A* 
b)/(2*a^2*(a + b*x^2)^(1/2)) - A/(2*a*x^2*(a + b*x^2)^(1/2)) + (3*A*b*atan 
h((a + b*x^2)^(1/2)/a^(1/2)))/(2*a^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.97 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {4 \sqrt {b \,x^{2}+a}\, a^{3} d \,x^{2}-\sqrt {b \,x^{2}+a}\, a^{2} b^{2}-2 \sqrt {b \,x^{2}+a}\, a^{2} b c \,x^{2}+2 \sqrt {b \,x^{2}+a}\, a^{2} b d \,x^{4}-\sqrt {b \,x^{2}+a}\, a \,b^{3} x^{2}-\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} x^{2}-\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} x^{4}+\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} x^{2}+\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} x^{4}}{2 a^{2} b^{2} x^{2} \left (b \,x^{2}+a \right )} \] Input:

int((D*x^6+C*x^4+B*x^2+A)/x^3/(b*x^2+a)^(3/2),x)
 

Output:

(4*sqrt(a + b*x**2)*a**3*d*x**2 - sqrt(a + b*x**2)*a**2*b**2 - 2*sqrt(a + 
b*x**2)*a**2*b*c*x**2 + 2*sqrt(a + b*x**2)*a**2*b*d*x**4 - sqrt(a + b*x**2 
)*a*b**3*x**2 - sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt( 
a))*a*b**3*x**2 - sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqr 
t(a))*b**4*x**4 + sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqr 
t(a))*a*b**3*x**2 + sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/s 
qrt(a))*b**4*x**4)/(2*a**2*b**2*x**2*(a + b*x**2))