\(\int \frac {(a+b x^2)^{3/2} (A+B x^2)}{x^7} \, dx\) [210]

Optimal result

Integrand size = 22, antiderivative size = 114 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx=-\frac {(A b+2 a B) \sqrt {a+b x^2}}{8 x^4}-\frac {b (A b+10 a B) \sqrt {a+b x^2}}{16 a x^2}-\frac {A \left (a+b x^2\right )^{3/2}}{6 x^6}+\frac {b^2 (A b-6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{3/2}} \] Output:

-1/8*(A*b+2*B*a)*(b*x^2+a)^(1/2)/x^4-1/16*b*(A*b+10*B*a)*(b*x^2+a)^(1/2)/a 
/x^2-1/6*A*(b*x^2+a)^(3/2)/x^6+1/16*b^2*(A*b-6*B*a)*arctanh((b*x^2+a)^(1/2 
)/a^(1/2))/a^(3/2)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx=\frac {\sqrt {a+b x^2} \left (-8 a^2 A-14 a A b x^2-12 a^2 B x^2-3 A b^2 x^4-30 a b B x^4\right )}{48 a x^6}-\frac {b^2 (-A b+6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{3/2}} \] Input:

Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^7,x]
 

Output:

(Sqrt[a + b*x^2]*(-8*a^2*A - 14*a*A*b*x^2 - 12*a^2*B*x^2 - 3*A*b^2*x^4 - 3 
0*a*b*B*x^4))/(48*a*x^6) - (b^2*(-(A*b) + 6*a*B)*ArcTanh[Sqrt[a + b*x^2]/S 
qrt[a]])/(16*a^(3/2))
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {354, 87, 51, 51, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^{3/2} \left (B x^2+A\right )}{x^8}dx^2\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {1}{2} \left (-\frac {(A b-6 a B) \int \frac {\left (b x^2+a\right )^{3/2}}{x^6}dx^2}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^6}\right )\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{2} \left (-\frac {(A b-6 a B) \left (\frac {3}{4} b \int \frac {\sqrt {b x^2+a}}{x^4}dx^2-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^6}\right )\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{2} \left (-\frac {(A b-6 a B) \left (\frac {3}{4} b \left (\frac {1}{2} b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^6}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{2} \left (-\frac {(A b-6 a B) \left (\frac {3}{4} b \left (\int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^6}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{2} \left (-\frac {(A b-6 a B) \left (\frac {3}{4} b \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^6}\right )\)

Input:

Int[((a + b*x^2)^(3/2)*(A + B*x^2))/x^7,x]
 

Output:

(-1/3*(A*(a + b*x^2)^(5/2))/(a*x^6) - ((A*b - 6*a*B)*(-1/2*(a + b*x^2)^(3/ 
2)/x^4 + (3*b*(-(Sqrt[a + b*x^2]/x^2) - (b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a] 
])/Sqrt[a]))/4))/(6*a))/2
 

Defintions of rubi rules used

rule 51

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

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 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

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 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81

method	result	size
pseudoelliptic	\(-\frac {-\frac {3 b^{2} x^{6} \left (A b -6 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{8}+\left (\frac {7 \left (\frac {15 x^{2} B}{7}+A \right ) b \,x^{2} a^{\frac {3}{2}}}{4}+\left (\frac {3 x^{2} B}{2}+A \right ) a^{\frac {5}{2}}+\frac {3 A \sqrt {a}\, b^{2} x^{4}}{8}\right ) \sqrt {b \,x^{2}+a}}{6 a^{\frac {3}{2}} x^{6}}\)	\(92\)
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (3 A \,b^{2} x^{4}+30 B a b \,x^{4}+14 a A b \,x^{2}+12 B \,a^{2} x^{2}+8 a^{2} A \right )}{48 x^{6} a}+\frac {\left (A b -6 B a \right ) b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{16 a^{\frac {3}{2}}}\)	\(99\)
default	\(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )\)	\(230\)

Input:

int((b*x^2+a)^(3/2)*(B*x^2+A)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/6*(-3/8*b^2*x^6*(A*b-6*B*a)*arctanh((b*x^2+a)^(1/2)/a^(1/2))+(7/4*(15/7 
*x^2*B+A)*b*x^2*a^(3/2)+(3/2*x^2*B+A)*a^(5/2)+3/8*A*a^(1/2)*b^2*x^4)*(b*x^ 
2+a)^(1/2))/a^(3/2)/x^6
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.97 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx=\left [-\frac {3 \, {\left (6 \, B a b^{2} - A b^{3}\right )} \sqrt {a} x^{6} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (3 \, {\left (10 \, B a^{2} b + A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{96 \, a^{2} x^{6}}, \frac {3 \, {\left (6 \, B a b^{2} - A b^{3}\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) - {\left (3 \, {\left (10 \, B a^{2} b + A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, a^{2} x^{6}}\right ] \] Input:

integrate((b*x^2+a)^(3/2)*(B*x^2+A)/x^7,x, algorithm="fricas")
 

Output:

[-1/96*(3*(6*B*a*b^2 - A*b^3)*sqrt(a)*x^6*log(-(b*x^2 + 2*sqrt(b*x^2 + a)* 
sqrt(a) + 2*a)/x^2) + 2*(3*(10*B*a^2*b + A*a*b^2)*x^4 + 8*A*a^3 + 2*(6*B*a 
^3 + 7*A*a^2*b)*x^2)*sqrt(b*x^2 + a))/(a^2*x^6), 1/48*(3*(6*B*a*b^2 - A*b^ 
3)*sqrt(-a)*x^6*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) - (3*(10*B*a^2*b + A*a* 
b^2)*x^4 + 8*A*a^3 + 2*(6*B*a^3 + 7*A*a^2*b)*x^2)*sqrt(b*x^2 + a))/(a^2*x^ 
6)]
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (104) = 208\).

Time = 60.29 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.22 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx=- \frac {A a^{2}}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {11 A a \sqrt {b}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {17 A b^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {A b^{\frac {5}{2}}}{16 a x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {3}{2}}} - \frac {B a^{2}}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 B a \sqrt {b}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {B b^{\frac {3}{2}}}{8 x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 B b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 \sqrt {a}} \] Input:

integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**7,x)
 

Output:

-A*a**2/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - 11*A*a*sqrt(b)/(24*x**5*sq 
rt(a/(b*x**2) + 1)) - 17*A*b**(3/2)/(48*x**3*sqrt(a/(b*x**2) + 1)) - A*b** 
(5/2)/(16*a*x*sqrt(a/(b*x**2) + 1)) + A*b**3*asinh(sqrt(a)/(sqrt(b)*x))/(1 
6*a**(3/2)) - B*a**2/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 3*B*a*sqrt(b) 
/(8*x**3*sqrt(a/(b*x**2) + 1)) - B*b**(3/2)*sqrt(a/(b*x**2) + 1)/(2*x) - B 
*b**(3/2)/(8*x*sqrt(a/(b*x**2) + 1)) - 3*B*b**2*asinh(sqrt(a)/(sqrt(b)*x)) 
/(8*sqrt(a))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (95) = 190\).

Time = 0.05 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.84 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx=-\frac {3 \, B b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, \sqrt {a}} + \frac {A b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {3}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{2}}{8 \, a^{2}} + \frac {3 \, \sqrt {b x^{2} + a} B b^{2}}{8 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{3}}{48 \, a^{3}} - \frac {\sqrt {b x^{2} + a} A b^{3}}{16 \, a^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b}{8 \, a^{2} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{2}}{48 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{4 \, a x^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{24 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{6 \, a x^{6}} \] Input:

integrate((b*x^2+a)^(3/2)*(B*x^2+A)/x^7,x, algorithm="maxima")
 

Output:

-3/8*B*b^2*arcsinh(a/(sqrt(a*b)*abs(x)))/sqrt(a) + 1/16*A*b^3*arcsinh(a/(s 
qrt(a*b)*abs(x)))/a^(3/2) + 1/8*(b*x^2 + a)^(3/2)*B*b^2/a^2 + 3/8*sqrt(b*x 
^2 + a)*B*b^2/a - 1/48*(b*x^2 + a)^(3/2)*A*b^3/a^3 - 1/16*sqrt(b*x^2 + a)* 
A*b^3/a^2 - 1/8*(b*x^2 + a)^(5/2)*B*b/(a^2*x^2) + 1/48*(b*x^2 + a)^(5/2)*A 
*b^2/(a^3*x^2) - 1/4*(b*x^2 + a)^(5/2)*B/(a*x^4) + 1/24*(b*x^2 + a)^(5/2)* 
A*b/(a^2*x^4) - 1/6*(b*x^2 + a)^(5/2)*A/(a*x^6)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx=\frac {1}{48} \, b^{3} {\left (\frac {3 \, {\left (6 \, B a - A b\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a b} - \frac {30 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a - 48 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} + 18 \, \sqrt {b x^{2} + a} B a^{3} + 3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b + 8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a b - 3 \, \sqrt {b x^{2} + a} A a^{2} b}{a b^{4} x^{6}}\right )} \] Input:

integrate((b*x^2+a)^(3/2)*(B*x^2+A)/x^7,x, algorithm="giac")
 

Output:

1/48*b^3*(3*(6*B*a - A*b)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a*b) 
- (30*(b*x^2 + a)^(5/2)*B*a - 48*(b*x^2 + a)^(3/2)*B*a^2 + 18*sqrt(b*x^2 + 
 a)*B*a^3 + 3*(b*x^2 + a)^(5/2)*A*b + 8*(b*x^2 + a)^(3/2)*A*a*b - 3*sqrt(b 
*x^2 + a)*A*a^2*b)/(a*b^4*x^6))
 

Mupad [B] (verification not implemented)

Time = 2.35 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx=\frac {A\,a\,\sqrt {b\,x^2+a}}{16\,x^6}-\frac {5\,B\,{\left (b\,x^2+a\right )}^{3/2}}{8\,x^4}-\frac {3\,B\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,\sqrt {a}}-\frac {A\,{\left (b\,x^2+a\right )}^{3/2}}{6\,x^6}+\frac {3\,B\,a\,\sqrt {b\,x^2+a}}{8\,x^4}-\frac {A\,{\left (b\,x^2+a\right )}^{5/2}}{16\,a\,x^6}-\frac {A\,b^3\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{16\,a^{3/2}} \] Input:

int(((A + B*x^2)*(a + b*x^2)^(3/2))/x^7,x)
 

Output:

(A*a*(a + b*x^2)^(1/2))/(16*x^6) - (5*B*(a + b*x^2)^(3/2))/(8*x^4) - (A*b^ 
3*atan(((a + b*x^2)^(1/2)*1i)/a^(1/2))*1i)/(16*a^(3/2)) - (3*B*b^2*atanh(( 
a + b*x^2)^(1/2)/a^(1/2)))/(8*a^(1/2)) - (A*(a + b*x^2)^(3/2))/(6*x^6) + ( 
3*B*a*(a + b*x^2)^(1/2))/(8*x^4) - (A*(a + b*x^2)^(5/2))/(16*a*x^6)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx=\frac {-8 \sqrt {b \,x^{2}+a}\, a^{3}-26 \sqrt {b \,x^{2}+a}\, a^{2} b \,x^{2}-33 \sqrt {b \,x^{2}+a}\, a \,b^{2} x^{4}+15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} x^{6}-15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} x^{6}}{48 a \,x^{6}} \] Input:

int((b*x^2+a)^(3/2)*(B*x^2+A)/x^7,x)
 

Output:

( - 8*sqrt(a + b*x**2)*a**3 - 26*sqrt(a + b*x**2)*a**2*b*x**2 - 33*sqrt(a 
+ b*x**2)*a*b**2*x**4 + 15*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt( 
b)*x)/sqrt(a))*b**3*x**6 - 15*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sq 
rt(b)*x)/sqrt(a))*b**3*x**6)/(48*a*x**6)