\(\int \frac {A+B x^2}{x^5 (a+b x^2)^{5/2}} \, dx\) [280]

Optimal result

Integrand size = 22, antiderivative size = 142 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^{5/2}} \, dx=\frac {b (A b-a B)}{3 a^3 \left (a+b x^2\right )^{3/2}}+\frac {b (3 A b-2 a B)}{a^4 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{4 a^3 x^4}+\frac {(11 A b-4 a B) \sqrt {a+b x^2}}{8 a^4 x^2}-\frac {5 b (7 A b-4 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{9/2}} \] Output:

1/3*b*(A*b-B*a)/a^3/(b*x^2+a)^(3/2)+b*(3*A*b-2*B*a)/a^4/(b*x^2+a)^(1/2)-1/ 
4*A*(b*x^2+a)^(1/2)/a^3/x^4+1/8*(11*A*b-4*B*a)*(b*x^2+a)^(1/2)/a^4/x^2-5/8 
*b*(7*A*b-4*B*a)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(9/2)
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^{5/2}} \, dx=\frac {105 A b^3 x^6+a^2 b x^2 \left (21 A-80 B x^2\right )+20 a b^2 x^4 \left (7 A-3 B x^2\right )-6 a^3 \left (A+2 B x^2\right )}{24 a^4 x^4 \left (a+b x^2\right )^{3/2}}+\frac {5 b (-7 A b+4 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{9/2}} \] Input:

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

Output:

(105*A*b^3*x^6 + a^2*b*x^2*(21*A - 80*B*x^2) + 20*a*b^2*x^4*(7*A - 3*B*x^2 
) - 6*a^3*(A + 2*B*x^2))/(24*a^4*x^4*(a + b*x^2)^(3/2)) + (5*b*(-7*A*b + 4 
*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(8*a^(9/2))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {354, 87, 52, 61, 61, 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.

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {1}{2} \left (-\frac {(7 A b-4 a B) \int \frac {1}{x^4 \left (b x^2+a\right )^{5/2}}dx^2}{4 a}-\frac {A}{2 a x^4 \left (a+b x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {1}{2} \left (-\frac {(7 A b-4 a B) \left (-\frac {5 b \int \frac {1}{x^2 \left (b x^2+a\right )^{5/2}}dx^2}{2 a}-\frac {1}{a x^2 \left (a+b x^2\right )^{3/2}}\right )}{4 a}-\frac {A}{2 a x^4 \left (a+b x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {1}{2} \left (-\frac {(7 A b-4 a B) \left (-\frac {5 b \left (\frac {\int \frac {1}{x^2 \left (b x^2+a\right )^{3/2}}dx^2}{a}+\frac {2}{3 a \left (a+b x^2\right )^{3/2}}\right )}{2 a}-\frac {1}{a x^2 \left (a+b x^2\right )^{3/2}}\right )}{4 a}-\frac {A}{2 a x^4 \left (a+b x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {1}{2} \left (-\frac {(7 A b-4 a B) \left (-\frac {5 b \left (\frac {\frac {\int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2}{a}+\frac {2}{a \sqrt {a+b x^2}}}{a}+\frac {2}{3 a \left (a+b x^2\right )^{3/2}}\right )}{2 a}-\frac {1}{a x^2 \left (a+b x^2\right )^{3/2}}\right )}{4 a}-\frac {A}{2 a x^4 \left (a+b x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{2} \left (-\frac {(7 A b-4 a B) \left (-\frac {5 b \left (\frac {\frac {2 \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{a b}+\frac {2}{a \sqrt {a+b x^2}}}{a}+\frac {2}{3 a \left (a+b x^2\right )^{3/2}}\right )}{2 a}-\frac {1}{a x^2 \left (a+b x^2\right )^{3/2}}\right )}{4 a}-\frac {A}{2 a x^4 \left (a+b x^2\right )^{3/2}}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{2} \left (-\frac {(7 A b-4 a B) \left (-\frac {5 b \left (\frac {\frac {2}{a \sqrt {a+b x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}}{a}+\frac {2}{3 a \left (a+b x^2\right )^{3/2}}\right )}{2 a}-\frac {1}{a x^2 \left (a+b x^2\right )^{3/2}}\right )}{4 a}-\frac {A}{2 a x^4 \left (a+b x^2\right )^{3/2}}\right )\)

Input:

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

Output:

(-1/2*A/(a*x^4*(a + b*x^2)^(3/2)) - ((7*A*b - 4*a*B)*(-(1/(a*x^2*(a + b*x^ 
2)^(3/2))) - (5*b*(2/(3*a*(a + b*x^2)^(3/2)) + (2/(a*Sqrt[a + b*x^2]) - (2 
*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(3/2))/a))/(2*a)))/(4*a))/2
 

Defintions of rubi rules used

rule 52

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

rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, 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 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 = 0.48 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(\frac {-\frac {35 b \left (A b -\frac {4 B a}{7}\right ) \left (b \,x^{2}+a \right )^{\frac {3}{2}} x^{4} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{8}+\frac {35 b^{2} \left (-\frac {3 x^{2} B}{7}+A \right ) x^{4} a^{\frac {3}{2}}}{6}+\frac {7 b \,x^{2} \left (-\frac {80 x^{2} B}{21}+A \right ) a^{\frac {5}{2}}}{8}+\frac {\left (-2 x^{2} B -A \right ) a^{\frac {7}{2}}}{4}+\frac {35 A \sqrt {a}\, b^{3} x^{6}}{8}}{a^{\frac {9}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}} x^{4}}\)	\(118\)
default	\(A \left (-\frac {1}{4 a \,x^{4} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {7 b \left (-\frac {1}{2 a \,x^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 b \left (\frac {1}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}\right )}{2 a}\right )}{4 a}\right )+B \left (-\frac {1}{2 a \,x^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 b \left (\frac {1}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}\right )}{2 a}\right )\)	\(200\)
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-11 A b \,x^{2}+4 B a \,x^{2}+2 A a \right )}{8 a^{4} x^{4}}-\frac {35 b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) A}{8 a^{\frac {9}{2}}}+\frac {5 b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) B}{2 a^{\frac {7}{2}}}+\frac {19 b^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, A}{12 a^{4} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {13 b \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, B}{12 a^{3} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {19 b^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, A}{12 a^{4} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {13 b \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, B}{12 a^{3} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {b \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, A}{12 a^{4} \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, B}{12 a^{3} \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {b \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, A}{12 a^{4} \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, B}{12 a^{3} \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}\)	\(618\)

Input:

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

Output:

7/8*(-5*b*(A*b-4/7*B*a)*(b*x^2+a)^(3/2)*x^4*arctanh((b*x^2+a)^(1/2)/a^(1/2 
))+20/3*b^2*(-3/7*x^2*B+A)*x^4*a^(3/2)+b*x^2*(-80/21*x^2*B+A)*a^(5/2)+2/7* 
(-2*B*x^2-A)*a^(7/2)+5*A*a^(1/2)*b^3*x^6)/(b*x^2+a)^(3/2)/a^(9/2)/x^4
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.89 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^{5/2}} \, dx=\left [-\frac {15 \, {\left ({\left (4 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} + 2 \, {\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} + {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4}\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (15 \, {\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} + 6 \, A a^{4} + 20 \, {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4} + 3 \, {\left (4 \, B a^{4} - 7 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, {\left (a^{5} b^{2} x^{8} + 2 \, a^{6} b x^{6} + a^{7} x^{4}\right )}}, -\frac {15 \, {\left ({\left (4 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} + 2 \, {\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} + {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (15 \, {\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} + 6 \, A a^{4} + 20 \, {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4} + 3 \, {\left (4 \, B a^{4} - 7 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{24 \, {\left (a^{5} b^{2} x^{8} + 2 \, a^{6} b x^{6} + a^{7} x^{4}\right )}}\right ] \] Input:

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

Output:

[-1/48*(15*((4*B*a*b^3 - 7*A*b^4)*x^8 + 2*(4*B*a^2*b^2 - 7*A*a*b^3)*x^6 + 
(4*B*a^3*b - 7*A*a^2*b^2)*x^4)*sqrt(a)*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqr 
t(a) + 2*a)/x^2) + 2*(15*(4*B*a^2*b^2 - 7*A*a*b^3)*x^6 + 6*A*a^4 + 20*(4*B 
*a^3*b - 7*A*a^2*b^2)*x^4 + 3*(4*B*a^4 - 7*A*a^3*b)*x^2)*sqrt(b*x^2 + a))/ 
(a^5*b^2*x^8 + 2*a^6*b*x^6 + a^7*x^4), -1/24*(15*((4*B*a*b^3 - 7*A*b^4)*x^ 
8 + 2*(4*B*a^2*b^2 - 7*A*a*b^3)*x^6 + (4*B*a^3*b - 7*A*a^2*b^2)*x^4)*sqrt( 
-a)*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (15*(4*B*a^2*b^2 - 7*A*a*b^3)*x^6 
 + 6*A*a^4 + 20*(4*B*a^3*b - 7*A*a^2*b^2)*x^4 + 3*(4*B*a^4 - 7*A*a^3*b)*x^ 
2)*sqrt(b*x^2 + a))/(a^5*b^2*x^8 + 2*a^6*b*x^6 + a^7*x^4)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1323 vs. \(2 (134) = 268\).

Time = 44.37 (sec) , antiderivative size = 1323, normalized size of antiderivative = 9.32 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((B*x**2+A)/x**5/(b*x**2+a)**(5/2),x)
 

Output:

A*(-6*a**(89/2)*b**75/(24*a**(93/2)*b**(151/2)*x**5*sqrt(a/(b*x**2) + 1) + 
 24*a**(91/2)*b**(153/2)*x**7*sqrt(a/(b*x**2) + 1)) + 21*a**(87/2)*b**76*x 
**2/(24*a**(93/2)*b**(151/2)*x**5*sqrt(a/(b*x**2) + 1) + 24*a**(91/2)*b**( 
153/2)*x**7*sqrt(a/(b*x**2) + 1)) + 140*a**(85/2)*b**77*x**4/(24*a**(93/2) 
*b**(151/2)*x**5*sqrt(a/(b*x**2) + 1) + 24*a**(91/2)*b**(153/2)*x**7*sqrt( 
a/(b*x**2) + 1)) + 105*a**(83/2)*b**78*x**6/(24*a**(93/2)*b**(151/2)*x**5* 
sqrt(a/(b*x**2) + 1) + 24*a**(91/2)*b**(153/2)*x**7*sqrt(a/(b*x**2) + 1)) 
- 105*a**42*b**(155/2)*x**5*sqrt(a/(b*x**2) + 1)*asinh(sqrt(a)/(sqrt(b)*x) 
)/(24*a**(93/2)*b**(151/2)*x**5*sqrt(a/(b*x**2) + 1) + 24*a**(91/2)*b**(15 
3/2)*x**7*sqrt(a/(b*x**2) + 1)) - 105*a**41*b**(157/2)*x**7*sqrt(a/(b*x**2 
) + 1)*asinh(sqrt(a)/(sqrt(b)*x))/(24*a**(93/2)*b**(151/2)*x**5*sqrt(a/(b* 
x**2) + 1) + 24*a**(91/2)*b**(153/2)*x**7*sqrt(a/(b*x**2) + 1))) + B*(-6*a 
**17*sqrt(1 + b*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**( 
35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) - 46*a**16*b*x**2*sqrt(1 + b*x** 
2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 + 1 
2*a**(33/2)*b**3*x**8) - 15*a**16*b*x**2*log(b*x**2/a)/(12*a**(39/2)*x**2 
+ 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) + 
 30*a**16*b*x**2*log(sqrt(1 + b*x**2/a) + 1)/(12*a**(39/2)*x**2 + 36*a**(3 
7/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) - 70*a**15* 
b**2*x**4*sqrt(1 + b*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 +...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^{5/2}} \, dx=\frac {5 \, B b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {7}{2}}} - \frac {35 \, A b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {9}{2}}} - \frac {5 \, B b}{2 \, \sqrt {b x^{2} + a} a^{3}} - \frac {5 \, B b}{6 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2}} + \frac {35 \, A b^{2}}{8 \, \sqrt {b x^{2} + a} a^{4}} + \frac {35 \, A b^{2}}{24 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} - \frac {B}{2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{2}} + \frac {7 \, A b}{8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x^{2}} - \frac {A}{4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{4}} \] Input:

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

Output:

5/2*B*b*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(7/2) - 35/8*A*b^2*arcsinh(a/(sqrt 
(a*b)*abs(x)))/a^(9/2) - 5/2*B*b/(sqrt(b*x^2 + a)*a^3) - 5/6*B*b/((b*x^2 + 
 a)^(3/2)*a^2) + 35/8*A*b^2/(sqrt(b*x^2 + a)*a^4) + 35/24*A*b^2/((b*x^2 + 
a)^(3/2)*a^3) - 1/2*B/((b*x^2 + a)^(3/2)*a*x^2) + 7/8*A*b/((b*x^2 + a)^(3/ 
2)*a^2*x^2) - 1/4*A/((b*x^2 + a)^(3/2)*a*x^4)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.16 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {5 \, {\left (4 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{4}} - \frac {6 \, {\left (b x^{2} + a\right )} B a b + B a^{2} b - 9 \, {\left (b x^{2} + a\right )} A b^{2} - A a b^{2}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a b - 4 \, \sqrt {b x^{2} + a} B a^{2} b - 11 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2} + 13 \, \sqrt {b x^{2} + a} A a b^{2}}{8 \, a^{4} b^{2} x^{4}} \] Input:

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

Output:

-5/8*(4*B*a*b - 7*A*b^2)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^4) - 
 1/3*(6*(b*x^2 + a)*B*a*b + B*a^2*b - 9*(b*x^2 + a)*A*b^2 - A*a*b^2)/((b*x 
^2 + a)^(3/2)*a^4) - 1/8*(4*(b*x^2 + a)^(3/2)*B*a*b - 4*sqrt(b*x^2 + a)*B* 
a^2*b - 11*(b*x^2 + a)^(3/2)*A*b^2 + 13*sqrt(b*x^2 + a)*A*a*b^2)/(a^4*b^2* 
x^4)
 

Mupad [B] (verification not implemented)

Time = 1.68 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.24 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^{5/2}} \, dx=\frac {35\,A\,b^2}{6\,a^3\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {10\,B\,b}{3\,a^2\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {35\,A\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{9/2}}-\frac {A}{4\,a\,x^4\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {B}{2\,a\,x^2\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {5\,B\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{7/2}}+\frac {7\,A\,b}{8\,a^2\,x^2\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {35\,A\,b^3\,x^2}{8\,a^4\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {5\,B\,b^2\,x^2}{2\,a^3\,{\left (b\,x^2+a\right )}^{3/2}} \] Input:

int((A + B*x^2)/(x^5*(a + b*x^2)^(5/2)),x)
 

Output:

(35*A*b^2)/(6*a^3*(a + b*x^2)^(3/2)) - (10*B*b)/(3*a^2*(a + b*x^2)^(3/2)) 
- (35*A*b^2*atanh((a + b*x^2)^(1/2)/a^(1/2)))/(8*a^(9/2)) - A/(4*a*x^4*(a 
+ b*x^2)^(3/2)) - B/(2*a*x^2*(a + b*x^2)^(3/2)) + (5*B*b*atanh((a + b*x^2) 
^(1/2)/a^(1/2)))/(2*a^(7/2)) + (7*A*b)/(8*a^2*x^2*(a + b*x^2)^(3/2)) + (35 
*A*b^3*x^2)/(8*a^4*(a + b*x^2)^(3/2)) - (5*B*b^2*x^2)/(2*a^3*(a + b*x^2)^( 
3/2))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.37 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^{5/2}} \, dx=\frac {-2 \sqrt {b \,x^{2}+a}\, a^{3}+5 \sqrt {b \,x^{2}+a}\, a^{2} b \,x^{2}+15 \sqrt {b \,x^{2}+a}\, a \,b^{2} x^{4}+15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} x^{4}+15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} x^{6}-15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} x^{4}-15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} x^{6}}{8 a^{4} x^{4} \left (b \,x^{2}+a \right )} \] Input:

int((B*x^2+A)/x^5/(b*x^2+a)^(5/2),x)
 

Output:

( - 2*sqrt(a + b*x**2)*a**3 + 5*sqrt(a + b*x**2)*a**2*b*x**2 + 15*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))*a*b**2*x**4 + 15*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + s 
qrt(b)*x)/sqrt(a))*b**3*x**6 - 15*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) 
+ sqrt(b)*x)/sqrt(a))*a*b**2*x**4 - 15*sqrt(a)*log((sqrt(a + b*x**2) + sqr 
t(a) + sqrt(b)*x)/sqrt(a))*b**3*x**6)/(8*a**4*x**4*(a + b*x**2))