\(\int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx\) [41]

Optimal result

Integrand size = 31, antiderivative size = 166 \[ \int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {a}{4 c^2 x^4 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {4 b c^2+5 a d^2}{4 c^4 x^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {3 \left (4 b c^2+5 a d^2\right ) \sqrt {-c+d x} \sqrt {c+d x}}{8 c^6 x^2}-\frac {3 d^2 \left (4 b c^2+5 a d^2\right ) \arctan \left (\frac {\sqrt {-c+d x} \sqrt {c+d x}}{c}\right )}{8 c^7} \] Output:

1/4*a/c^2/x^4/(d*x-c)^(1/2)/(d*x+c)^(1/2)-1/4*(5*a*d^2+4*b*c^2)/c^4/x^2/(d 
*x-c)^(1/2)/(d*x+c)^(1/2)-3/8*(5*a*d^2+4*b*c^2)*(d*x-c)^(1/2)*(d*x+c)^(1/2 
)/c^6/x^2-3/8*d^2*(5*a*d^2+4*b*c^2)*arctan((d*x-c)^(1/2)*(d*x+c)^(1/2)/c)/ 
c^7
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.73 \[ \int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {\frac {4 b c^3 x^2 \left (c^2-3 d^2 x^2\right )+a \left (2 c^5+5 c^3 d^2 x^2-15 c d^4 x^4\right )}{x^4 \sqrt {-c+d x} \sqrt {c+d x}}+6 d^2 \left (4 b c^2+5 a d^2\right ) \arctan \left (\frac {\sqrt {c+d x}}{\sqrt {-c+d x}}\right )}{8 c^7} \] Input:

Integrate[(a + b*x^2)/(x^5*(-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]
 

Output:

((4*b*c^3*x^2*(c^2 - 3*d^2*x^2) + a*(2*c^5 + 5*c^3*d^2*x^2 - 15*c*d^4*x^4) 
)/(x^4*Sqrt[-c + d*x]*Sqrt[c + d*x]) + 6*d^2*(4*b*c^2 + 5*a*d^2)*ArcTan[Sq 
rt[c + d*x]/Sqrt[-c + d*x]])/(8*c^7)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.88, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {956, 114, 27, 115, 27, 103, 218}

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.

\(\displaystyle \int \frac {a+b x^2}{x^5 (d x-c)^{3/2} (c+d x)^{3/2}} \, dx\)

\(\Big \downarrow \) 956

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 115

\(\displaystyle \frac {1}{4} \left (\frac {5 a d^2}{c^2}+4 b\right ) \left (\frac {3 d^2 \left (-\frac {\int \frac {d}{x \sqrt {d x-c} \sqrt {c+d x}}dx}{c^2 d}-\frac {1}{c^2 \sqrt {d x-c} \sqrt {c+d x}}\right )}{2 c^2}+\frac {1}{2 c^2 x^2 \sqrt {d x-c} \sqrt {c+d x}}\right )+\frac {a}{4 c^2 x^4 \sqrt {d x-c} \sqrt {c+d x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {5 a d^2}{c^2}+4 b\right ) \left (\frac {3 d^2 \left (-\frac {\int \frac {1}{x \sqrt {d x-c} \sqrt {c+d x}}dx}{c^2}-\frac {1}{c^2 \sqrt {d x-c} \sqrt {c+d x}}\right )}{2 c^2}+\frac {1}{2 c^2 x^2 \sqrt {d x-c} \sqrt {c+d x}}\right )+\frac {a}{4 c^2 x^4 \sqrt {d x-c} \sqrt {c+d x}}\)

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{4} \left (\frac {5 a d^2}{c^2}+4 b\right ) \left (\frac {3 d^2 \left (-\frac {\arctan \left (\frac {\sqrt {d x-c} \sqrt {c+d x}}{c}\right )}{c^3}-\frac {1}{c^2 \sqrt {d x-c} \sqrt {c+d x}}\right )}{2 c^2}+\frac {1}{2 c^2 x^2 \sqrt {d x-c} \sqrt {c+d x}}\right )+\frac {a}{4 c^2 x^4 \sqrt {d x-c} \sqrt {c+d x}}\)

Input:

Int[(a + b*x^2)/(x^5*(-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]
 

Output:

a/(4*c^2*x^4*Sqrt[-c + d*x]*Sqrt[c + d*x]) + ((4*b + (5*a*d^2)/c^2)*(1/(2* 
c^2*x^2*Sqrt[-c + d*x]*Sqrt[c + d*x]) + (3*d^2*(-(1/(c^2*Sqrt[-c + d*x]*Sq 
rt[c + d*x])) - ArcTan[(Sqrt[-c + d*x]*Sqrt[c + d*x])/c]/c^3))/(2*c^2)))/4
 

Defintions of rubi rules used

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 103

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ 
))), x_] :> Simp[b*f   Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq 
rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d 
*e - f*(b*c + a*d), 0]
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 115

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 
*n, 2*p]
 

rule 218

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

rule 956

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.) 
*(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^( 
m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(a1*a2*e*(m + 1 
))), x] + Simp[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(a1*a2*e^n*( 
m + 1))   Int[(e*x)^(m + n)*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] 
 /; FreeQ[{a1, b1, a2, b2, c, d, e, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + 
 a1*b2, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || ( 
LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.61

method	result	size
risch	\(\frac {\sqrt {d x +c}\, \left (-d x +c \right ) \left (7 a \,d^{2} x^{2}+4 b \,c^{2} x^{2}+2 a \,c^{2}\right )}{8 c^{6} x^{4} \sqrt {d x -c}}-\frac {d^{2} \left (-\frac {\left (15 a \,d^{2}+12 b \,c^{2}\right ) \ln \left (\frac {-2 c^{2}+2 \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}}{x}\right )}{\sqrt {-c^{2}}}+\frac {4 \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d^{2} \left (x -\frac {c}{d}\right )^{2}+2 c d \left (x -\frac {c}{d}\right )}}{d c \left (x -\frac {c}{d}\right )}-\frac {4 \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d^{2} \left (x +\frac {c}{d}\right )^{2}-2 c d \left (x +\frac {c}{d}\right )}}{d c \left (x +\frac {c}{d}\right )}\right ) \sqrt {\left (d x -c \right ) \left (d x +c \right )}}{8 c^{6} \sqrt {d x -c}\, \sqrt {d x +c}}\)	\(267\)
default	\(-\frac {-15 \ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right ) a \,d^{6} x^{6}-12 \ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right ) b \,c^{2} d^{4} x^{6}+15 \ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right ) a \,c^{2} d^{4} x^{4}+12 \ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right ) b \,c^{4} d^{2} x^{4}+15 \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\, a \,d^{4} x^{4}+12 \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\, b \,c^{2} d^{2} x^{4}-5 a \,c^{2} d^{2} x^{2} \sqrt {d^{2} x^{2}-c^{2}}\, \sqrt {-c^{2}}-4 b \,c^{4} x^{2} \sqrt {d^{2} x^{2}-c^{2}}\, \sqrt {-c^{2}}-2 a \,c^{4} \sqrt {d^{2} x^{2}-c^{2}}\, \sqrt {-c^{2}}}{8 \sqrt {d x -c}\, \sqrt {d^{2} x^{2}-c^{2}}\, \sqrt {d x +c}\, c^{6} \sqrt {-c^{2}}\, x^{4}}\)	\(387\)

Input:

int((b*x^2+a)/x^5/(d*x-c)^(3/2)/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/8*(d*x+c)^(1/2)*(-d*x+c)*(7*a*d^2*x^2+4*b*c^2*x^2+2*a*c^2)/c^6/x^4/(d*x- 
c)^(1/2)-1/8*d^2/c^6*(-(15*a*d^2+12*b*c^2)/(-c^2)^(1/2)*ln((-2*c^2+2*(-c^2 
)^(1/2)*(d^2*x^2-c^2)^(1/2))/x)+4*(a*d^2+b*c^2)/d/c/(x-c/d)*(d^2*(x-c/d)^2 
+2*c*d*(x-c/d))^(1/2)-4*(a*d^2+b*c^2)/d/c/(x+c/d)*(d^2*(x+c/d)^2-2*c*d*(x+ 
c/d))^(1/2))*((d*x-c)*(d*x+c))^(1/2)/(d*x-c)^(1/2)/(d*x+c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.99 \[ \int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {{\left (2 \, a c^{5} - 3 \, {\left (4 \, b c^{3} d^{2} + 5 \, a c d^{4}\right )} x^{4} + {\left (4 \, b c^{5} + 5 \, a c^{3} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {d x - c} - 6 \, {\left ({\left (4 \, b c^{2} d^{4} + 5 \, a d^{6}\right )} x^{6} - {\left (4 \, b c^{4} d^{2} + 5 \, a c^{2} d^{4}\right )} x^{4}\right )} \arctan \left (-\frac {d x - \sqrt {d x + c} \sqrt {d x - c}}{c}\right )}{8 \, {\left (c^{7} d^{2} x^{6} - c^{9} x^{4}\right )}} \] Input:

integrate((b*x^2+a)/x^5/(d*x-c)^(3/2)/(d*x+c)^(3/2),x, algorithm="fricas")
 

Output:

1/8*((2*a*c^5 - 3*(4*b*c^3*d^2 + 5*a*c*d^4)*x^4 + (4*b*c^5 + 5*a*c^3*d^2)* 
x^2)*sqrt(d*x + c)*sqrt(d*x - c) - 6*((4*b*c^2*d^4 + 5*a*d^6)*x^6 - (4*b*c 
^4*d^2 + 5*a*c^2*d^4)*x^4)*arctan(-(d*x - sqrt(d*x + c)*sqrt(d*x - c))/c)) 
/(c^7*d^2*x^6 - c^9*x^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)/x**5/(d*x-c)**(3/2)/(d*x+c)**(3/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.98 \[ \int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {3 \, b d^{2} \arcsin \left (\frac {c}{d {\left | x \right |}}\right )}{2 \, c^{5}} + \frac {15 \, a d^{4} \arcsin \left (\frac {c}{d {\left | x \right |}}\right )}{8 \, c^{7}} - \frac {3 \, b d^{2}}{2 \, \sqrt {d^{2} x^{2} - c^{2}} c^{4}} - \frac {15 \, a d^{4}}{8 \, \sqrt {d^{2} x^{2} - c^{2}} c^{6}} + \frac {b}{2 \, \sqrt {d^{2} x^{2} - c^{2}} c^{2} x^{2}} + \frac {5 \, a d^{2}}{8 \, \sqrt {d^{2} x^{2} - c^{2}} c^{4} x^{2}} + \frac {a}{4 \, \sqrt {d^{2} x^{2} - c^{2}} c^{2} x^{4}} \] Input:

integrate((b*x^2+a)/x^5/(d*x-c)^(3/2)/(d*x+c)^(3/2),x, algorithm="maxima")
 

Output:

3/2*b*d^2*arcsin(c/(d*abs(x)))/c^5 + 15/8*a*d^4*arcsin(c/(d*abs(x)))/c^7 - 
 3/2*b*d^2/(sqrt(d^2*x^2 - c^2)*c^4) - 15/8*a*d^4/(sqrt(d^2*x^2 - c^2)*c^6 
) + 1/2*b/(sqrt(d^2*x^2 - c^2)*c^2*x^2) + 5/8*a*d^2/(sqrt(d^2*x^2 - c^2)*c 
^4*x^2) + 1/4*a/(sqrt(d^2*x^2 - c^2)*c^2*x^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 402 vs. \(2 (142) = 284\).

Time = 0.25 (sec) , antiderivative size = 402, normalized size of antiderivative = 2.42 \[ \int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {3 \, {\left (4 \, b c^{2} d^{2} + 5 \, a d^{4}\right )} \arctan \left (\frac {{\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}}{2 \, c}\right )}{4 \, c^{7}} - \frac {{\left (b c^{2} d^{2} + a d^{4}\right )} \sqrt {d x + c}}{2 \, \sqrt {d x - c} c^{7}} + \frac {2 \, {\left (b c^{2} d^{2} + a d^{4}\right )}}{{\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 2 \, c\right )} c^{6}} + \frac {4 \, b c^{2} d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{14} + 7 \, a d^{4} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{14} + 16 \, b c^{4} d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{10} + 60 \, a c^{2} d^{4} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{10} - 64 \, b c^{6} d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{6} - 240 \, a c^{4} d^{4} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{6} - 256 \, b c^{8} d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} - 448 \, a c^{6} d^{4} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}}{2 \, {\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{4} c^{6}} \] Input:

integrate((b*x^2+a)/x^5/(d*x-c)^(3/2)/(d*x+c)^(3/2),x, algorithm="giac")
 

Output:

3/4*(4*b*c^2*d^2 + 5*a*d^4)*arctan(1/2*(sqrt(d*x + c) - sqrt(d*x - c))^2/c 
)/c^7 - 1/2*(b*c^2*d^2 + a*d^4)*sqrt(d*x + c)/(sqrt(d*x - c)*c^7) + 2*(b*c 
^2*d^2 + a*d^4)/(((sqrt(d*x + c) - sqrt(d*x - c))^2 + 2*c)*c^6) + 1/2*(4*b 
*c^2*d^2*(sqrt(d*x + c) - sqrt(d*x - c))^14 + 7*a*d^4*(sqrt(d*x + c) - sqr 
t(d*x - c))^14 + 16*b*c^4*d^2*(sqrt(d*x + c) - sqrt(d*x - c))^10 + 60*a*c^ 
2*d^4*(sqrt(d*x + c) - sqrt(d*x - c))^10 - 64*b*c^6*d^2*(sqrt(d*x + c) - s 
qrt(d*x - c))^6 - 240*a*c^4*d^4*(sqrt(d*x + c) - sqrt(d*x - c))^6 - 256*b* 
c^8*d^2*(sqrt(d*x + c) - sqrt(d*x - c))^2 - 448*a*c^6*d^4*(sqrt(d*x + c) - 
 sqrt(d*x - c))^2)/(((sqrt(d*x + c) - sqrt(d*x - c))^4 + 4*c^2)^4*c^6)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\int \frac {b\,x^2+a}{x^5\,{\left (c+d\,x\right )}^{3/2}\,{\left (d\,x-c\right )}^{3/2}} \,d x \] Input:

int((a + b*x^2)/(x^5*(c + d*x)^(3/2)*(d*x - c)^(3/2)),x)
 

Output:

int((a + b*x^2)/(x^5*(c + d*x)^(3/2)*(d*x - c)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 447, normalized size of antiderivative = 2.69 \[ \int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {-30 \sqrt {d x -c}\, \mathit {atan} \left (\frac {\sqrt {d x -c}+\sqrt {d x +c}-\sqrt {c}}{\sqrt {c}}\right ) a c \,d^{4} x^{4}-30 \sqrt {d x -c}\, \mathit {atan} \left (\frac {\sqrt {d x -c}+\sqrt {d x +c}-\sqrt {c}}{\sqrt {c}}\right ) a \,d^{5} x^{5}-24 \sqrt {d x -c}\, \mathit {atan} \left (\frac {\sqrt {d x -c}+\sqrt {d x +c}-\sqrt {c}}{\sqrt {c}}\right ) b \,c^{3} d^{2} x^{4}-24 \sqrt {d x -c}\, \mathit {atan} \left (\frac {\sqrt {d x -c}+\sqrt {d x +c}-\sqrt {c}}{\sqrt {c}}\right ) b \,c^{2} d^{3} x^{5}+30 \sqrt {d x -c}\, \mathit {atan} \left (\frac {\sqrt {d x -c}+\sqrt {d x +c}+\sqrt {c}}{\sqrt {c}}\right ) a c \,d^{4} x^{4}+30 \sqrt {d x -c}\, \mathit {atan} \left (\frac {\sqrt {d x -c}+\sqrt {d x +c}+\sqrt {c}}{\sqrt {c}}\right ) a \,d^{5} x^{5}+24 \sqrt {d x -c}\, \mathit {atan} \left (\frac {\sqrt {d x -c}+\sqrt {d x +c}+\sqrt {c}}{\sqrt {c}}\right ) b \,c^{3} d^{2} x^{4}+24 \sqrt {d x -c}\, \mathit {atan} \left (\frac {\sqrt {d x -c}+\sqrt {d x +c}+\sqrt {c}}{\sqrt {c}}\right ) b \,c^{2} d^{3} x^{5}+2 \sqrt {d x +c}\, a \,c^{5}+5 \sqrt {d x +c}\, a \,c^{3} d^{2} x^{2}-15 \sqrt {d x +c}\, a c \,d^{4} x^{4}+4 \sqrt {d x +c}\, b \,c^{5} x^{2}-12 \sqrt {d x +c}\, b \,c^{3} d^{2} x^{4}}{8 \sqrt {d x -c}\, c^{7} x^{4} \left (d x +c \right )} \] Input:

int((b*x^2+a)/x^5/(d*x-c)^(3/2)/(d*x+c)^(3/2),x)
 

Output:

( - 30*sqrt( - c + d*x)*atan((sqrt( - c + d*x) + sqrt(c + d*x) - sqrt(c))/ 
sqrt(c))*a*c*d**4*x**4 - 30*sqrt( - c + d*x)*atan((sqrt( - c + d*x) + sqrt 
(c + d*x) - sqrt(c))/sqrt(c))*a*d**5*x**5 - 24*sqrt( - c + d*x)*atan((sqrt 
( - c + d*x) + sqrt(c + d*x) - sqrt(c))/sqrt(c))*b*c**3*d**2*x**4 - 24*sqr 
t( - c + d*x)*atan((sqrt( - c + d*x) + sqrt(c + d*x) - sqrt(c))/sqrt(c))*b 
*c**2*d**3*x**5 + 30*sqrt( - c + d*x)*atan((sqrt( - c + d*x) + sqrt(c + d* 
x) + sqrt(c))/sqrt(c))*a*c*d**4*x**4 + 30*sqrt( - c + d*x)*atan((sqrt( - c 
 + d*x) + sqrt(c + d*x) + sqrt(c))/sqrt(c))*a*d**5*x**5 + 24*sqrt( - c + d 
*x)*atan((sqrt( - c + d*x) + sqrt(c + d*x) + sqrt(c))/sqrt(c))*b*c**3*d**2 
*x**4 + 24*sqrt( - c + d*x)*atan((sqrt( - c + d*x) + sqrt(c + d*x) + sqrt( 
c))/sqrt(c))*b*c**2*d**3*x**5 + 2*sqrt(c + d*x)*a*c**5 + 5*sqrt(c + d*x)*a 
*c**3*d**2*x**2 - 15*sqrt(c + d*x)*a*c*d**4*x**4 + 4*sqrt(c + d*x)*b*c**5* 
x**2 - 12*sqrt(c + d*x)*b*c**3*d**2*x**4)/(8*sqrt( - c + d*x)*c**7*x**4*(c 
 + d*x))