\(\int (a+\frac {b}{x^2}) (c+\frac {d}{x^2})^{3/2} x^5 \, dx\) [153]

Optimal result

Integrand size = 22, antiderivative size = 115 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^5 \, dx=\frac {d (10 b c+a d) \sqrt {c+\frac {d}{x^2}} x^2}{16 c}+\frac {1}{8} (2 b c+a d) \sqrt {c+\frac {d}{x^2}} x^4+\frac {1}{6} a \left (c+\frac {d}{x^2}\right )^{3/2} x^6+\frac {d^2 (6 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{16 c^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.07 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^5 \, dx=\frac {\sqrt {c+\frac {d}{x^2}} x \left (\sqrt {c} x \sqrt {d+c x^2} \left (6 b c \left (5 d+2 c x^2\right )+a \left (3 d^2+14 c d x^2+8 c^2 x^4\right )\right )+3 d^2 (-6 b c+a d) \log \left (-\sqrt {c} x+\sqrt {d+c x^2}\right )\right )}{48 c^{3/2} \sqrt {d+c x^2}} \] Input:

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

Output:

(Sqrt[c + d/x^2]*x*(Sqrt[c]*x*Sqrt[d + c*x^2]*(6*b*c*(5*d + 2*c*x^2) + a*( 
3*d^2 + 14*c*d*x^2 + 8*c^2*x^4)) + 3*d^2*(-6*b*c + a*d)*Log[-(Sqrt[c]*x) + 
 Sqrt[d + c*x^2]]))/(48*c^(3/2)*Sqrt[d + c*x^2])
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 110, 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 = {948, 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.

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

\(\Big \downarrow \) 948

\(\displaystyle -\frac {1}{2} \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^8d\frac {1}{x^2}\)

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

((a*(c + d/x^2)^(5/2)*x^6)/(3*c) - ((6*b*c - a*d)*(-1/2*((c + d/x^2)^(3/2) 
*x^4) + (3*d*(-(Sqrt[c + d/x^2]*x^2) - (d*ArcTanh[Sqrt[c + d/x^2]/Sqrt[c]] 
)/Sqrt[c]))/4))/(6*c))/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 948

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

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.01

method	result	size
risch	\(\frac {x^{2} \left (8 a \,c^{2} x^{4}+14 a d \,x^{2} c +12 b \,c^{2} x^{2}+3 a \,d^{2}+30 d b c \right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{48 c}-\frac {d^{2} \left (a d -6 c b \right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x}{16 c^{\frac {3}{2}} \sqrt {c \,x^{2}+d}}\)	\(116\)
default	\(\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} x^{3} \left (8 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {5}{2}} a x -2 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {3}{2}} a d x +12 c^{\frac {3}{2}} \left (c \,x^{2}+d \right )^{\frac {3}{2}} b x -3 \sqrt {c}\, \sqrt {c \,x^{2}+d}\, a \,d^{2} x +18 c^{\frac {3}{2}} \sqrt {c \,x^{2}+d}\, b d x -3 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) a \,d^{3}+18 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) b c \,d^{2}\right )}{48 \left (c \,x^{2}+d \right )^{\frac {3}{2}} c^{\frac {3}{2}}}\)	\(162\)

Input:

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

Output:

1/48/c*x^2*(8*a*c^2*x^4+14*a*c*d*x^2+12*b*c^2*x^2+3*a*d^2+30*b*c*d)*((c*x^ 
2+d)/x^2)^(1/2)-1/16*d^2*(a*d-6*b*c)/c^(3/2)*ln(c^(1/2)*x+(c*x^2+d)^(1/2)) 
*((c*x^2+d)/x^2)^(1/2)*x/(c*x^2+d)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.11 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^5 \, dx=\left [-\frac {3 \, {\left (6 \, b c d^{2} - a d^{3}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) - 2 \, {\left (8 \, a c^{3} x^{6} + 2 \, {\left (6 \, b c^{3} + 7 \, a c^{2} d\right )} x^{4} + 3 \, {\left (10 \, b c^{2} d + a c d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{96 \, c^{2}}, -\frac {3 \, {\left (6 \, b c d^{2} - a d^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - {\left (8 \, a c^{3} x^{6} + 2 \, {\left (6 \, b c^{3} + 7 \, a c^{2} d\right )} x^{4} + 3 \, {\left (10 \, b c^{2} d + a c d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{48 \, c^{2}}\right ] \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (102) = 204\).

Time = 68.92 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.20 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^5 \, dx=\frac {a c^{2} x^{7}}{6 \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {11 a c \sqrt {d} x^{5}}{24 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {17 a d^{\frac {3}{2}} x^{3}}{48 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {a d^{\frac {5}{2}} x}{16 c \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {a d^{3} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{16 c^{\frac {3}{2}}} + \frac {b c^{2} x^{5}}{4 \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 b c \sqrt {d} x^{3}}{8 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {b d^{\frac {3}{2}} x \sqrt {\frac {c x^{2}}{d} + 1}}{2} + \frac {b d^{\frac {3}{2}} x}{8 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 b d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{8 \sqrt {c}} \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (95) = 190\).

Time = 0.12 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.09 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^5 \, dx=\frac {1}{96} \, {\left (\frac {3 \, d^{3} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d^{3} + 8 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c d^{3} - 3 \, \sqrt {c + \frac {d}{x^{2}}} c^{2} d^{3}\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{3} c - 3 \, {\left (c + \frac {d}{x^{2}}\right )}^{2} c^{2} + 3 \, {\left (c + \frac {d}{x^{2}}\right )} c^{3} - c^{4}}\right )} a - \frac {1}{16} \, {\left (\frac {3 \, d^{2} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{\sqrt {c}} - \frac {2 \, {\left (5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d^{2} - 3 \, \sqrt {c + \frac {d}{x^{2}}} c d^{2}\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{2} - 2 \, {\left (c + \frac {d}{x^{2}}\right )} c + c^{2}}\right )} b \] Input:

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

Output:

1/96*(3*d^3*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c)))/c 
^(3/2) + 2*(3*(c + d/x^2)^(5/2)*d^3 + 8*(c + d/x^2)^(3/2)*c*d^3 - 3*sqrt(c 
 + d/x^2)*c^2*d^3)/((c + d/x^2)^3*c - 3*(c + d/x^2)^2*c^2 + 3*(c + d/x^2)* 
c^3 - c^4))*a - 1/16*(3*d^2*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^ 
2) + sqrt(c)))/sqrt(c) - 2*(5*(c + d/x^2)^(3/2)*d^2 - 3*sqrt(c + d/x^2)*c* 
d^2)/((c + d/x^2)^2 - 2*(c + d/x^2)*c + c^2))*b
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.25 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^5 \, dx=\frac {1}{48} \, {\left (2 \, {\left (4 \, a c x^{2} \mathrm {sgn}\left (x\right ) + \frac {6 \, b c^{5} \mathrm {sgn}\left (x\right ) + 7 \, a c^{4} d \mathrm {sgn}\left (x\right )}{c^{4}}\right )} x^{2} + \frac {3 \, {\left (10 \, b c^{4} d \mathrm {sgn}\left (x\right ) + a c^{3} d^{2} \mathrm {sgn}\left (x\right )\right )}}{c^{4}}\right )} \sqrt {c x^{2} + d} x - \frac {{\left (6 \, b c d^{2} \mathrm {sgn}\left (x\right ) - a d^{3} \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + d} \right |}\right )}{16 \, c^{\frac {3}{2}}} + \frac {{\left (6 \, b c d^{2} \log \left ({\left | d \right |}\right ) - a d^{3} \log \left ({\left | d \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{32 \, c^{\frac {3}{2}}} \] Input:

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

Output:

1/48*(2*(4*a*c*x^2*sgn(x) + (6*b*c^5*sgn(x) + 7*a*c^4*d*sgn(x))/c^4)*x^2 + 
 3*(10*b*c^4*d*sgn(x) + a*c^3*d^2*sgn(x))/c^4)*sqrt(c*x^2 + d)*x - 1/16*(6 
*b*c*d^2*sgn(x) - a*d^3*sgn(x))*log(abs(-sqrt(c)*x + sqrt(c*x^2 + d)))/c^( 
3/2) + 1/32*(6*b*c*d^2*log(abs(d)) - a*d^3*log(abs(d)))*sgn(x)/c^(3/2)
 

Mupad [B] (verification not implemented)

Time = 5.03 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.13 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^5 \, dx=\frac {a\,x^6\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{6}+\frac {5\,b\,x^4\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{8}+\frac {a\,x^6\,{\left (c+\frac {d}{x^2}\right )}^{5/2}}{16\,c}+\frac {3\,b\,d^2\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8\,\sqrt {c}}-\frac {a\,c\,x^6\,\sqrt {c+\frac {d}{x^2}}}{16}-\frac {3\,b\,c\,x^4\,\sqrt {c+\frac {d}{x^2}}}{8}+\frac {a\,d^3\,\mathrm {atan}\left (\frac {\sqrt {c+\frac {d}{x^2}}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{16\,c^{3/2}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.26 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^5 \, dx=\frac {8 \sqrt {c \,x^{2}+d}\, a \,c^{3} x^{5}+14 \sqrt {c \,x^{2}+d}\, a \,c^{2} d \,x^{3}+3 \sqrt {c \,x^{2}+d}\, a c \,d^{2} x +12 \sqrt {c \,x^{2}+d}\, b \,c^{3} x^{3}+30 \sqrt {c \,x^{2}+d}\, b \,c^{2} d x -3 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x}{\sqrt {d}}\right ) a \,d^{3}+18 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x}{\sqrt {d}}\right ) b c \,d^{2}}{48 c^{2}} \] Input:

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

Output:

(8*sqrt(c*x**2 + d)*a*c**3*x**5 + 14*sqrt(c*x**2 + d)*a*c**2*d*x**3 + 3*sq 
rt(c*x**2 + d)*a*c*d**2*x + 12*sqrt(c*x**2 + d)*b*c**3*x**3 + 30*sqrt(c*x* 
*2 + d)*b*c**2*d*x - 3*sqrt(c)*log((sqrt(c*x**2 + d) + sqrt(c)*x)/sqrt(d)) 
*a*d**3 + 18*sqrt(c)*log((sqrt(c*x**2 + d) + sqrt(c)*x)/sqrt(d))*b*c*d**2) 
/(48*c**2)