\(\int \frac {1}{x^5 \sqrt {\frac {e (a+b x^2)}{c+d x^2}}} \, dx\) [80]

Optimal result

Integrand size = 26, antiderivative size = 211 \[ \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {(3 b c+a d) \left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{8 a^2 c e x^2}-\frac {\left (c+d x^2\right )^2 \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{4 a c e x^4}-\frac {(b c-a d) (3 b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{\sqrt {a} \sqrt {e}}\right )}{8 a^{5/2} c^{3/2} \sqrt {e}} \] Output:

1/8*(a*d+3*b*c)*(d*x^2+c)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/a^2/c/e/x 
^2-1/4*(d*x^2+c)^2*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/a/c/e/x^4-1/8*(- 
a*d+b*c)*(a*d+3*b*c)*arctanh(c^(1/2)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2 
)/a^(1/2)/e^(1/2))/a^(5/2)/c^(3/2)/e^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {\sqrt {a} \sqrt {c} \left (a+b x^2\right ) \sqrt {c+d x^2} \left (3 b c x^2-a \left (2 c+d x^2\right )\right )-\left (3 b^2 c^2-2 a b c d-a^2 d^2\right ) x^4 \sqrt {a+b x^2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{5/2} c^{3/2} x^4 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}} \] Input:

Integrate[1/(x^5*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]),x]
 

Output:

(Sqrt[a]*Sqrt[c]*(a + b*x^2)*Sqrt[c + d*x^2]*(3*b*c*x^2 - a*(2*c + d*x^2)) 
 - (3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*x^4*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[c]* 
Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(8*a^(5/2)*c^(3/2)*x^4*Sqrt[( 
e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2])
 

Rubi [A] (warning: unable to verify)

Time = 0.56 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2053, 2052, 25, 298, 215, 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.

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

\(\Big \downarrow \) 2053

\(\displaystyle \frac {1}{2} \int \frac {1}{x^6 \sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}dx^2\)

\(\Big \downarrow \) 2052

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 298

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 221

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

Input:

Int[1/(x^5*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]),x]
 

Output:

(b*c - a*d)*e*(-1/4*((b*c - a*d)*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(a*c*( 
a*e - c*x^4)^2) - ((3*b*c + a*d)*(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/(2*a*e 
*(a*e - c*x^4)) + ArcTanh[(Sqrt[c]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqr 
t[a]*Sqrt[e])]/(2*a^(3/2)*Sqrt[c]*e^(3/2))))/(4*a*c))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

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 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
 

rule 2052

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

rule 2053

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

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.95

method	result	size
risch	\(-\frac {\left (b \,x^{2}+a \right ) \left (a d \,x^{2}-3 b c \,x^{2}+2 a c \right )}{8 a^{2} x^{4} c \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}+\frac {\left (a^{2} d^{2}+2 a b c d -3 b^{2} c^{2}\right ) \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {b d e \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) e}}{16 a^{2} c \sqrt {a c e}\, \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(200\)
default	\(-\frac {\left (b \,x^{2}+a \right ) \left (2 d^{2} b \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{6} a \sqrt {a c}+10 d \,b^{2} \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{6} c \sqrt {a c}-a^{3} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) d^{2} c \,x^{4}-2 \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) d b \,a^{2} c^{2} x^{4}+3 c^{3} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) b^{2} a \,x^{4}+2 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, d^{2} a^{2} x^{4} \sqrt {a c}+8 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a b c d \,x^{4}+10 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b^{2} c^{2} x^{4} \sqrt {a c}-2 \left (d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} d a \,x^{2} \sqrt {a c}-10 \left (d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} b c \,x^{2} \sqrt {a c}+4 \left (d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} a c \sqrt {a c}\right )}{16 \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a^{3} c^{2} x^{4} \sqrt {a c}}\)	\(559\)

Input:

int(1/x^5/(e*(b*x^2+a)/(d*x^2+c))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/8*(b*x^2+a)*(a*d*x^2-3*b*c*x^2+2*a*c)/a^2/x^4/c/(e*(b*x^2+a)/(d*x^2+c)) 
^(1/2)+1/16*(a^2*d^2+2*a*b*c*d-3*b^2*c^2)/a^2/c/(a*c*e)^(1/2)*ln((2*a*c*e+ 
(a*d*e+b*c*e)*x^2+2*(a*c*e)^(1/2)*(b*d*e*x^4+(a*d*e+b*c*e)*x^2+a*c*e)^(1/2 
))/x^2)/(e*(b*x^2+a)/(d*x^2+c))^(1/2)*((d*x^2+c)*(b*x^2+a)*e)^(1/2)/(d*x^2 
+c)
 

Fricas [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.10 \[ \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\left [-\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {a c e} x^{4} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e x^{4} + 8 \, a^{2} c^{2} e + 8 \, {\left (a b c^{2} + a^{2} c d\right )} e x^{2} + 4 \, {\left ({\left (b c d + a d^{2}\right )} x^{4} + 2 \, a c^{2} + {\left (b c^{2} + 3 \, a c d\right )} x^{2}\right )} \sqrt {a c e} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{x^{4}}\right ) + 4 \, {\left (2 \, a^{2} c^{3} - {\left (3 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{4} - 3 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{32 \, a^{3} c^{2} e x^{4}}, \frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {-a c e} x^{4} \arctan \left (\frac {\sqrt {-a c e} {\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{2 \, {\left (a b c e x^{2} + a^{2} c e\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{3} - {\left (3 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{4} - 3 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{16 \, a^{3} c^{2} e x^{4}}\right ] \] Input:

integrate(1/x^5/(e*(b*x^2+a)/(d*x^2+c))^(1/2),x, algorithm="fricas")
 

Output:

[-1/32*((3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*sqrt(a*c*e)*x^4*log(((b^2*c^2 + 
6*a*b*c*d + a^2*d^2)*e*x^4 + 8*a^2*c^2*e + 8*(a*b*c^2 + a^2*c*d)*e*x^2 + 4 
*((b*c*d + a*d^2)*x^4 + 2*a*c^2 + (b*c^2 + 3*a*c*d)*x^2)*sqrt(a*c*e)*sqrt( 
(b*e*x^2 + a*e)/(d*x^2 + c)))/x^4) + 4*(2*a^2*c^3 - (3*a*b*c^2*d - a^2*c*d 
^2)*x^4 - 3*(a*b*c^3 - a^2*c^2*d)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/ 
(a^3*c^2*e*x^4), 1/16*((3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*sqrt(-a*c*e)*x^4* 
arctan(1/2*sqrt(-a*c*e)*((b*c + a*d)*x^2 + 2*a*c)*sqrt((b*e*x^2 + a*e)/(d* 
x^2 + c))/(a*b*c*e*x^2 + a^2*c*e)) - 2*(2*a^2*c^3 - (3*a*b*c^2*d - a^2*c*d 
^2)*x^4 - 3*(a*b*c^3 - a^2*c^2*d)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/ 
(a^3*c^2*e*x^4)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\text {Timed out} \] Input:

integrate(1/x**5/(e*(b*x**2+a)/(d*x**2+c))**(1/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/x^5/(e*(b*x^2+a)/(d*x^2+c))^(1/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (187) = 374\).

Time = 0.18 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.54 \[ \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}}{\sqrt {-a c e}}\right )}{\sqrt {-a c e} a^{2} c} + \frac {5 \, {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )} a b^{2} c^{3} e + 10 \, {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )} a^{2} b c^{2} d e + {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )} a^{3} c d^{2} e + 8 \, \sqrt {b d e} a^{2} b c^{3} e - 3 \, {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )}^{3} b^{2} c^{2} + 2 \, {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )}^{3} a b c d + {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )}^{3} a^{2} d^{2} + 8 \, \sqrt {b d e} {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )}^{2} a^{2} c d}{{\left (a c e - {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )}^{2}\right )}^{2} a^{2} c}}{8 \, \mathrm {sgn}\left (d x^{2} + c\right )} \] Input:

integrate(1/x^5/(e*(b*x^2+a)/(d*x^2+c))^(1/2),x, algorithm="giac")
 

Output:

1/8*((3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*arctan(-(sqrt(b*d*e)*x^2 - sqrt(b*d 
*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))/sqrt(-a*c*e))/(sqrt(-a*c*e)*a^2*c 
) + (5*(sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e)) 
*a*b^2*c^3*e + 10*(sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^ 
2 + a*c*e))*a^2*b*c^2*d*e + (sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 
+ a*d*e*x^2 + a*c*e))*a^3*c*d^2*e + 8*sqrt(b*d*e)*a^2*b*c^3*e - 3*(sqrt(b* 
d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))^3*b^2*c^2 + 2* 
(sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))^3*a*b* 
c*d + (sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))^ 
3*a^2*d^2 + 8*sqrt(b*d*e)*(sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 + 
a*d*e*x^2 + a*c*e))^2*a^2*c*d)/((a*c*e - (sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 
 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))^2)^2*a^2*c))/sgn(d*x^2 + c)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\int \frac {1}{x^5\,\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}} \,d x \] Input:

int(1/(x^5*((e*(a + b*x^2))/(c + d*x^2))^(1/2)),x)
 

Output:

int(1/(x^5*((e*(a + b*x^2))/(c + d*x^2))^(1/2)), x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {\sqrt {e}\, \left (-2 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} c^{2}-\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} c d \,x^{2}+3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b \,c^{2} x^{2}+\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) a^{2} d^{2} x^{4}+2 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) a b c d \,x^{4}-3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) b^{2} c^{2} x^{4}-\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} d^{2} x^{4}-2 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) a b c d \,x^{4}+3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) b^{2} c^{2} x^{4}\right )}{8 a^{3} c^{2} e \,x^{4}} \] Input:

int(1/x^5/(e*(b*x^2+a)/(d*x^2+c))^(1/2),x)
 

Output:

(sqrt(e)*( - 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*c**2 - sqrt(c + d*x* 
*2)*sqrt(a + b*x**2)*a**2*c*d*x**2 + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a 
*b*c**2*x**2 + sqrt(c)*sqrt(a)*log( - sqrt(a)*sqrt(a + b*x**2)*c - sqrt(c) 
*sqrt(c + d*x**2)*a)*a**2*d**2*x**4 + 2*sqrt(c)*sqrt(a)*log( - sqrt(a)*sqr 
t(a + b*x**2)*c - sqrt(c)*sqrt(c + d*x**2)*a)*a*b*c*d*x**4 - 3*sqrt(c)*sqr 
t(a)*log( - sqrt(a)*sqrt(a + b*x**2)*c - sqrt(c)*sqrt(c + d*x**2)*a)*b**2* 
c**2*x**4 - sqrt(c)*sqrt(a)*log(x)*a**2*d**2*x**4 - 2*sqrt(c)*sqrt(a)*log( 
x)*a*b*c*d*x**4 + 3*sqrt(c)*sqrt(a)*log(x)*b**2*c**2*x**4))/(8*a**3*c**2*e 
*x**4)