\(\int \frac {(\frac {e (a+b x^2)}{c+d x^2})^{3/2}}{x^5} \, dx\) [60]

Optimal result

Integrand size = 26, antiderivative size = 252 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^5} \, dx=-\frac {d (b c-a d) e \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{c^3}-\frac {(5 b c-9 a d) e \left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{8 c^3 x^2}-\frac {a e \left (c+d x^2\right )^2 \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{4 c^3 x^4}-\frac {3 (b c-5 a d) (b c-a d) e^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{\sqrt {a} \sqrt {e}}\right )}{8 \sqrt {a} c^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 4.43 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.74 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^5} \, dx=-\frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {a} \sqrt {c} \sqrt {a+b x^2} \left (b c x^2 \left (5 c+13 d x^2\right )+a \left (2 c^2-5 c d x^2-15 d^2 x^4\right )\right )+3 \left (b^2 c^2-6 a b c d+5 a^2 d^2\right ) x^4 \sqrt {c+d x^2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )\right )}{8 \sqrt {a} c^{7/2} x^4 \sqrt {a+b x^2}} \] Input:

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

Output:

-1/8*(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x^2] 
*(b*c*x^2*(5*c + 13*d*x^2) + a*(2*c^2 - 5*c*d*x^2 - 15*d^2*x^4)) + 3*(b^2* 
c^2 - 6*a*b*c*d + 5*a^2*d^2)*x^4*Sqrt[c + d*x^2]*ArcTanh[(Sqrt[c]*Sqrt[a + 
 b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])]))/(Sqrt[a]*c^(7/2)*x^4*Sqrt[a + b*x^2] 
)
 

Rubi [A] (warning: unable to verify)

Time = 0.72 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.85, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {2053, 2052, 25, 360, 25, 1471, 27, 299, 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.

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

\(\Big \downarrow \) 2053

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

\(\Big \downarrow \) 2052

\(\displaystyle e (b c-a d) \int -\frac {x^8 \left (b e-d x^4\right )}{\left (a e-c x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\left (e (b c-a d) \int \frac {x^8 \left (b e-d x^4\right )}{\left (a e-c x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\right )\)

\(\Big \downarrow \) 360

\(\displaystyle e (b c-a d) \left (-\frac {\int -\frac {-4 c^2 d x^8+4 c (b c-a d) e x^4+a (b c-a d) e^2}{\left (a e-c x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 c^3}-\frac {a e^2 (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-c x^4\right )^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle e (b c-a d) \left (\frac {\int \frac {-4 c^2 d x^8+4 c (b c-a d) e x^4+a (b c-a d) e^2}{\left (a e-c x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 c^3}-\frac {a e^2 (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-c x^4\right )^2}\right )\)

\(\Big \downarrow \) 1471

\(\displaystyle e (b c-a d) \left (\frac {\frac {e (5 b c-9 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (a e-c x^4\right )}-\frac {\int \frac {a e \left ((3 b c-7 a d) e-8 c d x^4\right )}{a e-c x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{2 a e}}{4 c^3}-\frac {a e^2 (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-c x^4\right )^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle e (b c-a d) \left (\frac {\frac {e (5 b c-9 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (a e-c x^4\right )}-\frac {1}{2} \int \frac {(3 b c-7 a d) e-8 c d x^4}{a e-c x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 c^3}-\frac {a e^2 (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-c x^4\right )^2}\right )\)

\(\Big \downarrow \) 299

\(\displaystyle e (b c-a d) \left (\frac {\frac {1}{2} \left (-3 e (b c-5 a d) \int \frac {1}{a e-c x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}-8 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )+\frac {e (5 b c-9 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (a e-c x^4\right )}}{4 c^3}-\frac {a e^2 (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-c x^4\right )^2}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle e (b c-a d) \left (\frac {\frac {1}{2} \left (-\frac {3 \sqrt {e} (b c-5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {a} \sqrt {c}}-8 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )+\frac {e (5 b c-9 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (a e-c x^4\right )}}{4 c^3}-\frac {a e^2 (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-c x^4\right )^2}\right )\)

Input:

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

Output:

(b*c - a*d)*e*(-1/4*(a*(b*c - a*d)*e^2*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/ 
(c^3*(a*e - c*x^4)^2) + (((5*b*c - 9*a*d)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^ 
2)])/(2*(a*e - c*x^4)) + (-8*d*Sqrt[(e*(a + b*x^2))/(c + d*x^2)] - (3*(b*c 
 - 5*a*d)*Sqrt[e]*ArcTanh[(Sqrt[c]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqr 
t[a]*Sqrt[e])])/(Sqrt[a]*Sqrt[c]))/2)/(4*c^3))
 

Defintions of rubi rules used

rule 25

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

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 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 299

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

rule 360

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : 
> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p 
+ 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1))   Int[(a + b*x^2)^(p + 1)*Expan 
dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 
- 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; 
FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & 
& (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
 

rule 1471

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 
, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x 
, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q 
 + 1))   Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), 
x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 
2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

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 = 0.23 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.07

method	result	size
risch	\(-\frac {\left (d \,x^{2}+c \right ) \left (-7 a d \,x^{2}+5 b c \,x^{2}+2 a c \right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{8 c^{3} x^{4}}+\frac {\left (-\frac {\left (15 a^{2} d^{2}-18 a b c d +3 b^{2} c^{2}\right ) \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {b d e \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right )}{2 \sqrt {a c e}}+\frac {8 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b \,x^{2}+a \right )}{\left (a d -b c \right ) \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) e}}{8 c^{3} \left (b \,x^{2}+a \right )}\)	\(270\)
default	\(\text {Expression too large to display}\)	\(1042\)

Input:

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

Output:

-1/8*(d*x^2+c)*(-7*a*d*x^2+5*b*c*x^2+2*a*c)/c^3/x^4*e*(e*(b*x^2+a)/(d*x^2+ 
c))^(1/2)+1/8/c^3*(-1/2*(15*a^2*d^2-18*a*b*c*d+3*b^2*c^2)/(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)+8*d*(a^2*d^2-2*a*b*c*d+b^2*c^2)*(b*x^2+a)/(a*d-b*c)/(b*d 
*e*x^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2))*e/(b*x^2+a)*(e*(b*x^2+a)/(d*x^2+c 
))^(1/2)*((d*x^2+c)*(b*x^2+a)*e)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 1.96 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.73 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\left [\frac {3 \, {\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} e x^{4} \sqrt {\frac {e}{a c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e x^{4} + 8 \, a^{2} c^{2} e + 8 \, {\left (a b c^{2} + a^{2} c d\right )} e x^{2} - 4 \, {\left (2 \, a^{2} c^{3} + {\left (a b c^{2} d + a^{2} c d^{2}\right )} x^{4} + {\left (a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {e}{a c}}}{x^{4}}\right ) - 4 \, {\left ({\left (13 \, b c d - 15 \, a d^{2}\right )} e x^{4} + 2 \, a c^{2} e + 5 \, {\left (b c^{2} - a c d\right )} e x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{32 \, c^{3} x^{4}}, \frac {3 \, {\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} e x^{4} \sqrt {-\frac {e}{a c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {e}{a c}}}{2 \, {\left (b e x^{2} + a e\right )}}\right ) - 2 \, {\left ({\left (13 \, b c d - 15 \, a d^{2}\right )} e x^{4} + 2 \, a c^{2} e + 5 \, {\left (b c^{2} - a c d\right )} e x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{16 \, c^{3} x^{4}}\right ] \] Input:

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

Output:

[1/32*(3*(b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*e*x^4*sqrt(e/(a*c))*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*(2*a^2*c^3 + (a*b*c^2*d + a^2*c*d^2)*x^4 + (a*b*c^3 + 3*a^2*c^2*d)*x 
^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))*sqrt(e/(a*c)))/x^4) - 4*((13*b*c*d - 
 15*a*d^2)*e*x^4 + 2*a*c^2*e + 5*(b*c^2 - a*c*d)*e*x^2)*sqrt((b*e*x^2 + a* 
e)/(d*x^2 + c)))/(c^3*x^4), 1/16*(3*(b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*e*x^ 
4*sqrt(-e/(a*c))*arctan(1/2*((b*c + a*d)*x^2 + 2*a*c)*sqrt((b*e*x^2 + a*e) 
/(d*x^2 + c))*sqrt(-e/(a*c))/(b*e*x^2 + a*e)) - 2*((13*b*c*d - 15*a*d^2)*e 
*x^4 + 2*a*c^2*e + 5*(b*c^2 - a*c*d)*e*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + 
c)))/(c^3*x^4)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\text {Exception raised: ValueError} \] Input:

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

Exception generated. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{2,[4,1,4]%%%},[2,7,0]%%%}+%%%{%%%{-8,[3,2,4]%%%},[2,6, 
1]%%%}+%%
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 515, normalized size of antiderivative = 2.04 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\frac {\sqrt {e}\, e \left (-2 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} c^{3}+5 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} c^{2} d \,x^{2}+15 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} c \,d^{2} x^{4}-5 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b \,c^{3} x^{2}-13 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b \,c^{2} d \,x^{4}+15 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) a^{2} c \,d^{2} x^{4}+15 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) a^{2} d^{3} x^{6}-18 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) a b \,c^{2} d \,x^{4}-18 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) a b c \,d^{2} x^{6}+3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) b^{2} c^{3} x^{4}+3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) b^{2} c^{2} d \,x^{6}-15 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} c \,d^{2} x^{4}-15 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} d^{3} x^{6}+18 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) a b \,c^{2} d \,x^{4}+18 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) a b c \,d^{2} x^{6}-3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) b^{2} c^{3} x^{4}-3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) b^{2} c^{2} d \,x^{6}\right )}{8 a \,c^{4} x^{4} \left (d \,x^{2}+c \right )} \] Input:

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

Output:

(sqrt(e)*e*( - 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*c**3 + 5*sqrt(c + 
d*x**2)*sqrt(a + b*x**2)*a**2*c**2*d*x**2 + 15*sqrt(c + d*x**2)*sqrt(a + b 
*x**2)*a**2*c*d**2*x**4 - 5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c**3*x** 
2 - 13*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c**2*d*x**4 + 15*sqrt(c)*sqrt 
(a)*log(sqrt(a)*sqrt(a + b*x**2)*c - sqrt(c)*sqrt(c + d*x**2)*a)*a**2*c*d* 
*2*x**4 + 15*sqrt(c)*sqrt(a)*log(sqrt(a)*sqrt(a + b*x**2)*c - sqrt(c)*sqrt 
(c + d*x**2)*a)*a**2*d**3*x**6 - 18*sqrt(c)*sqrt(a)*log(sqrt(a)*sqrt(a + b 
*x**2)*c - sqrt(c)*sqrt(c + d*x**2)*a)*a*b*c**2*d*x**4 - 18*sqrt(c)*sqrt(a 
)*log(sqrt(a)*sqrt(a + b*x**2)*c - sqrt(c)*sqrt(c + d*x**2)*a)*a*b*c*d**2* 
x**6 + 3*sqrt(c)*sqrt(a)*log(sqrt(a)*sqrt(a + b*x**2)*c - sqrt(c)*sqrt(c + 
 d*x**2)*a)*b**2*c**3*x**4 + 3*sqrt(c)*sqrt(a)*log(sqrt(a)*sqrt(a + b*x**2 
)*c - sqrt(c)*sqrt(c + d*x**2)*a)*b**2*c**2*d*x**6 - 15*sqrt(c)*sqrt(a)*lo 
g(x)*a**2*c*d**2*x**4 - 15*sqrt(c)*sqrt(a)*log(x)*a**2*d**3*x**6 + 18*sqrt 
(c)*sqrt(a)*log(x)*a*b*c**2*d*x**4 + 18*sqrt(c)*sqrt(a)*log(x)*a*b*c*d**2* 
x**6 - 3*sqrt(c)*sqrt(a)*log(x)*b**2*c**3*x**4 - 3*sqrt(c)*sqrt(a)*log(x)* 
b**2*c**2*d*x**6))/(8*a*c**4*x**4*(c + d*x**2))