\(\int \frac {x^5}{(\frac {e (a+b x^2)}{c+d x^2})^{3/2}} \, dx\) [86]

Optimal result

Integrand size = 26, antiderivative size = 350 \[ \int \frac {x^5}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=-\frac {a^2 (b c-a d)}{b^4 e \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}-\frac {\left (b^2 c^2+10 a b c d-19 a^2 d^2\right ) \left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{16 b^4 d e^2}-\frac {(b c+11 a d) \left (c+d x^2\right )^2 \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{24 b^3 d e^2}+\frac {\left (c+d x^2\right )^3 \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{6 b^2 d e^2}-\frac {(b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{\sqrt {b} \sqrt {e}}\right )}{16 b^{9/2} d^{3/2} e^{3/2}} \] Output:

-a^2*(-a*d+b*c)/b^4/e/(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)-1/16*(-19*a^2 
*d^2+10*a*b*c*d+b^2*c^2)*(d*x^2+c)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/ 
b^4/d/e^2-1/24*(11*a*d+b*c)*(d*x^2+c)^2*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^( 
1/2)/b^3/d/e^2+1/6*(d*x^2+c)^3*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/b^2/ 
d/e^2-1/16*(-a*d+b*c)*(-35*a^2*d^2+10*a*b*c*d+b^2*c^2)*arctanh(d^(1/2)*(b* 
e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/b^(1/2)/e^(1/2))/b^(9/2)/d^(3/2)/e^(3/ 
2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 4.60 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.71 \[ \int \frac {x^5}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\frac {\sqrt {d} \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}} \left (105 a^3 d^2+5 a^2 b d \left (-20 c+7 d x^2\right )+a b^2 \left (3 c^2-38 c d x^2-14 d^2 x^4\right )+b^3 x^2 \left (3 c^2+14 c d x^2+8 d^2 x^4\right )\right )-3 \sqrt {b c-a d} \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt {a+b x^2} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{48 b^4 d^{3/2} e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}}} \] Input:

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

Output:

(Sqrt[d]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*(105*a^3*d^2 + 5*a^2*b*d*(-20*c 
 + 7*d*x^2) + a*b^2*(3*c^2 - 38*c*d*x^2 - 14*d^2*x^4) + b^3*x^2*(3*c^2 + 1 
4*c*d*x^2 + 8*d^2*x^4)) - 3*Sqrt[b*c - a*d]*(b^2*c^2 + 10*a*b*c*d - 35*a^2 
*d^2)*Sqrt[a + b*x^2]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/ 
(48*b^4*d^(3/2)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[(b*(c + d*x^2))/( 
b*c - a*d)])
 

Rubi [A] (warning: unable to verify)

Time = 0.76 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.83, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {2053, 2052, 365, 25, 27, 298, 215, 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.

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

\(\Big \downarrow \) 2053

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

\(\Big \downarrow \) 2052

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

\(\Big \downarrow \) 365

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 298

\(\displaystyle e (b c-a d) \left (-\frac {\frac {1}{6} \left (\frac {5 a (2 b c-7 a d)}{b}+\frac {b c^2}{d}\right ) \int \frac {1}{\left (b e-d x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}-\frac {\left (\frac {b c^2}{d}-\frac {a (2 b c-7 a d)}{b}\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{6 \left (b e-d x^4\right )^3}}{b}-\frac {a^2 e}{b x^2 \left (b e-d x^4\right )^3}\right )\)

\(\Big \downarrow \) 215

\(\displaystyle e (b c-a d) \left (-\frac {\frac {1}{6} \left (\frac {5 a (2 b c-7 a d)}{b}+\frac {b c^2}{d}\right ) \left (\frac {3 \int \frac {1}{\left (b e-d x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 b e}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 b e \left (b e-d x^4\right )^2}\right )-\frac {\left (\frac {b c^2}{d}-\frac {a (2 b c-7 a d)}{b}\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{6 \left (b e-d x^4\right )^3}}{b}-\frac {a^2 e}{b x^2 \left (b e-d x^4\right )^3}\right )\)

\(\Big \downarrow \) 215

\(\displaystyle e (b c-a d) \left (-\frac {\frac {1}{6} \left (\frac {5 a (2 b c-7 a d)}{b}+\frac {b c^2}{d}\right ) \left (\frac {3 \left (\frac {\int \frac {1}{b e-d x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{2 b e}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 b e \left (b e-d x^4\right )}\right )}{4 b e}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 b e \left (b e-d x^4\right )^2}\right )-\frac {\left (\frac {b c^2}{d}-\frac {a (2 b c-7 a d)}{b}\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{6 \left (b e-d x^4\right )^3}}{b}-\frac {a^2 e}{b x^2 \left (b e-d x^4\right )^3}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle e (b c-a d) \left (-\frac {a^2 e}{b x^2 \left (b e-d x^4\right )^3}-\frac {\frac {1}{6} \left (\frac {5 a (2 b c-7 a d)}{b}+\frac {b c^2}{d}\right ) \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{2 b^{3/2} \sqrt {d} e^{3/2}}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 b e \left (b e-d x^4\right )}\right )}{4 b e}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 b e \left (b e-d x^4\right )^2}\right )-\frac {\left (\frac {b c^2}{d}-\frac {a (2 b c-7 a d)}{b}\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{6 \left (b e-d x^4\right )^3}}{b}\right )\)

Input:

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

Output:

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

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 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 365

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x 
_Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] 
- Simp[1/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p 
+ 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, 
 b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -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 = 0.22 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.91

method	result	size
risch	\(\frac {\left (8 b^{2} d^{2} x^{4}-22 a b \,d^{2} x^{2}+14 b^{2} c \,x^{2} d +57 a^{2} d^{2}-52 a b c d +3 b^{2} c^{2}\right ) \left (b \,x^{2}+a \right )}{48 d \,b^{4} e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}-\frac {\left (\frac {\left (35 a^{2} d^{2}-10 a b c d -b^{2} c^{2}\right ) \left (a d -b c \right ) \ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +b d \,x^{2} e}{\sqrt {b d e}}+\sqrt {b d e \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right )}{2 \sqrt {b d e}}-\frac {16 a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d \left (d \,x^{2}+c \right )}{\left (a d -b c \right ) \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) e}}{16 b^{4} d e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(318\)
default	\(\text {Expression too large to display}\)	\(1027\)

Input:

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

Output:

1/48/d*(8*b^2*d^2*x^4-22*a*b*d^2*x^2+14*b^2*c*d*x^2+57*a^2*d^2-52*a*b*c*d+ 
3*b^2*c^2)*(b*x^2+a)/b^4/e/(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/16/b^4/d*(1/2*( 
35*a^2*d^2-10*a*b*c*d-b^2*c^2)*(a*d-b*c)*ln((1/2*a*d*e+1/2*b*c*e+b*d*x^2*e 
)/(b*d*e)^(1/2)+(b*d*e*x^4+(a*d*e+b*c*e)*x^2+a*c*e)^(1/2))/(b*d*e)^(1/2)-1 
6*a^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)*d*(d*x^2+c)/(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/(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 = 0.89 (sec) , antiderivative size = 781, normalized size of antiderivative = 2.23 \[ \int \frac {x^5}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/192*(3*(a*b^3*c^3 + 9*a^2*b^2*c^2*d - 45*a^3*b*c*d^2 + 35*a^4*d^3 + (b^ 
4*c^3 + 9*a*b^3*c^2*d - 45*a^2*b^2*c*d^2 + 35*a^3*b*d^3)*x^2)*sqrt(b*d*e)* 
log(8*b^2*d^2*e*x^4 + 8*(b^2*c*d + a*b*d^2)*e*x^2 + (b^2*c^2 + 6*a*b*c*d + 
 a^2*d^2)*e - 4*(2*b*d^2*x^4 + b*c^2 + a*c*d + (3*b*c*d + a*d^2)*x^2)*sqrt 
(b*d*e)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))) + 4*(8*b^4*d^4*x^8 + 3*a*b^3*c^ 
3*d - 100*a^2*b^2*c^2*d^2 + 105*a^3*b*c*d^3 + 2*(11*b^4*c*d^3 - 7*a*b^3*d^ 
4)*x^6 + (17*b^4*c^2*d^2 - 52*a*b^3*c*d^3 + 35*a^2*b^2*d^4)*x^4 + (3*b^4*c 
^3*d - 35*a*b^3*c^2*d^2 - 65*a^2*b^2*c*d^3 + 105*a^3*b*d^4)*x^2)*sqrt((b*e 
*x^2 + a*e)/(d*x^2 + c)))/(b^6*d^2*e^2*x^2 + a*b^5*d^2*e^2), 1/96*(3*(a*b^ 
3*c^3 + 9*a^2*b^2*c^2*d - 45*a^3*b*c*d^2 + 35*a^4*d^3 + (b^4*c^3 + 9*a*b^3 
*c^2*d - 45*a^2*b^2*c*d^2 + 35*a^3*b*d^3)*x^2)*sqrt(-b*d*e)*arctan(1/2*(2* 
b*d*x^2 + b*c + a*d)*sqrt(-b*d*e)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))/(b^2*d 
*e*x^2 + a*b*d*e)) + 2*(8*b^4*d^4*x^8 + 3*a*b^3*c^3*d - 100*a^2*b^2*c^2*d^ 
2 + 105*a^3*b*c*d^3 + 2*(11*b^4*c*d^3 - 7*a*b^3*d^4)*x^6 + (17*b^4*c^2*d^2 
 - 52*a*b^3*c*d^3 + 35*a^2*b^2*d^4)*x^4 + (3*b^4*c^3*d - 35*a*b^3*c^2*d^2 
- 65*a^2*b^2*c*d^3 + 105*a^3*b*d^4)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)) 
)/(b^6*d^2*e^2*x^2 + a*b^5*d^2*e^2)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

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

integrate(x^5/(e*(b*x^2+a)/(d*x^2+c))^(3/2),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,[1,0,0]%%%},[2,1,0]%%%}+%%%{%%{[-4,0]:[1,0,%%%{-1,[1 
,1,1]%%%}
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.74 \[ \int \frac {x^5}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\frac {\sqrt {e}\, \left (105 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{3} b \,d^{3}-100 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,d^{2}+35 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d^{3} x^{2}+3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a \,b^{3} c^{2} d -38 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,d^{2} x^{2}-14 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a \,b^{3} d^{3} x^{4}+3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{4} c^{2} d \,x^{2}+14 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{4} c \,d^{2} x^{4}+8 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{4} d^{3} x^{6}+105 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{4} d^{3}-135 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{3} b c \,d^{2}+105 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{3} b \,d^{3} x^{2}+27 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{2} b^{2} c^{2} d -135 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{2} b^{2} c \,d^{2} x^{2}+3 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a \,b^{3} c^{3}+27 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a \,b^{3} c^{2} d \,x^{2}+3 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) b^{4} c^{3} x^{2}\right )}{48 b^{5} d^{2} e^{2} \left (b \,x^{2}+a \right )} \] Input:

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

Output:

(sqrt(e)*(105*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*b*d**3 - 100*sqrt(c + 
 d*x**2)*sqrt(a + b*x**2)*a**2*b**2*c*d**2 + 35*sqrt(c + d*x**2)*sqrt(a + 
b*x**2)*a**2*b**2*d**3*x**2 + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**3*c 
**2*d - 38*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**3*c*d**2*x**2 - 14*sqrt( 
c + d*x**2)*sqrt(a + b*x**2)*a*b**3*d**3*x**4 + 3*sqrt(c + d*x**2)*sqrt(a 
+ b*x**2)*b**4*c**2*d*x**2 + 14*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**4*c*d 
**2*x**4 + 8*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**4*d**3*x**6 + 105*sqrt(d 
)*sqrt(b)*log( - sqrt(b)*sqrt(a + b*x**2)*d + sqrt(d)*sqrt(c + d*x**2)*b)* 
a**4*d**3 - 135*sqrt(d)*sqrt(b)*log( - sqrt(b)*sqrt(a + b*x**2)*d + sqrt(d 
)*sqrt(c + d*x**2)*b)*a**3*b*c*d**2 + 105*sqrt(d)*sqrt(b)*log( - sqrt(b)*s 
qrt(a + b*x**2)*d + sqrt(d)*sqrt(c + d*x**2)*b)*a**3*b*d**3*x**2 + 27*sqrt 
(d)*sqrt(b)*log( - sqrt(b)*sqrt(a + b*x**2)*d + sqrt(d)*sqrt(c + d*x**2)*b 
)*a**2*b**2*c**2*d - 135*sqrt(d)*sqrt(b)*log( - sqrt(b)*sqrt(a + b*x**2)*d 
 + sqrt(d)*sqrt(c + d*x**2)*b)*a**2*b**2*c*d**2*x**2 + 3*sqrt(d)*sqrt(b)*l 
og( - sqrt(b)*sqrt(a + b*x**2)*d + sqrt(d)*sqrt(c + d*x**2)*b)*a*b**3*c**3 
 + 27*sqrt(d)*sqrt(b)*log( - sqrt(b)*sqrt(a + b*x**2)*d + sqrt(d)*sqrt(c + 
 d*x**2)*b)*a*b**3*c**2*d*x**2 + 3*sqrt(d)*sqrt(b)*log( - sqrt(b)*sqrt(a + 
 b*x**2)*d + sqrt(d)*sqrt(c + d*x**2)*b)*b**4*c**3*x**2))/(48*b**5*d**2*e* 
*2*(a + b*x**2))