\(\int \frac {1}{x^5 (\frac {e (a+b x^2)}{c+d x^2})^{3/2}} \, dx\) [91]

Optimal result

Integrand size = 26, antiderivative size = 257 \[ \int \frac {1}{x^5 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\frac {b (b c-a d)}{a^3 e \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}+\frac {(7 b c-3 a d) \left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{8 a^3 e^2 x^2}-\frac {\left (c+d x^2\right )^2 \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{4 a^2 e^2 x^4}-\frac {3 (b c-a d) (5 b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{\sqrt {a} \sqrt {e}}\right )}{8 a^{7/2} \sqrt {c} e^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 4.38 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^5 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\frac {\sqrt {a} \sqrt {c} \sqrt {c+d x^2} \left (15 b^2 c x^4+a b x^2 \left (5 c-13 d x^2\right )-a^2 \left (2 c+5 d x^2\right )\right )-3 \left (5 b^2 c^2-6 a b c d+a^2 d^2\right ) x^4 \sqrt {a+b x^2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{7/2} \sqrt {c} e x^4 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}} \] Input:

Integrate[1/(x^5*((e*(a + b*x^2))/(c + d*x^2))^(3/2)),x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.69 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.81, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {2053, 2052, 25, 361, 25, 27, 361, 25, 27, 359, 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.

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

\(\Big \downarrow \) 2053

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

\(\Big \downarrow \) 2052

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 361

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 361

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 359

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

\(\Big \downarrow \) 221

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

Input:

Int[1/(x^5*((e*(a + b*x^2))/(c + d*x^2))^(3/2)),x]
 

Output:

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

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 359

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

rule 361

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[x^m*(a + b*x^2)^(p + 1)*E 
xpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c 
- a*d)*x^(-m + 2))/(a + b*x^2)] - ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], 
 x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ILtQ[m/ 
2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 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 = 0.22 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.07

method	result	size
risch	\(-\frac {\left (b \,x^{2}+a \right ) \left (5 a d \,x^{2}-7 b c \,x^{2}+2 a c \right )}{8 a^{3} x^{4} e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}+\frac {\left (\frac {\left (-3 a^{2} d^{2}+18 a b c d -15 b^{2} c^{2}\right ) \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {b d e \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right )}{2 \sqrt {a c e}}-\frac {8 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (d \,x^{2}+c \right )}{\left (a d -b c \right ) \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) e}}{8 a^{3} e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(274\)
default	\(\text {Expression too large to display}\)	\(1042\)

Input:

int(1/x^5/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/8*(b*x^2+a)*(5*a*d*x^2-7*b*c*x^2+2*a*c)/a^3/x^4/e/(e*(b*x^2+a)/(d*x^2+c 
))^(1/2)+1/8/a^3*(1/2*(-3*a^2*d^2+18*a*b*c*d-15*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*b*(a^2*d^2-2*a*b*c*d+b^2*c^2)*(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 = 3.43 (sec) , antiderivative size = 613, normalized size of antiderivative = 2.39 \[ \int \frac {1}{x^5 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\left [\frac {3 \, {\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{6} + {\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{4}\right )} \sqrt {a c e} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e x^{4} + 8 \, a^{2} c^{2} e + 8 \, {\left (a b c^{2} + a^{2} c d\right )} e x^{2} - 4 \, {\left ({\left (b c d + a d^{2}\right )} x^{4} + 2 \, a c^{2} + {\left (b c^{2} + 3 \, a c d\right )} x^{2}\right )} \sqrt {a c e} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{x^{4}}\right ) + 4 \, {\left ({\left (15 \, a b^{2} c^{2} d - 13 \, a^{2} b c d^{2}\right )} x^{6} - 2 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d - 5 \, a^{3} c d^{2}\right )} x^{4} + {\left (5 \, a^{2} b c^{3} - 7 \, a^{3} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{32 \, {\left (a^{4} b c e^{2} x^{6} + a^{5} c e^{2} x^{4}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{6} + {\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{4}\right )} \sqrt {-a c e} \arctan \left (\frac {\sqrt {-a c e} {\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{2 \, {\left (a b c e x^{2} + a^{2} c e\right )}}\right ) + 2 \, {\left ({\left (15 \, a b^{2} c^{2} d - 13 \, a^{2} b c d^{2}\right )} x^{6} - 2 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d - 5 \, a^{3} c d^{2}\right )} x^{4} + {\left (5 \, a^{2} b c^{3} - 7 \, a^{3} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{16 \, {\left (a^{4} b c e^{2} x^{6} + a^{5} c e^{2} x^{4}\right )}}\right ] \] Input:

integrate(1/x^5/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="fricas")
 

Output:

[1/32*(3*((5*b^3*c^2 - 6*a*b^2*c*d + a^2*b*d^2)*x^6 + (5*a*b^2*c^2 - 6*a^2 
*b*c*d + a^3*d^2)*x^4)*sqrt(a*c*e)*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*((15*a*b^2*c^2*d - 13*a^2*b*c*d^2)*x^6 - 2*a^3*c^3 + (15*a 
*b^2*c^3 - 8*a^2*b*c^2*d - 5*a^3*c*d^2)*x^4 + (5*a^2*b*c^3 - 7*a^3*c^2*d)* 
x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(a^4*b*c*e^2*x^6 + a^5*c*e^2*x^4), 
 1/16*(3*((5*b^3*c^2 - 6*a*b^2*c*d + a^2*b*d^2)*x^6 + (5*a*b^2*c^2 - 6*a^2 
*b*c*d + a^3*d^2)*x^4)*sqrt(-a*c*e)*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 
*((15*a*b^2*c^2*d - 13*a^2*b*c*d^2)*x^6 - 2*a^3*c^3 + (15*a*b^2*c^3 - 8*a^ 
2*b*c^2*d - 5*a^3*c*d^2)*x^4 + (5*a^2*b*c^3 - 7*a^3*c^2*d)*x^2)*sqrt((b*e* 
x^2 + a*e)/(d*x^2 + c)))/(a^4*b*c*e^2*x^6 + a^5*c*e^2*x^4)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^5 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/x**5/(e*(b*x**2+a)/(d*x**2+c))**(3/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^5 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/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. \[ \int \frac {1}{x^5 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/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,[1,4,4]%%%},[2,1,7,0]%%%}+%%%{%%%{-8,[2,3,4]%%%},[2, 
1,6,1]%%%
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^5 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\int \frac {1}{x^5\,{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}} \,d x \] Input:

int(1/(x^5*((e*(a + b*x^2))/(c + d*x^2))^(3/2)),x)
 

Output:

int(1/(x^5*((e*(a + b*x^2))/(c + d*x^2))^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.01 \[ \int \frac {1}{x^5 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\frac {\sqrt {e}\, \left (-2 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{3} c^{2}-5 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{3} c d \,x^{2}+5 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} b \,c^{2} x^{2}-13 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} b c d \,x^{4}+15 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a \,b^{2} c^{2} x^{4}+3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) a^{3} d^{2} x^{4}-18 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) a^{2} b c d \,x^{4}+3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) a^{2} b \,d^{2} x^{6}+15 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) a \,b^{2} c^{2} x^{4}-18 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) a \,b^{2} c d \,x^{6}+15 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) b^{3} c^{2} x^{6}-3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) a^{3} d^{2} x^{4}+18 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} b c d \,x^{4}-3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} b \,d^{2} x^{6}-15 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{2} c^{2} x^{4}+18 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{2} c d \,x^{6}-15 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) b^{3} c^{2} x^{6}\right )}{8 a^{4} c \,e^{2} x^{4} \left (b \,x^{2}+a \right )} \] Input:

int(1/x^5/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x)
 

Output:

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