\(\int \frac {\sqrt {\frac {e (a+b x^2)}{c+d x^2}}}{x^7} \, dx\) [48]

Optimal result

Integrand size = 26, antiderivative size = 294 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^7} \, dx=\frac {\left (b^2 c^2+2 a b c d-11 a^2 d^2\right ) \left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{16 a^2 c^3 x^2}-\frac {(b c-13 a d) \left (c+d x^2\right )^2 \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{24 a c^3 x^4}-\frac {\left (c+d x^2\right )^3 \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{6 c^3 x^6}-\frac {(b c-a d) \left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{\sqrt {a} \sqrt {e}}\right )}{16 a^{5/2} c^{7/2}} \] Output:

1/16*(-11*a^2*d^2+2*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)/a^2/c^3/x^2-1/24*(-13*a*d+b*c)*(d*x^2+c)^2*(b*e/d-(-a*d+b*c)*e 
/d/(d*x^2+c))^(1/2)/a/c^3/x^4-1/6*(d*x^2+c)^3*(b*e/d-(-a*d+b*c)*e/d/(d*x^2 
+c))^(1/2)/c^3/x^6-1/16*(-a*d+b*c)*(5*a^2*d^2+2*a*b*c*d+b^2*c^2)*e^(1/2)*a 
rctanh(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^(7/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.48 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^7} \, dx=\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2} \left (\sqrt {a} \sqrt {c} \sqrt {a+b x^2} \sqrt {c+d x^2} \left (3 b^2 c^2 x^4-2 a b c x^2 \left (c-2 d x^2\right )+a^2 \left (-8 c^2+10 c d x^2-15 d^2 x^4\right )\right )-3 \left (b^3 c^3+a b^2 c^2 d+3 a^2 b c d^2-5 a^3 d^3\right ) x^6 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )\right )}{48 a^{5/2} c^{7/2} x^6 \sqrt {a+b x^2}} \] Input:

Integrate[Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/x^7,x]
 

Output:

(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2]*(Sqrt[a]*Sqrt[c]*Sqrt[a 
 + b*x^2]*Sqrt[c + d*x^2]*(3*b^2*c^2*x^4 - 2*a*b*c*x^2*(c - 2*d*x^2) + a^2 
*(-8*c^2 + 10*c*d*x^2 - 15*d^2*x^4)) - 3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b* 
c*d^2 - 5*a^3*d^3)*x^6*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + 
 d*x^2])]))/(48*a^(5/2)*c^(7/2)*x^6*Sqrt[a + b*x^2])
 

Rubi [A] (warning: unable to verify)

Time = 0.80 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.91, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2053, 2052, 366, 27, 360, 27, 298, 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.

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

\(\Big \downarrow \) 2053

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

\(\Big \downarrow \) 2052

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

\(\Big \downarrow \) 366

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 360

\(\displaystyle e (b c-a d) \left (\frac {\frac {e (b c-a d) (3 a d+b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c \left (a e-c x^4\right )^2}-\frac {\int \frac {c \left ((b c-a d) (b c+3 a d) e-8 a c d^2 x^4\right )}{\left (a e-c x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 c^2}}{2 a c^2}+\frac {e x^6 (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle e (b c-a d) \left (\frac {\frac {e (b c-a d) (3 a d+b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c \left (a e-c x^4\right )^2}-\frac {\int \frac {(b c-a d) (b c+3 a d) e-8 a c d^2 x^4}{\left (a e-c x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 c}}{2 a c^2}+\frac {e x^6 (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\)

\(\Big \downarrow \) 298

\(\displaystyle e (b c-a d) \left (\frac {\frac {e (b c-a d) (3 a d+b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c \left (a e-c x^4\right )^2}-\frac {\frac {\left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \int \frac {1}{a e-c x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{2 a}+\frac {\left (-11 a^2 d^2+2 a b c d+b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 a \left (a e-c x^4\right )}}{4 c}}{2 a c^2}+\frac {e x^6 (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle e (b c-a d) \left (\frac {\frac {e (b c-a d) (3 a d+b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c \left (a e-c x^4\right )^2}-\frac {\frac {\left (-11 a^2 d^2+2 a b c d+b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 a \left (a e-c x^4\right )}+\frac {\left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{2 a^{3/2} \sqrt {c} \sqrt {e}}}{4 c}}{2 a c^2}+\frac {e x^6 (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\)

Input:

Int[Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/x^7,x]
 

Output:

(b*c - a*d)*e*(((b*c - a*d)^2*e*x^6)/(6*a*c^2*(a*e - c*x^4)^3) + (((b*c - 
a*d)*(b*c + 3*a*d)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(4*c*(a*e - c*x^4) 
^2) - (((b^2*c^2 + 2*a*b*c*d - 11*a^2*d^2)*Sqrt[(e*(a + b*x^2))/(c + d*x^2 
)])/(2*a*(a*e - c*x^4)) + ((b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt 
[c]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqrt[a]*Sqrt[e])])/(2*a^(3/2)*Sqrt 
[c]*Sqrt[e]))/(4*c))/(2*a*c^2))
 

Defintions of rubi rules used

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 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 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 366

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, 
x_Symbol] :> Simp[(-(b*c - a*d)^2)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a* 
b^2*e*(p + 1))), x] + Simp[1/(2*a*b^2*(p + 1))   Int[(e*x)^m*(a + b*x^2)^(p 
 + 1)*Simp[(b*c - a*d)^2*(m + 1) + 2*b^2*c^2*(p + 1) + 2*a*b*d^2*(p + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p 
, -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 = 0.19 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.87

method	result	size
risch	\(-\frac {\left (d \,x^{2}+c \right ) \left (15 a^{2} d^{2} x^{4}-4 a b c d \,x^{4}-3 b^{2} c^{2} x^{4}-10 a^{2} c d \,x^{2}+2 a b \,c^{2} x^{2}+8 a^{2} c^{2}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{48 c^{3} x^{6} a^{2}}+\frac {\left (5 a^{3} d^{3}-3 a^{2} b c \,d^{2}-a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {b d e \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) e}}{32 a^{2} c^{3} \sqrt {a c e}\, \left (b \,x^{2}+a \right )}\)	\(257\)
default	\(-\frac {\sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-66 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a^{2} b \,d^{3} x^{8}-24 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a \,b^{2} c \,d^{2} x^{8}-6 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b^{3} c^{2} d \,x^{8}-15 \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 ) a^{4} c \,d^{3} x^{6}+9 \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 ) a^{3} b \,c^{2} d^{2} x^{6}+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 ) a^{2} b^{2} c^{3} d \,x^{6}+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 ) a \,b^{3} c^{4} x^{6}-66 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a^{3} d^{3} x^{6}-54 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a^{2} b c \,d^{2} x^{6}-18 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a \,b^{2} c^{2} d \,x^{6}-6 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b^{3} c^{3} x^{6}+66 \left (d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a^{2} d^{2} x^{4}+24 \left (d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a b c d \,x^{4}+6 \left (d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, b^{2} c^{2} x^{4}-36 \left (d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a^{2} c d \,x^{2}-12 \left (d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a b \,c^{2} x^{2}+16 \left (d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} a^{2} c^{2} \sqrt {a c}\right )}{96 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, c^{4} a^{3} x^{6} \sqrt {a c}}\)	\(849\)

Input:

int((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/48*(d*x^2+c)*(15*a^2*d^2*x^4-4*a*b*c*d*x^4-3*b^2*c^2*x^4-10*a^2*c*d*x^2 
+2*a*b*c^2*x^2+8*a^2*c^2)/c^3/x^6/a^2*(e*(b*x^2+a)/(d*x^2+c))^(1/2)+1/32*( 
5*a^3*d^3-3*a^2*b*c*d^2-a*b^2*c^2*d-b^3*c^3)/a^2/c^3/(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)/( 
b*x^2+a)
 

Fricas [A] (verification not implemented)

Time = 1.56 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.91 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^7} \, dx=\left [-\frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} x^{6} \sqrt {\frac {e}{a c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e x^{4} + 8 \, a^{2} c^{2} e + 8 \, {\left (a b c^{2} + a^{2} c d\right )} e x^{2} + 4 \, {\left (2 \, a^{2} c^{3} + {\left (a b c^{2} d + a^{2} c d^{2}\right )} x^{4} + {\left (a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {e}{a c}}}{x^{4}}\right ) - 4 \, {\left ({\left (3 \, b^{2} c^{2} d + 4 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{6} - 8 \, a^{2} c^{3} + {\left (3 \, b^{2} c^{3} + 2 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} x^{4} - 2 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{192 \, a^{2} c^{3} x^{6}}, \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} x^{6} \sqrt {-\frac {e}{a c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {e}{a c}}}{2 \, {\left (b e x^{2} + a e\right )}}\right ) + 2 \, {\left ({\left (3 \, b^{2} c^{2} d + 4 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{6} - 8 \, a^{2} c^{3} + {\left (3 \, b^{2} c^{3} + 2 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} x^{4} - 2 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{96 \, a^{2} c^{3} x^{6}}\right ] \] Input:

integrate((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^7,x, algorithm="fricas")
 

Output:

[-1/192*(3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*x^6*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*((3*b^2*c^2*d + 4*a*b*c*d^2 - 15*a^2*d^3)*x^6 - 8*a^2*c^3 + (3*b^2*c 
^3 + 2*a*b*c^2*d - 5*a^2*c*d^2)*x^4 - 2*(a*b*c^3 - a^2*c^2*d)*x^2)*sqrt((b 
*e*x^2 + a*e)/(d*x^2 + c)))/(a^2*c^3*x^6), 1/96*(3*(b^3*c^3 + a*b^2*c^2*d 
+ 3*a^2*b*c*d^2 - 5*a^3*d^3)*x^6*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*((3*b^2*c^2*d + 4*a*b*c*d^2 - 15*a^2*d^3)*x^6 - 8*a^2*c^3 + (3*b^2* 
c^3 + 2*a*b*c^2*d - 5*a^2*c*d^2)*x^4 - 2*(a*b*c^3 - a^2*c^2*d)*x^2)*sqrt(( 
b*e*x^2 + a*e)/(d*x^2 + c)))/(a^2*c^3*x^6)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^7} \, dx=\text {Timed out} \] Input:

integrate((e*(b*x**2+a)/(d*x**2+c))**(1/2)/x**7,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^7} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^7,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. 1001 vs. \(2 (266) = 532\).

Time = 0.21 (sec) , antiderivative size = 1001, normalized size of antiderivative = 3.40 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^7} \, dx =\text {Too large to display} \] Input:

integrate((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^7,x, algorithm="giac")
 

Output:

1/48*(3*(b^3*c^3*e + a*b^2*c^2*d*e + 3*a^2*b*c*d^2*e - 5*a^3*d^3*e)*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^3) - (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))*a^2*b^3*c^5*e^3 + 51*(sqrt(b*d*e)*x^2 - sq 
rt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))*a^3*b^2*c^4*d*e^3 + 105*(sq 
rt(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^4*b*c^3 
*d^2*e^3 + 33*(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^5*c^2*d^3*e^3 + 16*sqrt(b*d*e)*a^4*b*c^4*d*e^3 + 48*sqrt(b*d*e)* 
a^5*c^3*d^2*e^3 + 8*(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^3*c^4*e^2 + 72*(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*b^2*c^3*d*e^2 + 24*(sqrt(b*d*e)*x^2 - s 
qrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))^3*a^3*b*c^2*d^2*e^2 - 40*( 
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^4*c 
*d^3*e^2 + 48*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*b^2*c^4*e^2 + 144*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^3*b*c^3*d*e^2 - 3*(s 
qrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))^5*b^3*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) 
)^5*a*b^2*c^2*d*e - 9*(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))^5*a^2*b*c*d^2*e + 15*(sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^7} \, dx=\int \frac {\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}}{x^7} \,d x \] Input:

int(((e*(a + b*x^2))/(c + d*x^2))^(1/2)/x^7,x)
 

Output:

int(((e*(a + b*x^2))/(c + d*x^2))^(1/2)/x^7, x)
 

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^7} \, dx=\frac {\sqrt {e}\, \left (-8 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{3} c^{3}+10 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{3} c^{2} d \,x^{2}-15 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{3} c \,d^{2} x^{4}-2 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} b \,c^{3} x^{2}+4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} b \,c^{2} d \,x^{4}+3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a \,b^{2} c^{3} x^{4}+15 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c +\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) a^{3} d^{3} x^{6}-9 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c +\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) a^{2} b c \,d^{2} x^{6}-3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c +\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) a \,b^{2} c^{2} d \,x^{6}-3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c +\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) b^{3} c^{3} x^{6}-15 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) a^{3} d^{3} x^{6}+9 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} b c \,d^{2} x^{6}+3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{2} c^{2} d \,x^{6}+3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) b^{3} c^{3} x^{6}\right )}{48 a^{3} c^{4} x^{6}} \] Input:

int((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^7,x)
 

Output:

(sqrt(e)*( - 8*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*c**3 + 10*sqrt(c + d 
*x**2)*sqrt(a + b*x**2)*a**3*c**2*d*x**2 - 15*sqrt(c + d*x**2)*sqrt(a + b* 
x**2)*a**3*c*d**2*x**4 - 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*c**3*x 
**2 + 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*c**2*d*x**4 + 3*sqrt(c + 
d*x**2)*sqrt(a + b*x**2)*a*b**2*c**3*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**3*d**3*x**6 - 9*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**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)*a*b**2*c**2*d*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**3*c**3*x**6 - 15*sqrt 
(c)*sqrt(a)*log(x)*a**3*d**3*x**6 + 9*sqrt(c)*sqrt(a)*log(x)*a**2*b*c*d**2 
*x**6 + 3*sqrt(c)*sqrt(a)*log(x)*a*b**2*c**2*d*x**6 + 3*sqrt(c)*sqrt(a)*lo 
g(x)*b**3*c**3*x**6))/(48*a**3*c**4*x**6)