\(\int \frac {(\frac {e (a+b x^2)}{c+d x^2})^{3/2}}{x^7} \, dx\) [61]

Optimal result

Integrand size = 26, antiderivative size = 346 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\frac {d^2 (b c-a d) e \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{c^4}-\frac {\left (b^2 c^2-22 a b c d+29 a^2 d^2\right ) e \left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{16 a c^4 x^2}-\frac {(7 b c-19 a d) e \left (c+d x^2\right )^2 \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{24 c^4 x^4}-\frac {a e \left (c+d x^2\right )^3 \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{6 c^4 x^6}+\frac {(b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{\sqrt {a} \sqrt {e}}\right )}{16 a^{3/2} c^{9/2}} \] Output:

d^2*(-a*d+b*c)*e*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/c^4-1/16*(29*a^2*d 
^2-22*a*b*c*d+b^2*c^2)*e*(d*x^2+c)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/ 
a/c^4/x^2-1/24*(-19*a*d+7*b*c)*e*(d*x^2+c)^2*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+ 
c))^(1/2)/c^4/x^4-1/6*a*e*(d*x^2+c)^3*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/ 
2)/c^4/x^6+1/16*(-a*d+b*c)*(-35*a^2*d^2+10*a*b*c*d+b^2*c^2)*e^(3/2)*arctan 
h(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^(3/2)/ 
c^(9/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 4.88 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.71 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\sqrt {a} \sqrt {c} \sqrt {a+b x^2} \left (3 b^2 c^2 x^4 \left (c+d x^2\right )+2 a b c x^2 \left (7 c^2-19 c d x^2-50 d^2 x^4\right )+a^2 \left (8 c^3-14 c^2 d x^2+35 c d^2 x^4+105 d^3 x^6\right )\right )+3 \left (b^3 c^3+9 a b^2 c^2 d-45 a^2 b c d^2+35 a^3 d^3\right ) x^6 \sqrt {c+d x^2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )\right )}{48 a^{3/2} c^{9/2} x^6 \sqrt {a+b x^2}} \] Input:

Integrate[((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^7,x]
 

Output:

(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(-(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x^2]*(3 
*b^2*c^2*x^4*(c + d*x^2) + 2*a*b*c*x^2*(7*c^2 - 19*c*d*x^2 - 50*d^2*x^4) + 
 a^2*(8*c^3 - 14*c^2*d*x^2 + 35*c*d^2*x^4 + 105*d^3*x^6))) + 3*(b^3*c^3 + 
9*a*b^2*c^2*d - 45*a^2*b*c*d^2 + 35*a^3*d^3)*x^6*Sqrt[c + d*x^2]*ArcTanh[( 
Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])]))/(48*a^(3/2)*c^(9/2)* 
x^6*Sqrt[a + b*x^2])
 

Rubi [A] (warning: unable to verify)

Time = 0.94 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.87, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {2053, 2052, 366, 25, 27, 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.

\(\displaystyle \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx\)

\(\Big \downarrow \) 2053

\(\displaystyle \frac {1}{2} \int \frac {\left (\frac {e \left (b x^2+a\right )}{d x^2+c}\right )^{3/2}}{x^8}dx^2\)

\(\Big \downarrow \) 2052

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

\(\Big \downarrow \) 366

\(\displaystyle e (b c-a d) \left (\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}-\frac {\int -\frac {x^8 \left (-6 a c d^2 e x^4+6 b^2 c^2 e^2-5 (b c e-a d e)^2\right )}{\left (a e-c x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{6 a c^2 e}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle e (b c-a d) \left (\frac {\int \frac {e x^8 \left (\left (b^2 c^2+10 a b d c-5 a^2 d^2\right ) e-6 a c d^2 x^4\right )}{\left (a e-c x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{6 a c^2 e}+\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle e (b c-a d) \left (\frac {\int \frac {x^8 \left (\left (b^2 c^2+10 a b d c-5 a^2 d^2\right ) e-6 a c d^2 x^4\right )}{\left (a e-c x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{6 a c^2}+\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\)

\(\Big \downarrow \) 360

\(\displaystyle e (b c-a d) \left (\frac {\frac {\int -\frac {-24 a c^3 d^2 x^8+4 c^2 (b c-a d) (b c+11 a d) e x^4+a c (b c-a d) (b c+11 a d) e^2}{\left (a e-c x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 c^3}+\frac {a e^2 (b c-a d) (11 a d+b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^2 \left (a e-c x^4\right )^2}}{6 a c^2}+\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle e (b c-a d) \left (\frac {\frac {a e^2 (b c-a d) (11 a d+b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^2 \left (a e-c x^4\right )^2}-\frac {\int \frac {-24 a c^3 d^2 x^8+4 c^2 (b c-a d) (b c+11 a d) e x^4+a c (b c-a d) (b c+11 a d) e^2}{\left (a e-c x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 c^3}}{6 a c^2}+\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\)

\(\Big \downarrow \) 1471

\(\displaystyle e (b c-a d) \left (\frac {\frac {a e^2 (b c-a d) (11 a d+b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^2 \left (a e-c x^4\right )^2}-\frac {\frac {c e \left (-79 a^2 d^2+50 a b c d+5 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (a e-c x^4\right )}-\frac {\int \frac {3 a c e \left (\left (b^2 c^2+10 a b d c-19 a^2 d^2\right ) e-16 a c d^2 x^4\right )}{a e-c x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{2 a e}}{4 c^3}}{6 a c^2}+\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle e (b c-a d) \left (\frac {\frac {a e^2 (b c-a d) (11 a d+b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^2 \left (a e-c x^4\right )^2}-\frac {\frac {c e \left (-79 a^2 d^2+50 a b c d+5 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (a e-c x^4\right )}-\frac {3}{2} c \int \frac {\left (b^2 c^2+10 a b d c-19 a^2 d^2\right ) e-16 a c d^2 x^4}{a e-c x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 c^3}}{6 a c^2}+\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\)

\(\Big \downarrow \) 299

\(\displaystyle e (b c-a d) \left (\frac {\frac {a e^2 (b c-a d) (11 a d+b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^2 \left (a e-c x^4\right )^2}-\frac {\frac {c e \left (-79 a^2 d^2+50 a b c d+5 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (a e-c x^4\right )}-\frac {3}{2} c \left (e \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \int \frac {1}{a e-c x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}+16 a d^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{4 c^3}}{6 a c^2}+\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle e (b c-a d) \left (\frac {\frac {a e^2 (b c-a d) (11 a d+b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^2 \left (a e-c x^4\right )^2}-\frac {\frac {c e \left (-79 a^2 d^2+50 a b c d+5 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (a e-c x^4\right )}-\frac {3}{2} c \left (\frac {\sqrt {e} \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {a} \sqrt {c}}+16 a d^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{4 c^3}}{6 a c^2}+\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\)

Input:

Int[((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^7,x]
 

Output:

(b*c - a*d)*e*(((b*c - a*d)^2*e*x^10)/(6*a*c^2*(a*e - c*x^4)^3) + ((a*(b*c 
 - a*d)*(b*c + 11*a*d)*e^2*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(4*c^2*(a*e 
- c*x^4)^2) - ((c*(5*b^2*c^2 + 50*a*b*c*d - 79*a^2*d^2)*e*Sqrt[(e*(a + b*x 
^2))/(c + d*x^2)])/(2*(a*e - c*x^4)) - (3*c*(16*a*d^2*Sqrt[(e*(a + b*x^2)) 
/(c + d*x^2)] + ((b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*Sqrt[e]*ArcTanh[(Sqrt 
[c]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqrt[a]*Sqrt[e])])/(Sqrt[a]*Sqrt[c 
])))/2)/(4*c^3))/(6*a*c^2))
 

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 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 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 = 0.24 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.96

method	result	size
risch	\(-\frac {\left (d \,x^{2}+c \right ) \left (57 a^{2} d^{2} x^{4}-52 a b c d \,x^{4}+3 b^{2} c^{2} x^{4}-22 a^{2} c d \,x^{2}+14 a b \,c^{2} x^{2}+8 a^{2} c^{2}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{48 c^{4} x^{6} a}-\frac {\left (-\frac {\left (35 a^{3} d^{3}-45 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {b d e \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right )}{2 \sqrt {a c e}}+\frac {16 a \,d^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b \,x^{2}+a \right )}{\left (a d -b c \right ) \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) e}}{16 a \,c^{4} \left (b \,x^{2}+a \right )}\)	\(333\)
default	\(\text {Expression too large to display}\)	\(1498\)

Input:

int((e*(b*x^2+a)/(d*x^2+c))^(3/2)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/48*(d*x^2+c)*(57*a^2*d^2*x^4-52*a*b*c*d*x^4+3*b^2*c^2*x^4-22*a^2*c*d*x^ 
2+14*a*b*c^2*x^2+8*a^2*c^2)/c^4/x^6/a*e*(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/16 
/a/c^4*(-1/2*(35*a^3*d^3-45*a^2*b*c*d^2+9*a*b^2*c^2*d+b^3*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)+16*a*d^2*(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 = 7.92 (sec) , antiderivative size = 573, normalized size of antiderivative = 1.66 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\left [\frac {3 \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 45 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3}\right )} e x^{6} \sqrt {\frac {e}{a c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e x^{4} + 8 \, a^{2} c^{2} e + 8 \, {\left (a b c^{2} + a^{2} c d\right )} e x^{2} + 4 \, {\left (2 \, a^{2} c^{3} + {\left (a b c^{2} d + a^{2} c d^{2}\right )} x^{4} + {\left (a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {e}{a c}}}{x^{4}}\right ) - 4 \, {\left ({\left (3 \, b^{2} c^{2} d - 100 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} e x^{6} + 8 \, a^{2} c^{3} e + {\left (3 \, b^{2} c^{3} - 38 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} e x^{4} + 14 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} e x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{192 \, a c^{4} x^{6}}, -\frac {3 \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 45 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3}\right )} e x^{6} \sqrt {-\frac {e}{a c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {e}{a c}}}{2 \, {\left (b e x^{2} + a e\right )}}\right ) + 2 \, {\left ({\left (3 \, b^{2} c^{2} d - 100 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} e x^{6} + 8 \, a^{2} c^{3} e + {\left (3 \, b^{2} c^{3} - 38 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} e x^{4} + 14 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} e x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{96 \, a c^{4} x^{6}}\right ] \] Input:

integrate((e*(b*x^2+a)/(d*x^2+c))^(3/2)/x^7,x, algorithm="fricas")
 

Output:

[1/192*(3*(b^3*c^3 + 9*a*b^2*c^2*d - 45*a^2*b*c*d^2 + 35*a^3*d^3)*e*x^6*sq 
rt(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 - 100*a*b*c*d^2 + 105*a^2*d^3)*e*x^6 + 8*a^2*c^3* 
e + (3*b^2*c^3 - 38*a*b*c^2*d + 35*a^2*c*d^2)*e*x^4 + 14*(a*b*c^3 - a^2*c^ 
2*d)*e*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(a*c^4*x^6), -1/96*(3*(b^3* 
c^3 + 9*a*b^2*c^2*d - 45*a^2*b*c*d^2 + 35*a^3*d^3)*e*x^6*sqrt(-e/(a*c))*ar 
ctan(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 - 100*a*b*c*d^2 + 105*a^2*d^3 
)*e*x^6 + 8*a^2*c^3*e + (3*b^2*c^3 - 38*a*b*c^2*d + 35*a^2*c*d^2)*e*x^4 + 
14*(a*b*c^3 - a^2*c^2*d)*e*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(a*c^4* 
x^6)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\text {Timed out} \] Input:

integrate((e*(b*x**2+a)/(d*x**2+c))**(3/2)/x**7,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\text {Exception raised: ValueError} \] Input:

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

Exception generated. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*(b*x^2+a)/(d*x^2+c))^(3/2)/x^7,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,[5,1,5]%%%},[2,9,0]%%%}+%%%{%%%{-10,[4,2,5]%%%},[2,8 
,1]%%%}+%
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\int \frac {{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}}{x^7} \,d x \] Input:

int(((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^7,x)
 

Output:

int(((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^7, x)
 

Reduce [B] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 771, normalized size of antiderivative = 2.23 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx =\text {Too large to display} \] Input:

int((e*(b*x^2+a)/(d*x^2+c))^(3/2)/x^7,x)
 

Output:

(sqrt(e)*e*( - 8*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*c**4 + 14*sqrt(c + 
 d*x**2)*sqrt(a + b*x**2)*a**3*c**3*d*x**2 - 35*sqrt(c + d*x**2)*sqrt(a + 
b*x**2)*a**3*c**2*d**2*x**4 - 105*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*c 
*d**3*x**6 - 14*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*c**4*x**2 + 38*sq 
rt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*c**3*d*x**4 + 100*sqrt(c + d*x**2)* 
sqrt(a + b*x**2)*a**2*b*c**2*d**2*x**6 - 3*sqrt(c + d*x**2)*sqrt(a + b*x** 
2)*a*b**2*c**4*x**4 - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c**3*d*x* 
*6 + 105*sqrt(c)*sqrt(a)*log(sqrt(a)*sqrt(a + b*x**2)*c + sqrt(c)*sqrt(c + 
 d*x**2)*a)*a**3*c*d**3*x**6 + 105*sqrt(c)*sqrt(a)*log(sqrt(a)*sqrt(a + b* 
x**2)*c + sqrt(c)*sqrt(c + d*x**2)*a)*a**3*d**4*x**8 - 135*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**2* 
d**2*x**6 - 135*sqrt(c)*sqrt(a)*log(sqrt(a)*sqrt(a + b*x**2)*c + sqrt(c)*s 
qrt(c + d*x**2)*a)*a**2*b*c*d**3*x**8 + 27*sqrt(c)*sqrt(a)*log(sqrt(a)*sqr 
t(a + b*x**2)*c + sqrt(c)*sqrt(c + d*x**2)*a)*a*b**2*c**3*d*x**6 + 27*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**2*x**8 + 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**4*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*d*x**8 - 105*s 
qrt(c)*sqrt(a)*log(x)*a**3*c*d**3*x**6 - 105*sqrt(c)*sqrt(a)*log(x)*a**3*d 
**4*x**8 + 135*sqrt(c)*sqrt(a)*log(x)*a**2*b*c**2*d**2*x**6 + 135*sqrt(...