\(\int \frac {1}{(e x)^{5/2} (a+b x)^2 (a c-b c x)^2} \, dx\) [68]

Optimal result

Integrand size = 26, antiderivative size = 147 \[ \int \frac {1}{(e x)^{5/2} (a+b x)^2 (a c-b c x)^2} \, dx=-\frac {7}{6 a^4 c^2 e (e x)^{3/2}}+\frac {1}{2 a^2 c^2 e (e x)^{3/2} \left (a^2-b^2 x^2\right )}+\frac {7 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{11/2} c^2 e^{5/2}}+\frac {7 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{11/2} c^2 e^{5/2}} \] Output:

-7/6/a^4/c^2/e/(e*x)^(3/2)+1/2/a^2/c^2/e/(e*x)^(3/2)/(-b^2*x^2+a^2)+7/4*b^ 
(3/2)*arctan(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1/2))/a^(11/2)/c^2/e^(5/2)+7/4 
*b^(3/2)*arctanh(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1/2))/a^(11/2)/c^2/e^(5/2)
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(e x)^{5/2} (a+b x)^2 (a c-b c x)^2} \, dx=\frac {x \left (\frac {2 a^{3/2} \left (-4 a^2+7 b^2 x^2\right )}{a^2-b^2 x^2}+21 b^{3/2} x^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+21 b^{3/2} x^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{12 a^{11/2} c^2 (e x)^{5/2}} \] Input:

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

Output:

(x*((2*a^(3/2)*(-4*a^2 + 7*b^2*x^2))/(a^2 - b^2*x^2) + 21*b^(3/2)*x^(3/2)* 
ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]] + 21*b^(3/2)*x^(3/2)*ArcTanh[(Sqrt[b]*Sq 
rt[x])/Sqrt[a]]))/(12*a^(11/2)*c^2*(e*x)^(5/2))
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {82, 253, 27, 264, 266, 756, 218, 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.

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

\(\Big \downarrow \) 82

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

\(\Big \downarrow \) 253

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 264

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 756

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

1/(2*a^2*c^2*e*(e*x)^(3/2)*(a^2 - b^2*x^2)) + (7*(-2/(3*a^2*e*(e*x)^(3/2)) 
 + (2*b^2*((Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*x])/(Sqrt[a]*Sqrt[e])])/(2*a^(3 
/2)*Sqrt[b]) + (Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*x])/(Sqrt[a]*Sqrt[e])])/(2 
*a^(3/2)*Sqrt[b])))/(a^2*e^3)))/(4*a^2*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 82

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) 
)^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, 
 e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

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 253

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

rule 264

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

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(\frac {2 e^{3} \left (-\frac {1}{3 a^{4} e^{4} \left (e x \right )^{\frac {3}{2}}}+\frac {b^{2} \left (\frac {\sqrt {e x}}{2 b e x +2 a e}+\frac {7 \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}\right )}{4 a^{5} e^{5}}+\frac {b^{2} \left (\frac {\sqrt {e x}}{-2 b e x +2 a e}+\frac {7 \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}\right )}{4 a^{5} e^{5}}\right )}{c^{2}}\)	\(126\)
default	\(\frac {2 e^{3} \left (-\frac {1}{3 a^{4} e^{4} \left (e x \right )^{\frac {3}{2}}}+\frac {b^{2} \left (\frac {\sqrt {e x}}{2 b e x +2 a e}+\frac {7 \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}\right )}{4 a^{5} e^{5}}+\frac {b^{2} \left (\frac {\sqrt {e x}}{-2 b e x +2 a e}+\frac {7 \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}\right )}{4 a^{5} e^{5}}\right )}{c^{2}}\)	\(126\)
pseudoelliptic	\(\frac {\frac {7 \left (b x +a \right ) \left (-b x +a \right ) \left (e x \right )^{\frac {3}{2}} b^{2} \operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{4}+\frac {7 \left (b x +a \right ) \left (-b x +a \right ) \left (e x \right )^{\frac {3}{2}} b^{2} \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{4}-\frac {2 a \left (-\frac {7 b^{2} x^{2}}{4}+a^{2}\right ) e \sqrt {a e b}}{3}}{e^{2} c^{2} \left (b x +a \right ) a^{5} \sqrt {a e b}\, \left (-b x +a \right ) \left (e x \right )^{\frac {3}{2}}}\)	\(129\)
risch	\(-\frac {2}{3 a^{4} x \sqrt {e x}\, e^{2} c^{2}}+\frac {\frac {b^{2} \sqrt {e x}}{4 a^{5} \left (b e x +a e \right )}+\frac {7 b^{2} \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{4 a^{5} \sqrt {a e b}}-\frac {b^{2} \sqrt {e x}}{4 a^{5} \left (b e x -a e \right )}+\frac {7 b^{2} \operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{4 a^{5} \sqrt {a e b}}}{e^{2} c^{2}}\)	\(132\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.36 \[ \int \frac {1}{(e x)^{5/2} (a+b x)^2 (a c-b c x)^2} \, dx=\left [\frac {42 \, {\left (b^{3} e x^{4} - a^{2} b e x^{2}\right )} \sqrt {\frac {b}{a e}} \arctan \left (\sqrt {e x} \sqrt {\frac {b}{a e}}\right ) + 21 \, {\left (b^{3} e x^{4} - a^{2} b e x^{2}\right )} \sqrt {\frac {b}{a e}} \log \left (\frac {b x + 2 \, \sqrt {e x} a \sqrt {\frac {b}{a e}} + a}{b x - a}\right ) - 4 \, {\left (7 \, a b^{2} x^{2} - 4 \, a^{3}\right )} \sqrt {e x}}{24 \, {\left (a^{5} b^{2} c^{2} e^{3} x^{4} - a^{7} c^{2} e^{3} x^{2}\right )}}, -\frac {42 \, {\left (b^{3} e x^{4} - a^{2} b e x^{2}\right )} \sqrt {-\frac {b}{a e}} \arctan \left (\sqrt {e x} \sqrt {-\frac {b}{a e}}\right ) - 21 \, {\left (b^{3} e x^{4} - a^{2} b e x^{2}\right )} \sqrt {-\frac {b}{a e}} \log \left (\frac {b x + 2 \, \sqrt {e x} a \sqrt {-\frac {b}{a e}} - a}{b x + a}\right ) + 4 \, {\left (7 \, a b^{2} x^{2} - 4 \, a^{3}\right )} \sqrt {e x}}{24 \, {\left (a^{5} b^{2} c^{2} e^{3} x^{4} - a^{7} c^{2} e^{3} x^{2}\right )}}\right ] \] Input:

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

Output:

[1/24*(42*(b^3*e*x^4 - a^2*b*e*x^2)*sqrt(b/(a*e))*arctan(sqrt(e*x)*sqrt(b/ 
(a*e))) + 21*(b^3*e*x^4 - a^2*b*e*x^2)*sqrt(b/(a*e))*log((b*x + 2*sqrt(e*x 
)*a*sqrt(b/(a*e)) + a)/(b*x - a)) - 4*(7*a*b^2*x^2 - 4*a^3)*sqrt(e*x))/(a^ 
5*b^2*c^2*e^3*x^4 - a^7*c^2*e^3*x^2), -1/24*(42*(b^3*e*x^4 - a^2*b*e*x^2)* 
sqrt(-b/(a*e))*arctan(sqrt(e*x)*sqrt(-b/(a*e))) - 21*(b^3*e*x^4 - a^2*b*e* 
x^2)*sqrt(-b/(a*e))*log((b*x + 2*sqrt(e*x)*a*sqrt(-b/(a*e)) - a)/(b*x + a) 
) + 4*(7*a*b^2*x^2 - 4*a^3)*sqrt(e*x))/(a^5*b^2*c^2*e^3*x^4 - a^7*c^2*e^3* 
x^2)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(1/(e*x)^(5/2)/(b*x+a)^2/(-b*c*x+a*c)^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 [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(e x)^{5/2} (a+b x)^2 (a c-b c x)^2} \, dx=-\frac {\sqrt {e x} b^{2}}{2 \, {\left (b^{2} e^{2} x^{2} - a^{2} e^{2}\right )} a^{4} c^{2} e} + \frac {7 \, b^{2} \arctan \left (\frac {\sqrt {e x} b}{\sqrt {a b e}}\right )}{4 \, \sqrt {a b e} a^{5} c^{2} e^{2}} - \frac {7 \, b^{2} \arctan \left (\frac {\sqrt {e x} b}{\sqrt {-a b e}}\right )}{4 \, \sqrt {-a b e} a^{5} c^{2} e^{2}} - \frac {2}{3 \, \sqrt {e x} a^{4} c^{2} e^{2} x} \] Input:

integrate(1/(e*x)^(5/2)/(b*x+a)^2/(-b*c*x+a*c)^2,x, algorithm="giac")
 

Output:

-1/2*sqrt(e*x)*b^2/((b^2*e^2*x^2 - a^2*e^2)*a^4*c^2*e) + 7/4*b^2*arctan(sq 
rt(e*x)*b/sqrt(a*b*e))/(sqrt(a*b*e)*a^5*c^2*e^2) - 7/4*b^2*arctan(sqrt(e*x 
)*b/sqrt(-a*b*e))/(sqrt(-a*b*e)*a^5*c^2*e^2) - 2/3/(sqrt(e*x)*a^4*c^2*e^2* 
x)
 

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(e x)^{5/2} (a+b x)^2 (a c-b c x)^2} \, dx=\frac {\frac {2\,e}{3\,a^2}-\frac {7\,b^2\,e\,x^2}{6\,a^4}}{b^2\,c^2\,{\left (e\,x\right )}^{7/2}-a^2\,c^2\,e^2\,{\left (e\,x\right )}^{3/2}}+\frac {7\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{4\,a^{11/2}\,c^2\,e^{5/2}}+\frac {7\,b^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{4\,a^{11/2}\,c^2\,e^{5/2}} \] Input:

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

Output:

((2*e)/(3*a^2) - (7*b^2*e*x^2)/(6*a^4))/(b^2*c^2*(e*x)^(7/2) - a^2*c^2*e^2 
*(e*x)^(3/2)) + (7*b^(3/2)*atan((b^(1/2)*(e*x)^(1/2))/(a^(1/2)*e^(1/2))))/ 
(4*a^(11/2)*c^2*e^(5/2)) + (7*b^(3/2)*atanh((b^(1/2)*(e*x)^(1/2))/(a^(1/2) 
*e^(1/2))))/(4*a^(11/2)*c^2*e^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(e x)^{5/2} (a+b x)^2 (a c-b c x)^2} \, dx=\frac {\sqrt {e}\, \left (42 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b x -42 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) b^{3} x^{3}-21 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) a^{2} b x +21 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) b^{3} x^{3}+21 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) a^{2} b x -21 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) b^{3} x^{3}-16 a^{4}+28 a^{2} b^{2} x^{2}\right )}{24 \sqrt {x}\, a^{6} c^{2} e^{3} x \left (-b^{2} x^{2}+a^{2}\right )} \] Input:

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

Output:

(sqrt(e)*(42*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a 
**2*b*x - 42*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*b 
**3*x**3 - 21*sqrt(x)*sqrt(b)*sqrt(a)*log( - sqrt(a) + sqrt(x)*sqrt(b))*a* 
*2*b*x + 21*sqrt(x)*sqrt(b)*sqrt(a)*log( - sqrt(a) + sqrt(x)*sqrt(b))*b**3 
*x**3 + 21*sqrt(x)*sqrt(b)*sqrt(a)*log(sqrt(a) + sqrt(x)*sqrt(b))*a**2*b*x 
 - 21*sqrt(x)*sqrt(b)*sqrt(a)*log(sqrt(a) + sqrt(x)*sqrt(b))*b**3*x**3 - 1 
6*a**4 + 28*a**2*b**2*x**2))/(24*sqrt(x)*a**6*c**2*e**3*x*(a**2 - b**2*x** 
2))