\(\int \frac {1}{\sqrt {e x} (a+b x)^3 (a c-b c x)^3} \, dx\) [75]

Optimal result

Integrand size = 26, antiderivative size = 161 \[ \int \frac {1}{\sqrt {e x} (a+b x)^3 (a c-b c x)^3} \, dx=\frac {\sqrt {e x}}{4 a^2 c^3 e \left (a^2-b^2 x^2\right )^2}+\frac {7 \sqrt {e x}}{16 a^4 c^3 e \left (a^2-b^2 x^2\right )}+\frac {21 \arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{32 a^{11/2} \sqrt {b} c^3 \sqrt {e}}+\frac {21 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{32 a^{11/2} \sqrt {b} c^3 \sqrt {e}} \] Output:

1/4*(e*x)^(1/2)/a^2/c^3/e/(-b^2*x^2+a^2)^2+7/16*(e*x)^(1/2)/a^4/c^3/e/(-b^ 
2*x^2+a^2)+21/32*arctan(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1/2))/a^(11/2)/b^(1 
/2)/c^3/e^(1/2)+21/32*arctanh(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1/2))/a^(11/2 
)/b^(1/2)/c^3/e^(1/2)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt {e x} (a+b x)^3 (a c-b c x)^3} \, dx=\frac {x \left (11 a^2-7 b^2 x^2\right )}{16 a^4 c^3 \sqrt {e x} \left (a^2-b^2 x^2\right )^2}+\frac {21 \sqrt {x} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{32 a^{11/2} \sqrt {b} c^3 \sqrt {e x}}+\frac {21 \sqrt {x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{32 a^{11/2} \sqrt {b} c^3 \sqrt {e x}} \] Input:

Integrate[1/(Sqrt[e*x]*(a + b*x)^3*(a*c - b*c*x)^3),x]
 

Output:

(x*(11*a^2 - 7*b^2*x^2))/(16*a^4*c^3*Sqrt[e*x]*(a^2 - b^2*x^2)^2) + (21*Sq 
rt[x]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(32*a^(11/2)*Sqrt[b]*c^3*Sqrt[e*x 
]) + (21*Sqrt[x]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(32*a^(11/2)*Sqrt[b]* 
c^3*Sqrt[e*x])
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.08, 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, 253, 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.

\(\displaystyle \int \frac {1}{\sqrt {e x} (a+b x)^3 (a c-b c x)^3} \, dx\)

\(\Big \downarrow \) 82

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

\(\Big \downarrow \) 253

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 253

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 756

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

Input:

Int[1/(Sqrt[e*x]*(a + b*x)^3*(a*c - b*c*x)^3),x]
 

Output:

Sqrt[e*x]/(4*a^2*c^3*e*(a^2 - b^2*x^2)^2) + (7*(Sqrt[e*x]/(2*a^2*e*(a^2 - 
b^2*x^2)) + (3*((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])))/(2*a^2*e)))/(8*a^2*c^3)
 

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 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 = 0.36 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(-\frac {2 e^{5} \left (-\frac {\frac {\frac {9 b \left (e x \right )^{\frac {3}{2}}}{4}+\frac {11 a e \sqrt {e x}}{4}}{\left (b e x +a e \right )^{2}}+\frac {21 \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{4 \sqrt {a e b}}}{16 a^{5} e^{5}}-\frac {\frac {-\frac {9 b \left (e x \right )^{\frac {3}{2}}}{4}+\frac {11 a e \sqrt {e x}}{4}}{\left (-b e x +a e \right )^{2}}+\frac {21 \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{4 \sqrt {a e b}}}{16 a^{5} e^{5}}\right )}{c^{3}}\)	\(131\)
default	\(\frac {2 e^{5} \left (\frac {\frac {\frac {9 b \left (e x \right )^{\frac {3}{2}}}{4}+\frac {11 a e \sqrt {e x}}{4}}{\left (b e x +a e \right )^{2}}+\frac {21 \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{4 \sqrt {a e b}}}{16 a^{5} e^{5}}+\frac {\frac {-\frac {9 b \left (e x \right )^{\frac {3}{2}}}{4}+\frac {11 a e \sqrt {e x}}{4}}{\left (-b e x +a e \right )^{2}}+\frac {21 \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{4 \sqrt {a e b}}}{16 a^{5} e^{5}}\right )}{c^{3}}\)	\(131\)
pseudoelliptic	\(-\frac {e^{5} \left (-\frac {9 b \left (e x \right )^{\frac {3}{2}}}{e^{7} \left (b x +a \right )^{2} a^{5}}-\frac {11 \sqrt {e x}}{e^{6} \left (b x +a \right )^{2} a^{4}}-\frac {21 \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{\sqrt {a e b}\, a^{5} e^{5}}+\frac {-\frac {\sqrt {e x}\, \left (-9 b x +11 a \right )}{e \left (-b x +a \right )^{2}}-\frac {21 \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{\sqrt {a e b}}}{a^{5} e^{5}}\right )}{32 c^{3}}\)	\(135\)

Input:

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

Output:

-2*e^5/c^3*(-1/16/a^5/e^5*((9/4*b*(e*x)^(3/2)+11/4*a*e*(e*x)^(1/2))/(b*e*x 
+a*e)^2+21/4/(a*e*b)^(1/2)*arctan(b*(e*x)^(1/2)/(a*e*b)^(1/2)))-1/16/a^5/e 
^5*((-9/4*b*(e*x)^(3/2)+11/4*a*e*(e*x)^(1/2))/(-b*e*x+a*e)^2+21/4/(a*e*b)^ 
(1/2)*arctanh(b*(e*x)^(1/2)/(a*e*b)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.30 \[ \int \frac {1}{\sqrt {e x} (a+b x)^3 (a c-b c x)^3} \, dx=\left [-\frac {42 \, {\left (b^{4} x^{4} - 2 \, a^{2} b^{2} x^{2} + a^{4}\right )} \sqrt {a b e} \arctan \left (\frac {\sqrt {a b e} \sqrt {e x}}{b e x}\right ) - 21 \, {\left (b^{4} x^{4} - 2 \, a^{2} b^{2} x^{2} + a^{4}\right )} \sqrt {a b e} \log \left (\frac {b e x + a e + 2 \, \sqrt {a b e} \sqrt {e x}}{b x - a}\right ) + 4 \, {\left (7 \, a^{2} b^{3} x^{2} - 11 \, a^{4} b\right )} \sqrt {e x}}{64 \, {\left (a^{6} b^{5} c^{3} e x^{4} - 2 \, a^{8} b^{3} c^{3} e x^{2} + a^{10} b c^{3} e\right )}}, -\frac {42 \, {\left (b^{4} x^{4} - 2 \, a^{2} b^{2} x^{2} + a^{4}\right )} \sqrt {-a b e} \arctan \left (\frac {\sqrt {-a b e} \sqrt {e x}}{b e x}\right ) + 21 \, {\left (b^{4} x^{4} - 2 \, a^{2} b^{2} x^{2} + a^{4}\right )} \sqrt {-a b e} \log \left (\frac {b e x - a e - 2 \, \sqrt {-a b e} \sqrt {e x}}{b x + a}\right ) + 4 \, {\left (7 \, a^{2} b^{3} x^{2} - 11 \, a^{4} b\right )} \sqrt {e x}}{64 \, {\left (a^{6} b^{5} c^{3} e x^{4} - 2 \, a^{8} b^{3} c^{3} e x^{2} + a^{10} b c^{3} e\right )}}\right ] \] Input:

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

Output:

[-1/64*(42*(b^4*x^4 - 2*a^2*b^2*x^2 + a^4)*sqrt(a*b*e)*arctan(sqrt(a*b*e)* 
sqrt(e*x)/(b*e*x)) - 21*(b^4*x^4 - 2*a^2*b^2*x^2 + a^4)*sqrt(a*b*e)*log((b 
*e*x + a*e + 2*sqrt(a*b*e)*sqrt(e*x))/(b*x - a)) + 4*(7*a^2*b^3*x^2 - 11*a 
^4*b)*sqrt(e*x))/(a^6*b^5*c^3*e*x^4 - 2*a^8*b^3*c^3*e*x^2 + a^10*b*c^3*e), 
 -1/64*(42*(b^4*x^4 - 2*a^2*b^2*x^2 + a^4)*sqrt(-a*b*e)*arctan(sqrt(-a*b*e 
)*sqrt(e*x)/(b*e*x)) + 21*(b^4*x^4 - 2*a^2*b^2*x^2 + a^4)*sqrt(-a*b*e)*log 
((b*e*x - a*e - 2*sqrt(-a*b*e)*sqrt(e*x))/(b*x + a)) + 4*(7*a^2*b^3*x^2 - 
11*a^4*b)*sqrt(e*x))/(a^6*b^5*c^3*e*x^4 - 2*a^8*b^3*c^3*e*x^2 + a^10*b*c^3 
*e)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {e x} (a+b x)^3 (a c-b c x)^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {e x} (a+b x)^3 (a c-b c x)^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(e*x)^(1/2)/(b*x+a)^3/(-b*c*x+a*c)^3,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 = 118, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\sqrt {e x} (a+b x)^3 (a c-b c x)^3} \, dx=\frac {21 \, \arctan \left (\frac {\sqrt {e x} b}{\sqrt {a b e}}\right )}{32 \, \sqrt {a b e} a^{5} c^{3}} - \frac {21 \, \arctan \left (\frac {\sqrt {e x} b}{\sqrt {-a b e}}\right )}{32 \, \sqrt {-a b e} a^{5} c^{3}} - \frac {7 \, \sqrt {e x} b^{2} e^{3} x^{2} - 11 \, \sqrt {e x} a^{2} e^{3}}{16 \, {\left (b^{2} e^{2} x^{2} - a^{2} e^{2}\right )}^{2} a^{4} c^{3}} \] Input:

integrate(1/(e*x)^(1/2)/(b*x+a)^3/(-b*c*x+a*c)^3,x, algorithm="giac")
 

Output:

21/32*arctan(sqrt(e*x)*b/sqrt(a*b*e))/(sqrt(a*b*e)*a^5*c^3) - 21/32*arctan 
(sqrt(e*x)*b/sqrt(-a*b*e))/(sqrt(-a*b*e)*a^5*c^3) - 1/16*(7*sqrt(e*x)*b^2* 
e^3*x^2 - 11*sqrt(e*x)*a^2*e^3)/((b^2*e^2*x^2 - a^2*e^2)^2*a^4*c^3)
 

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {e x} (a+b x)^3 (a c-b c x)^3} \, dx=\frac {\frac {11\,e^3\,\sqrt {e\,x}}{16\,a^2}-\frac {7\,b^2\,e\,{\left (e\,x\right )}^{5/2}}{16\,a^4}}{a^4\,c^3\,e^4-2\,a^2\,b^2\,c^3\,e^4\,x^2+b^4\,c^3\,e^4\,x^4}+\frac {21\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{32\,a^{11/2}\,\sqrt {b}\,c^3\,\sqrt {e}}+\frac {21\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{32\,a^{11/2}\,\sqrt {b}\,c^3\,\sqrt {e}} \] Input:

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

Output:

((11*e^3*(e*x)^(1/2))/(16*a^2) - (7*b^2*e*(e*x)^(5/2))/(16*a^4))/(a^4*c^3* 
e^4 + b^4*c^3*e^4*x^4 - 2*a^2*b^2*c^3*e^4*x^2) + (21*atan((b^(1/2)*(e*x)^( 
1/2))/(a^(1/2)*e^(1/2))))/(32*a^(11/2)*b^(1/2)*c^3*e^(1/2)) + (21*atanh((b 
^(1/2)*(e*x)^(1/2))/(a^(1/2)*e^(1/2))))/(32*a^(11/2)*b^(1/2)*c^3*e^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.67 \[ \int \frac {1}{\sqrt {e x} (a+b x)^3 (a c-b c x)^3} \, dx=\frac {\sqrt {e}\, \left (42 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{4}-84 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{2} x^{2}+42 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) b^{4} x^{4}-21 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) a^{4}+42 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) a^{2} b^{2} x^{2}-21 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) b^{4} x^{4}+21 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) a^{4}-42 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) a^{2} b^{2} x^{2}+21 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) b^{4} x^{4}+44 \sqrt {x}\, a^{4} b -28 \sqrt {x}\, a^{2} b^{3} x^{2}\right )}{64 a^{6} b \,c^{3} e \left (b^{4} x^{4}-2 a^{2} b^{2} x^{2}+a^{4}\right )} \] Input:

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

Output:

(sqrt(e)*(42*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**4 - 84 
*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**2*b**2*x**2 + 42*s 
qrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*b**4*x**4 - 21*sqrt(b)* 
sqrt(a)*log( - sqrt(a) + sqrt(x)*sqrt(b))*a**4 + 42*sqrt(b)*sqrt(a)*log( - 
 sqrt(a) + sqrt(x)*sqrt(b))*a**2*b**2*x**2 - 21*sqrt(b)*sqrt(a)*log( - sqr 
t(a) + sqrt(x)*sqrt(b))*b**4*x**4 + 21*sqrt(b)*sqrt(a)*log(sqrt(a) + sqrt( 
x)*sqrt(b))*a**4 - 42*sqrt(b)*sqrt(a)*log(sqrt(a) + sqrt(x)*sqrt(b))*a**2* 
b**2*x**2 + 21*sqrt(b)*sqrt(a)*log(sqrt(a) + sqrt(x)*sqrt(b))*b**4*x**4 + 
44*sqrt(x)*a**4*b - 28*sqrt(x)*a**2*b**3*x**2))/(64*a**6*b*c**3*e*(a**4 - 
2*a**2*b**2*x**2 + b**4*x**4))