\(\int \frac {1}{(e x)^{3/2} (a+b x)^2 (a c-b c x)^2} \, dx\) [67]

Optimal result

Integrand size = 26, antiderivative size = 147 \[ \int \frac {1}{(e x)^{3/2} (a+b x)^2 (a c-b c x)^2} \, dx=-\frac {5}{2 a^4 c^2 e \sqrt {e x}}+\frac {1}{2 a^2 c^2 e \sqrt {e x} \left (a^2-b^2 x^2\right )}-\frac {5 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{9/2} c^2 e^{3/2}}+\frac {5 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{9/2} c^2 e^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(e x)^{3/2} (a+b x)^2 (a c-b c x)^2} \, dx=\frac {x \left (\frac {2 \sqrt {a} \left (-4 a^2+5 b^2 x^2\right )}{a^2-b^2 x^2}-5 \sqrt {b} \sqrt {x} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+5 \sqrt {b} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{4 a^{9/2} c^2 (e x)^{3/2}} \] Input:

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

Output:

(x*((2*Sqrt[a]*(-4*a^2 + 5*b^2*x^2))/(a^2 - b^2*x^2) - 5*Sqrt[b]*Sqrt[x]*A 
rcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]] + 5*Sqrt[b]*Sqrt[x]*ArcTanh[(Sqrt[b]*Sqrt 
[x])/Sqrt[a]]))/(4*a^(9/2)*c^2*(e*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {82, 253, 27, 264, 266, 27, 827, 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.

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

\(\Big \downarrow \) 82

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

\(\Big \downarrow \) 253

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 264

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 827

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {5 \left (\frac {2 b^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 \sqrt {a} b^{3/2} \sqrt {e}}-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 \sqrt {a} b^{3/2} \sqrt {e}}\right )}{a^2 e}-\frac {2}{a^2 e \sqrt {e x}}\right )}{4 a^2 c^2}+\frac {1}{2 a^2 c^2 e \sqrt {e x} \left (a^2-b^2 x^2\right )}\)

Input:

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

Output:

1/(2*a^2*c^2*e*Sqrt[e*x]*(a^2 - b^2*x^2)) + (5*(-2/(a^2*e*Sqrt[e*x]) + (2* 
b^2*(-1/2*ArcTan[(Sqrt[b]*Sqrt[e*x])/(Sqrt[a]*Sqrt[e])]/(Sqrt[a]*b^(3/2)*S 
qrt[e]) + ArcTanh[(Sqrt[b]*Sqrt[e*x])/(Sqrt[a]*Sqrt[e])]/(2*Sqrt[a]*b^(3/2 
)*Sqrt[e])))/(a^2*e)))/(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 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   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 = 113, normalized size of antiderivative = 0.77

method	result	size
pseudoelliptic	\(\frac {e^{3} \left (-\frac {b \sqrt {e x}}{4 e^{5} \left (b x +a \right )}-\frac {5 \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right ) b}{4 \sqrt {a e b}\, e^{4}}-\frac {b \left (-\frac {\sqrt {e x}}{e \left (-b x +a \right )}-\frac {5 \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{\sqrt {a e b}}\right )}{4 e^{4}}-\frac {2}{e^{4} \sqrt {e x}}\right )}{c^{2} a^{4}}\)	\(113\)
risch	\(-\frac {2}{a^{4} c^{2} e \sqrt {e x}}+\frac {-\frac {b \sqrt {e x}}{4 a^{4} \left (b e x +a e \right )}-\frac {5 b \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{4 a^{4} \sqrt {a e b}}-\frac {b \sqrt {e x}}{4 a^{4} \left (b e x -a e \right )}+\frac {5 b \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{4 a^{4} \sqrt {a e b}}}{e \,c^{2}}\)	\(121\)
derivativedivides	\(\frac {2 e^{3} \left (-\frac {b \left (\frac {\sqrt {e x}}{2 b e x +2 a e}+\frac {5 \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}\right )}{4 a^{4} e^{4}}-\frac {1}{a^{4} e^{4} \sqrt {e x}}+\frac {b \left (\frac {\sqrt {e x}}{-2 b e x +2 a e}+\frac {5 \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}\right )}{4 a^{4} e^{4}}\right )}{c^{2}}\)	\(122\)
default	\(\frac {2 e^{3} \left (-\frac {b \left (\frac {\sqrt {e x}}{2 b e x +2 a e}+\frac {5 \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}\right )}{4 a^{4} e^{4}}-\frac {1}{a^{4} e^{4} \sqrt {e x}}+\frac {b \left (\frac {\sqrt {e x}}{-2 b e x +2 a e}+\frac {5 \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}\right )}{4 a^{4} e^{4}}\right )}{c^{2}}\)	\(122\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.24 \[ \int \frac {1}{(e x)^{3/2} (a+b x)^2 (a c-b c x)^2} \, dx=\left [-\frac {10 \, {\left (b^{2} e x^{3} - a^{2} e x\right )} \sqrt {\frac {b}{a e}} \arctan \left (\sqrt {e x} \sqrt {\frac {b}{a e}}\right ) - 5 \, {\left (b^{2} e x^{3} - a^{2} e x\right )} \sqrt {\frac {b}{a e}} \log \left (\frac {b x + 2 \, \sqrt {e x} a \sqrt {\frac {b}{a e}} + a}{b x - a}\right ) + 4 \, {\left (5 \, b^{2} x^{2} - 4 \, a^{2}\right )} \sqrt {e x}}{8 \, {\left (a^{4} b^{2} c^{2} e^{2} x^{3} - a^{6} c^{2} e^{2} x\right )}}, -\frac {10 \, {\left (b^{2} e x^{3} - a^{2} e x\right )} \sqrt {-\frac {b}{a e}} \arctan \left (\sqrt {e x} \sqrt {-\frac {b}{a e}}\right ) - 5 \, {\left (b^{2} e x^{3} - a^{2} e x\right )} \sqrt {-\frac {b}{a e}} \log \left (\frac {b x - 2 \, \sqrt {e x} a \sqrt {-\frac {b}{a e}} - a}{b x + a}\right ) + 4 \, {\left (5 \, b^{2} x^{2} - 4 \, a^{2}\right )} \sqrt {e x}}{8 \, {\left (a^{4} b^{2} c^{2} e^{2} x^{3} - a^{6} c^{2} e^{2} x\right )}}\right ] \] Input:

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

Output:

[-1/8*(10*(b^2*e*x^3 - a^2*e*x)*sqrt(b/(a*e))*arctan(sqrt(e*x)*sqrt(b/(a*e 
))) - 5*(b^2*e*x^3 - a^2*e*x)*sqrt(b/(a*e))*log((b*x + 2*sqrt(e*x)*a*sqrt( 
b/(a*e)) + a)/(b*x - a)) + 4*(5*b^2*x^2 - 4*a^2)*sqrt(e*x))/(a^4*b^2*c^2*e 
^2*x^3 - a^6*c^2*e^2*x), -1/8*(10*(b^2*e*x^3 - a^2*e*x)*sqrt(-b/(a*e))*arc 
tan(sqrt(e*x)*sqrt(-b/(a*e))) - 5*(b^2*e*x^3 - a^2*e*x)*sqrt(-b/(a*e))*log 
((b*x - 2*sqrt(e*x)*a*sqrt(-b/(a*e)) - a)/(b*x + a)) + 4*(5*b^2*x^2 - 4*a^ 
2)*sqrt(e*x))/(a^4*b^2*c^2*e^2*x^3 - a^6*c^2*e^2*x)]
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.82 (sec) , antiderivative size = 1239, normalized size of antiderivative = 8.43 \[ \int \frac {1}{(e x)^{3/2} (a+b x)^2 (a c-b c x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((10*a**(7/2)*b**2*x**(3/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(8*a** 
8*b**(3/2)*c**2*e**(3/2)*x**(3/2) - 8*a**6*b**(7/2)*c**2*e**(3/2)*x**(7/2) 
) - 10*a**(7/2)*b**2*x**(3/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(8*a**8*b**(3/ 
2)*c**2*e**(3/2)*x**(3/2) - 8*a**6*b**(7/2)*c**2*e**(3/2)*x**(7/2)) + 5*pi 
*a**(7/2)*b**2*x**(3/2)/(8*a**8*b**(3/2)*c**2*e**(3/2)*x**(3/2) - 8*a**6*b 
**(7/2)*c**2*e**(3/2)*x**(7/2)) - 10*a**(3/2)*b**4*x**(7/2)*acoth(sqrt(b)* 
sqrt(x)/sqrt(a))/(8*a**8*b**(3/2)*c**2*e**(3/2)*x**(3/2) - 8*a**6*b**(7/2) 
*c**2*e**(3/2)*x**(7/2)) + 10*a**(3/2)*b**4*x**(7/2)*atan(sqrt(b)*sqrt(x)/ 
sqrt(a))/(8*a**8*b**(3/2)*c**2*e**(3/2)*x**(3/2) - 8*a**6*b**(7/2)*c**2*e* 
*(3/2)*x**(7/2)) - 5*pi*a**(3/2)*b**4*x**(7/2)/(8*a**8*b**(3/2)*c**2*e**(3 
/2)*x**(3/2) - 8*a**6*b**(7/2)*c**2*e**(3/2)*x**(7/2)) - 16*a**4*b**(3/2)* 
x/(8*a**8*b**(3/2)*c**2*e**(3/2)*x**(3/2) - 8*a**6*b**(7/2)*c**2*e**(3/2)* 
x**(7/2)) + 20*a**2*b**(7/2)*x**3/(8*a**8*b**(3/2)*c**2*e**(3/2)*x**(3/2) 
- 8*a**6*b**(7/2)*c**2*e**(3/2)*x**(7/2)), Abs(b*x/a) > 1), (a**(7/2)*b**2 
*x**(3/2)*(-5 - 5*I)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(a**8*b**(3/2)*c**2*e** 
(3/2)*x**(3/2)*(4 + 4*I) + a**6*b**(7/2)*c**2*e**(3/2)*x**(7/2)*(-4 - 4*I) 
) + a**(7/2)*b**2*x**(3/2)*(5 + 5*I)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(a**8* 
b**(3/2)*c**2*e**(3/2)*x**(3/2)*(4 + 4*I) + a**6*b**(7/2)*c**2*e**(3/2)*x* 
*(7/2)*(-4 - 4*I)) + 5*I*pi*a**(7/2)*b**2*x**(3/2)/(a**8*b**(3/2)*c**2*e** 
(3/2)*x**(3/2)*(4 + 4*I) + a**6*b**(7/2)*c**2*e**(3/2)*x**(7/2)*(-4 - 4...
 

Maxima [F(-2)]

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

integrate(1/(e*x)^(3/2)/(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 = 0.13 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(e x)^{3/2} (a+b x)^2 (a c-b c x)^2} \, dx=-\frac {\frac {5 \, b \arctan \left (\frac {\sqrt {e x} b}{\sqrt {a b e}}\right )}{\sqrt {a b e} a^{4} c^{2}} + \frac {5 \, b \arctan \left (\frac {\sqrt {e x} b}{\sqrt {-a b e}}\right )}{\sqrt {-a b e} a^{4} c^{2}} + \frac {2 \, {\left (5 \, b^{2} e^{2} x^{2} - 4 \, a^{2} e^{2}\right )}}{{\left (\sqrt {e x} b^{2} e^{2} x^{2} - \sqrt {e x} a^{2} e^{2}\right )} a^{4} c^{2}}}{4 \, e} \] Input:

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

Output:

-1/4*(5*b*arctan(sqrt(e*x)*b/sqrt(a*b*e))/(sqrt(a*b*e)*a^4*c^2) + 5*b*arct 
an(sqrt(e*x)*b/sqrt(-a*b*e))/(sqrt(-a*b*e)*a^4*c^2) + 2*(5*b^2*e^2*x^2 - 4 
*a^2*e^2)/((sqrt(e*x)*b^2*e^2*x^2 - sqrt(e*x)*a^2*e^2)*a^4*c^2))/e
 

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(e x)^{3/2} (a+b x)^2 (a c-b c x)^2} \, dx=\frac {\frac {2\,e}{a^2}-\frac {5\,b^2\,e\,x^2}{2\,a^4}}{b^2\,c^2\,{\left (e\,x\right )}^{5/2}-a^2\,c^2\,e^2\,\sqrt {e\,x}}-\frac {5\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{4\,a^{9/2}\,c^2\,e^{3/2}}+\frac {5\,\sqrt {b}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{4\,a^{9/2}\,c^2\,e^{3/2}} \] Input:

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

Output:

((2*e)/a^2 - (5*b^2*e*x^2)/(2*a^4))/(b^2*c^2*(e*x)^(5/2) - a^2*c^2*e^2*(e* 
x)^(1/2)) - (5*b^(1/2)*atan((b^(1/2)*(e*x)^(1/2))/(a^(1/2)*e^(1/2))))/(4*a 
^(9/2)*c^2*e^(3/2)) + (5*b^(1/2)*atanh((b^(1/2)*(e*x)^(1/2))/(a^(1/2)*e^(1 
/2))))/(4*a^(9/2)*c^2*e^(3/2))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(e x)^{3/2} (a+b x)^2 (a c-b c x)^2} \, dx=\frac {\sqrt {e}\, \left (-10 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{2}+10 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) b^{2} x^{2}-5 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) a^{2}+5 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) b^{2} x^{2}+5 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) a^{2}-5 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) b^{2} x^{2}-16 a^{3}+20 a \,b^{2} x^{2}\right )}{8 \sqrt {x}\, a^{5} c^{2} e^{2} \left (-b^{2} x^{2}+a^{2}\right )} \] Input:

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

Output:

(sqrt(e)*( - 10*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)) 
)*a**2 + 10*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*b* 
*2*x**2 - 5*sqrt(x)*sqrt(b)*sqrt(a)*log( - sqrt(a) + sqrt(x)*sqrt(b))*a**2 
 + 5*sqrt(x)*sqrt(b)*sqrt(a)*log( - sqrt(a) + sqrt(x)*sqrt(b))*b**2*x**2 + 
 5*sqrt(x)*sqrt(b)*sqrt(a)*log(sqrt(a) + sqrt(x)*sqrt(b))*a**2 - 5*sqrt(x) 
*sqrt(b)*sqrt(a)*log(sqrt(a) + sqrt(x)*sqrt(b))*b**2*x**2 - 16*a**3 + 20*a 
*b**2*x**2))/(8*sqrt(x)*a**5*c**2*e**2*(a**2 - b**2*x**2))