\(\int \frac {1}{x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx\) [128]

Optimal result

Integrand size = 26, antiderivative size = 115 \[ \int \frac {1}{x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {1}{a^2 c x^2 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {3 \sqrt {a+b x} \sqrt {a c-b c x}}{2 a^4 c^2 x^2}-\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )}{2 a^5 c^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=-\frac {a^3-3 a b^2 x^2+6 b^2 x^2 \sqrt {a-b x} \sqrt {a+b x} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{2 a^5 c x^2 \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:

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

Output:

-1/2*(a^3 - 3*a*b^2*x^2 + 6*b^2*x^2*Sqrt[a - b*x]*Sqrt[a + b*x]*ArcTanh[Sq 
rt[a + b*x]/Sqrt[a - b*x]])/(a^5*c*x^2*Sqrt[c*(a - b*x)]*Sqrt[a + b*x])
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {114, 27, 115, 27, 35, 103, 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.

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 115

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 35

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

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/2*1/(a^2*c*x^2*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + (3*b^2*(1/(a^2*c*Sqrt 
[a + b*x]*Sqrt[a*c - b*c*x]) - ArcTanh[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/( 
a*Sqrt[c])]/(a^3*c^(3/2))))/(2*a^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 35

Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> 
 Simp[(b/d)^m   Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} 
, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] &&  !(IntegerQ[n] && SimplerQ[a + 
b*x, c + d*x])
 

rule 103

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ 
))), x_] :> Simp[b*f   Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq 
rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d 
*e - f*(b*c + a*d), 0]
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 115

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 
*n, 2*p]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(211\) vs. \(2(95)=190\).

Time = 0.32 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.84

method	result	size
default	\(-\frac {\sqrt {c \left (-b x +a \right )}\, \left (-3 \ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}{x}\right ) b^{4} c \,x^{4}+3 \ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}{x}\right ) a^{2} b^{2} c \,x^{2}-3 b^{2} x^{2} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {a^{2} c}+\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {a^{2} c}\, a^{2}\right )}{2 a^{4} c^{2} \left (-b x +a \right ) \sqrt {a^{2} c}\, x^{2} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b x +a}}\)	\(212\)
risch	\(-\frac {\left (-b x +a \right ) \sqrt {b x +a}}{2 a^{4} x^{2} \sqrt {-c \left (b x -a \right )}\, c}+\frac {\left (-\frac {3 b^{2} \ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {-b^{2} c \,x^{2}+a^{2} c}}{x}\right )}{2 a^{4} \sqrt {a^{2} c}}-\frac {b \sqrt {-b^{2} c \left (x -\frac {a}{b}\right )^{2}-2 a b c \left (x -\frac {a}{b}\right )}}{2 a^{5} c \left (x -\frac {a}{b}\right )}+\frac {b \sqrt {-b^{2} c \left (x +\frac {a}{b}\right )^{2}+2 a b c \left (x +\frac {a}{b}\right )}}{2 a^{5} c \left (x +\frac {a}{b}\right )}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}}{\sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}\, c}\)	\(231\)

Input:

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

Output:

-1/2*(c*(-b*x+a))^(1/2)/a^4/c^2*(-3*ln(2*(a^2*c+(a^2*c)^(1/2)*(c*(-b^2*x^2 
+a^2))^(1/2))/x)*b^4*c*x^4+3*ln(2*(a^2*c+(a^2*c)^(1/2)*(c*(-b^2*x^2+a^2))^ 
(1/2))/x)*a^2*b^2*c*x^2-3*b^2*x^2*(c*(-b^2*x^2+a^2))^(1/2)*(a^2*c)^(1/2)+( 
c*(-b^2*x^2+a^2))^(1/2)*(a^2*c)^(1/2)*a^2)/(-b*x+a)/(a^2*c)^(1/2)/x^2/(c*( 
-b^2*x^2+a^2))^(1/2)/(b*x+a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.33 \[ \int \frac {1}{x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\left [\frac {3 \, {\left (b^{4} x^{4} - a^{2} b^{2} x^{2}\right )} \sqrt {c} \log \left (-\frac {b^{2} c x^{2} - 2 \, a^{2} c + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {c}}{x^{2}}\right ) - 2 \, {\left (3 \, a b^{2} x^{2} - a^{3}\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{4 \, {\left (a^{5} b^{2} c^{2} x^{4} - a^{7} c^{2} x^{2}\right )}}, -\frac {3 \, {\left (b^{4} x^{4} - a^{2} b^{2} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {-c}}{b^{2} c x^{2} - a^{2} c}\right ) + {\left (3 \, a b^{2} x^{2} - a^{3}\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{2 \, {\left (a^{5} b^{2} c^{2} x^{4} - a^{7} c^{2} x^{2}\right )}}\right ] \] Input:

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

Output:

[1/4*(3*(b^4*x^4 - a^2*b^2*x^2)*sqrt(c)*log(-(b^2*c*x^2 - 2*a^2*c + 2*sqrt 
(-b*c*x + a*c)*sqrt(b*x + a)*a*sqrt(c))/x^2) - 2*(3*a*b^2*x^2 - a^3)*sqrt( 
-b*c*x + a*c)*sqrt(b*x + a))/(a^5*b^2*c^2*x^4 - a^7*c^2*x^2), -1/2*(3*(b^4 
*x^4 - a^2*b^2*x^2)*sqrt(-c)*arctan(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*a*sqr 
t(-c)/(b^2*c*x^2 - a^2*c)) + (3*a*b^2*x^2 - a^3)*sqrt(-b*c*x + a*c)*sqrt(b 
*x + a))/(a^5*b^2*c^2*x^4 - a^7*c^2*x^2)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {3 \, b^{2}}{2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{4} c} - \frac {3 \, b^{2} \log \left (\frac {2 \, a^{2} c}{{\left | x \right |}} + \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a \sqrt {c}}{{\left | x \right |}}\right )}{2 \, a^{5} c^{\frac {3}{2}}} - \frac {1}{2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{2} c x^{2}} \] Input:

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

Output:

3/2*b^2/(sqrt(-b^2*c*x^2 + a^2*c)*a^4*c) - 3/2*b^2*log(2*a^2*c/abs(x) + 2* 
sqrt(-b^2*c*x^2 + a^2*c)*a*sqrt(c)/abs(x))/(a^5*c^(3/2)) - 1/2/(sqrt(-b^2* 
c*x^2 + a^2*c)*a^2*c*x^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (95) = 190\).

Time = 0.17 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.47 \[ \int \frac {1}{x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {1}{2} \, b^{2} {\left (\frac {4 \, {\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{6} - 4 \, a^{2} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} c^{2}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} + 4 \, a^{2} c^{2}\right )}^{2} a^{4} \sqrt {-c}} + \frac {4}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} - 2 \, a c\right )} a^{4} \sqrt {-c}} + \frac {6 \, \arctan \left (\frac {{\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2}}{2 \, a c}\right )}{a^{5} \sqrt {-c} c} - \frac {\sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}}{{\left ({\left (b x + a\right )} c - 2 \, a c\right )} a^{5} c}\right )} \] Input:

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

Output:

1/2*b^2*(4*((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^6 - 4*a^ 
2*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*c^2)/(((sqrt(b*x 
 + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4 + 4*a^2*c^2)^2*a^4*sqrt(-c) 
) + 4/(((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2 - 2*a*c)*a 
^4*sqrt(-c)) + 6*arctan(1/2*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 
2*a*c))^2/(a*c))/(a^5*sqrt(-c)*c) - sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + 
a)/(((b*x + a)*c - 2*a*c)*a^5*c))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.03 \[ \int \frac {1}{x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {\sqrt {c}\, \left (-3 \sqrt {b x +a}\, \sqrt {-b x +a}\, \mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right ) b^{2} x^{2}+3 \sqrt {b x +a}\, \sqrt {-b x +a}\, \mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right ) b^{2} x^{2}-3 \sqrt {b x +a}\, \sqrt {-b x +a}\, \mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right ) b^{2} x^{2}+3 \sqrt {b x +a}\, \sqrt {-b x +a}\, \mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right ) b^{2} x^{2}-a^{3}+3 a \,b^{2} x^{2}\right )}{2 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{5} c^{2} x^{2}} \] Input:

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

Output:

(sqrt(c)*( - 3*sqrt(a + b*x)*sqrt(a - b*x)*log( - sqrt(2) + tan(asin(sqrt( 
a - b*x)/(sqrt(a)*sqrt(2)))/2) - 1)*b**2*x**2 + 3*sqrt(a + b*x)*sqrt(a - b 
*x)*log( - sqrt(2) + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2) + 1)*b** 
2*x**2 - 3*sqrt(a + b*x)*sqrt(a - b*x)*log(sqrt(2) + tan(asin(sqrt(a - b*x 
)/(sqrt(a)*sqrt(2)))/2) - 1)*b**2*x**2 + 3*sqrt(a + b*x)*sqrt(a - b*x)*log 
(sqrt(2) + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2) + 1)*b**2*x**2 - a 
**3 + 3*a*b**2*x**2))/(2*sqrt(a + b*x)*sqrt(a - b*x)*a**5*c**2*x**2)