Integrand size = 26, antiderivative size = 72 \[ \int \frac {1}{x (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {1}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )}{a^3 c^{3/2}} \] Output:
1/a^2/c/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-arctanh((b*x+a)^(1/2)*(-b*c*x+a*c )^(1/2)/a/c^(1/2))/a^3/c^(3/2)
Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {a-2 \sqrt {a-b x} \sqrt {a+b x} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{a^3 c \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:
Integrate[1/(x*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)),x]
Output:
(a - 2*Sqrt[a - b*x]*Sqrt[a + b*x]*ArcTanh[Sqrt[a + b*x]/Sqrt[a - b*x]])/( a^3*c*Sqrt[c*(a - b*x)]*Sqrt[a + b*x])
Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {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.
\(\displaystyle \int \frac {1}{x (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 115 |
\(\displaystyle \frac {\int \frac {b c (a-b x)}{x \sqrt {a+b x} (a c-b c x)^{3/2}}dx}{a^2 b c}+\frac {1}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {a-b x}{x \sqrt {a+b x} (a c-b c x)^{3/2}}dx}{a^2}+\frac {1}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 35 |
\(\displaystyle \frac {\int \frac {1}{x \sqrt {a+b x} \sqrt {a c-b c x}}dx}{a^2 c}+\frac {1}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle \frac {b \int \frac {1}{b (a+b x) (a c-b c x)-a^2 b c}d\left (\sqrt {a+b x} \sqrt {a c-b c x}\right )}{a^2 c}+\frac {1}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )}{a^3 c^{3/2}}\) |
Input:
Int[1/(x*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)),x]
Output:
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))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(166\) vs. \(2(60)=120\).
Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.32
method | result | size |
default | \(\frac {\left (\ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}{x}\right ) b^{2} c \,x^{2}-a^{2} c \ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}{x}\right )+\sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\right ) \sqrt {c \left (-b x +a \right )}}{c^{2} a^{2} \left (-b x +a \right ) \sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b x +a}}\) | \(167\) |
Input:
int(1/x/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x,method=_RETURNVERBOSE)
Output:
(ln(2*(a^2*c+(a^2*c)^(1/2)*(c*(-b^2*x^2+a^2))^(1/2))/x)*b^2*c*x^2-a^2*c*ln (2*(a^2*c+(a^2*c)^(1/2)*(c*(-b^2*x^2+a^2))^(1/2))/x)+(a^2*c)^(1/2)*(c*(-b^ 2*x^2+a^2))^(1/2))*(c*(-b*x+a))^(1/2)/c^2/a^2/(-b*x+a)/(a^2*c)^(1/2)/(c*(- b^2*x^2+a^2))^(1/2)/(b*x+a)^(1/2)
Time = 0.11 (sec) , antiderivative size = 220, normalized size of antiderivative = 3.06 \[ \int \frac {1}{x (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\left [\frac {{\left (b^{2} x^{2} - a^{2}\right )} \sqrt {c} \log \left (-\frac {b^{2} c x^{2} - 2 \, a^{2} c + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {c}}{x^{2}}\right ) - 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} a}{2 \, {\left (a^{3} b^{2} c^{2} x^{2} - a^{5} c^{2}\right )}}, -\frac {{\left (b^{2} x^{2} - a^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {-c}}{b^{2} c x^{2} - a^{2} c}\right ) + \sqrt {-b c x + a c} \sqrt {b x + a} a}{a^{3} b^{2} c^{2} x^{2} - a^{5} c^{2}}\right ] \] Input:
integrate(1/x/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="fricas")
Output:
[1/2*((b^2*x^2 - a^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*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*a) /(a^3*b^2*c^2*x^2 - a^5*c^2), -((b^2*x^2 - a^2)*sqrt(-c)*arctan(sqrt(-b*c* x + a*c)*sqrt(b*x + a)*a*sqrt(-c)/(b^2*c*x^2 - a^2*c)) + sqrt(-b*c*x + a*c )*sqrt(b*x + a)*a)/(a^3*b^2*c^2*x^2 - a^5*c^2)]
Result contains complex when optimal does not.
Time = 77.00 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=- \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & 1, 2, \frac {5}{2} \\\frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2, \frac {5}{2} & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} a^{3} c^{\frac {3}{2}}} - \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} 0, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, 1 & \\\frac {3}{4}, \frac {5}{4} & 0, \frac {1}{2}, \frac {3}{2}, 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} a^{3} c^{\frac {3}{2}}} \] Input:
integrate(1/x/(b*x+a)**(3/2)/(-b*c*x+a*c)**(3/2),x)
Output:
-I*meijerg(((5/4, 7/4, 1), (1, 2, 5/2)), ((5/4, 3/2, 7/4, 2, 5/2), (0,)), a**2/(b**2*x**2))/(2*pi**(3/2)*a**3*c**(3/2)) - meijerg(((0, 1/2, 3/4, 1, 5/4, 1), ()), ((3/4, 5/4), (0, 1/2, 3/2, 0)), a**2*exp_polar(-2*I*pi)/(b** 2*x**2))/(2*pi**(3/2)*a**3*c**(3/2))
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {1}{\sqrt {-b^{2} c x^{2} + a^{2} c} a^{2} c} - \frac {\log \left (\frac {2 \, a^{2} c}{{\left | x \right |}} + \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a \sqrt {c}}{{\left | x \right |}}\right )}{a^{3} c^{\frac {3}{2}}} \] Input:
integrate(1/x/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="maxima")
Output:
1/(sqrt(-b^2*c*x^2 + a^2*c)*a^2*c) - log(2*a^2*c/abs(x) + 2*sqrt(-b^2*c*x^ 2 + a^2*c)*a*sqrt(c)/abs(x))/(a^3*c^(3/2))
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (60) = 120\).
Time = 0.15 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.08 \[ \int \frac {1}{x (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {2}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} - 2 \, a c\right )} a^{2} \sqrt {-c}} + \frac {2 \, \arctan \left (\frac {{\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2}}{2 \, a c}\right )}{a^{3} \sqrt {-c} c} - \frac {\sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}}{2 \, {\left ({\left (b x + a\right )} c - 2 \, a c\right )} a^{3} c} \] Input:
integrate(1/x/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="giac")
Output:
2/(((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2 - 2*a*c)*a^2*s qrt(-c)) + 2*arctan(1/2*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a* c))^2/(a*c))/(a^3*sqrt(-c)*c) - 1/2*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a)/(((b*x + a)*c - 2*a*c)*a^3*c)
Timed out. \[ \int \frac {1}{x (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\int \frac {1}{x\,{\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2}} \,d x \] Input:
int(1/(x*(a*c - b*c*x)^(3/2)*(a + b*x)^(3/2)),x)
Output:
int(1/(x*(a*c - b*c*x)^(3/2)*(a + b*x)^(3/2)), x)
Time = 0.18 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.64 \[ \int \frac {1}{x (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {\sqrt {c}\, \left (-\sqrt {b x +a}\, \sqrt {-b x +a}\, \mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right )+\sqrt {b x +a}\, \sqrt {-b x +a}\, \mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right )-\sqrt {b x +a}\, \sqrt {-b x +a}\, \mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right )+\sqrt {b x +a}\, \sqrt {-b x +a}\, \mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right )+a \right )}{\sqrt {b x +a}\, \sqrt {-b x +a}\, a^{3} c^{2}} \] Input:
int(1/x/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x)
Output:
(sqrt(c)*( - sqrt(a + b*x)*sqrt(a - b*x)*log( - sqrt(2) + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2) - 1) + sqrt(a + b*x)*sqrt(a - b*x)*log( - sqr t(2) + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2) + 1) - sqrt(a + b*x)*s qrt(a - b*x)*log(sqrt(2) + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2) - 1) + sqrt(a + b*x)*sqrt(a - b*x)*log(sqrt(2) + tan(asin(sqrt(a - b*x)/(sqr t(a)*sqrt(2)))/2) + 1) + a))/(sqrt(a + b*x)*sqrt(a - b*x)*a**3*c**2)