Integrand size = 26, antiderivative size = 83 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{2 a^2 c x^2}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )}{2 a^3 \sqrt {c}} \] Output:
-1/2*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/a^2/c/x^2-1/2*b^2*arctanh((b*x+a)^(1 /2)*(-b*c*x+a*c)^(1/2)/a/c^(1/2))/a^3/c^(1/2)
Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {a (-a+b x) \sqrt {a+b x}-2 b^2 x^2 \sqrt {a-b x} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{2 a^3 x^2 \sqrt {c (a-b x)}} \] Input:
Integrate[1/(x^3*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]
Output:
(a*(-a + b*x)*Sqrt[a + b*x] - 2*b^2*x^2*Sqrt[a - b*x]*ArcTanh[Sqrt[a + b*x ]/Sqrt[a - b*x]])/(2*a^3*x^2*Sqrt[c*(a - b*x)])
Time = 0.19 (sec) , antiderivative size = 83, 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 = {114, 25, 27, 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.
\(\displaystyle \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {\int -\frac {b^2 c}{x \sqrt {a+b x} \sqrt {a c-b c x}}dx}{2 a^2 c}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{2 a^2 c x^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {b^2 c}{x \sqrt {a+b x} \sqrt {a c-b c x}}dx}{2 a^2 c}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{2 a^2 c x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b^2 \int \frac {1}{x \sqrt {a+b x} \sqrt {a c-b c x}}dx}{2 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{2 a^2 c x^2}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle \frac {b^3 \int \frac {1}{b (a+b x) (a c-b c x)-a^2 b c}d\left (\sqrt {a+b x} \sqrt {a c-b c x}\right )}{2 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{2 a^2 c x^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )}{2 a^3 \sqrt {c}}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{2 a^2 c x^2}\) |
Input:
Int[1/(x^3*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]
Output:
-1/2*(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(a^2*c*x^2) - (b^2*ArcTanh[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(a*Sqrt[c])])/(2*a^3*Sqrt[c])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
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]
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.46
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (\ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}{x}\right ) b^{2} c \,x^{2}+\sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\right )}{2 c \,a^{2} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, x^{2} \sqrt {a^{2} c}}\) | \(121\) |
risch | \(-\frac {\left (-b x +a \right ) \sqrt {b x +a}}{2 a^{2} x^{2} \sqrt {-c \left (b x -a \right )}}-\frac {b^{2} \ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {-b^{2} c \,x^{2}+a^{2} c}}{x}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}}{2 a^{2} \sqrt {a^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\) | \(124\) |
Input:
int(1/x^3/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2*(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)/c/a^2*(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)^(1/2)*(c*(-b^2*x^2+a^2))^(1/2) )/(c*(-b^2*x^2+a^2))^(1/2)/x^2/(a^2*c)^(1/2)
Time = 0.10 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.12 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\left [\frac {b^{2} \sqrt {c} x^{2} \log \left (-\frac {b^{2} c x^{2} - 2 \, a^{2} c + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {c}}{x^{2}}\right ) - 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} a}{4 \, a^{3} c x^{2}}, -\frac {b^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {-c}}{b^{2} c x^{2} - a^{2} c}\right ) + \sqrt {-b c x + a c} \sqrt {b x + a} a}{2 \, a^{3} c x^{2}}\right ] \] Input:
integrate(1/x^3/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="fricas")
Output:
[1/4*(b^2*sqrt(c)*x^2*log(-(b^2*c*x^2 - 2*a^2*c + 2*sqrt(-b*c*x + a*c)*sqr t(b*x + a)*a*sqrt(c))/x^2) - 2*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*a)/(a^3*c* x^2), -1/2*(b^2*sqrt(-c)*x^2*arctan(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*a*sqr t(-c)/(b^2*c*x^2 - a^2*c)) + sqrt(-b*c*x + a*c)*sqrt(b*x + a)*a)/(a^3*c*x^ 2)]
Timed out. \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\text {Timed out} \] Input:
integrate(1/x**3/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {b^{2} \log \left (\frac {2 \, a^{2} c}{{\left | x \right |}} + \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a \sqrt {c}}{{\left | x \right |}}\right )}{2 \, a^{3} \sqrt {c}} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c}}{2 \, a^{2} c x^{2}} \] Input:
integrate(1/x^3/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="maxima")
Output:
-1/2*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^3*sqrt(c)) - 1/2*sqrt(-b^2*c*x^2 + a^2*c)/(a^2*c*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (67) = 134\).
Time = 0.14 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.29 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-b^{2} \sqrt {-c} c^{2} {\left (\frac {2 \, {\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{6} - 4 \, a^{2} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} c^{2}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} + 4 \, a^{2} c^{2}\right )}^{2} a^{2} c^{2}} + \frac {\arctan \left (\frac {{\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2}}{2 \, a c}\right )}{a^{3} c^{3}}\right )} \] Input:
integrate(1/x^3/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="giac")
Output:
-b^2*sqrt(-c)*c^2*(2*((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^ 2*c^2) + arctan(1/2*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^ 2/(a*c))/(a^3*c^3))
Time = 3.59 (sec) , antiderivative size = 395, normalized size of antiderivative = 4.76 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {b^2\,\ln \left (\frac {{\left (\sqrt {c\,\left (a-b\,x\right )}-\sqrt {a\,c}\right )}^2}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}-c\right )\,{\left (a\,c\right )}^{3/2}}{2\,a^{9/2}\,c^2}-\frac {\frac {b^2\,{\left (a\,c\right )}^{3/2}\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^2}{16\,a^{9/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}-\frac {b^2\,c\,{\left (a\,c\right )}^{3/2}}{32\,a^{9/2}}+\frac {15\,b^2\,{\left (a\,c\right )}^{3/2}\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^4}{32\,a^{9/2}\,c\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}}{\frac {{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^6}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}-\frac {2\,c\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^4}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}+\frac {c^2\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^2}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}}-\frac {b^2\,\ln \left (\frac {\sqrt {c\,\left (a-b\,x\right )}-\sqrt {a\,c}}{\sqrt {a+b\,x}-\sqrt {a}}\right )\,{\left (a\,c\right )}^{3/2}}{2\,a^{9/2}\,c^2}+\frac {b^2\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^2}{32\,a^{3/2}\,{\left (a\,c\right )}^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2} \] Input:
int(1/(x^3*(a*c - b*c*x)^(1/2)*(a + b*x)^(1/2)),x)
Output:
(b^2*log(((c*(a - b*x))^(1/2) - (a*c)^(1/2))^2/((a + b*x)^(1/2) - a^(1/2)) ^2 - c)*(a*c)^(3/2))/(2*a^(9/2)*c^2) - ((b^2*(a*c)^(3/2)*((a*c - b*c*x)^(1 /2) - (a*c)^(1/2))^2)/(16*a^(9/2)*((a + b*x)^(1/2) - a^(1/2))^2) - (b^2*c* (a*c)^(3/2))/(32*a^(9/2)) + (15*b^2*(a*c)^(3/2)*((a*c - b*c*x)^(1/2) - (a* c)^(1/2))^4)/(32*a^(9/2)*c*((a + b*x)^(1/2) - a^(1/2))^4))/(((a*c - b*c*x) ^(1/2) - (a*c)^(1/2))^6/((a + b*x)^(1/2) - a^(1/2))^6 - (2*c*((a*c - b*c*x )^(1/2) - (a*c)^(1/2))^4)/((a + b*x)^(1/2) - a^(1/2))^4 + (c^2*((a*c - b*c *x)^(1/2) - (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/2))^2) - (b^2*log(((c* (a - b*x))^(1/2) - (a*c)^(1/2))/((a + b*x)^(1/2) - a^(1/2)))*(a*c)^(3/2))/ (2*a^(9/2)*c^2) + (b^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(32*a^(3/2)* (a*c)^(3/2)*((a + b*x)^(1/2) - a^(1/2))^2)
Time = 0.16 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.98 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {\sqrt {c}\, \left (-\sqrt {b x +a}\, \sqrt {-b x +a}\, a -\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right ) b^{2} x^{2}+\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right ) b^{2} x^{2}-\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right ) b^{2} x^{2}+\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right ) b^{2} x^{2}\right )}{2 a^{3} c \,x^{2}} \] Input:
int(1/x^3/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x)
Output:
(sqrt(c)*( - sqrt(a + b*x)*sqrt(a - b*x)*a - log( - sqrt(2) + tan(asin(sqr t(a - b*x)/(sqrt(a)*sqrt(2)))/2) - 1)*b**2*x**2 + log( - sqrt(2) + tan(asi n(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2) + 1)*b**2*x**2 - log(sqrt(2) + tan(a sin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2) - 1)*b**2*x**2 + log(sqrt(2) + tan (asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2) + 1)*b**2*x**2))/(2*a**3*c*x**2)