Integrand size = 26, antiderivative size = 77 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^3} \, dx=-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{2 x^2}+\frac {b^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )}{2 a} \] Output:
-1/2*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^2+1/2*b^2*c^(1/2)*arctanh((b*x+a)^ (1/2)*(-b*c*x+a*c)^(1/2)/a/c^(1/2))/a
Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^3} \, dx=\frac {1}{2} \sqrt {c (a-b x)} \left (-\frac {\sqrt {a+b x}}{x^2}+\frac {2 b^2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{a \sqrt {a-b x}}\right ) \] Input:
Integrate[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/x^3,x]
Output:
(Sqrt[c*(a - b*x)]*(-(Sqrt[a + b*x]/x^2) + (2*b^2*ArcTanh[Sqrt[a + b*x]/Sq rt[a - b*x]])/(a*Sqrt[a - b*x])))/2
Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.48, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {105, 105, 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.
\(\displaystyle \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^3} \, dx\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {1}{2} b \int \frac {\sqrt {a c-b c x}}{x^2 \sqrt {a+b x}}dx-\frac {\sqrt {a+b x} (a c-b c x)^{3/2}}{2 a c x^2}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {1}{2} b \left (-b c \int \frac {1}{x \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a x}\right )-\frac {\sqrt {a+b x} (a c-b c x)^{3/2}}{2 a c x^2}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle \frac {1}{2} b \left (b^2 (-c) \int \frac {1}{b (a+b x) (a c-b c x)-a^2 b c}d\left (\sqrt {a+b x} \sqrt {a c-b c x}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a x}\right )-\frac {\sqrt {a+b x} (a c-b c x)^{3/2}}{2 a c x^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} b \left (\frac {b \sqrt {c} \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )}{a}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a x}\right )-\frac {\sqrt {a+b x} (a c-b c x)^{3/2}}{2 a c x^2}\) |
Input:
Int[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/x^3,x]
Output:
-1/2*(Sqrt[a + b*x]*(a*c - b*c*x)^(3/2))/(a*c*x^2) + (b*(-((Sqrt[a + b*x]* Sqrt[a*c - b*c*x])/(a*x)) + (b*Sqrt[c]*ArcTanh[(Sqrt[a + b*x]*Sqrt[a*c - b *c*x])/(a*Sqrt[c])])/a))/2
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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
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 = 0.22 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.51
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (\ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}{x}\right ) b^{2} c \,x^{2}-\sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\right )}{2 \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, x^{2} \sqrt {a^{2} c}}\) | \(116\) |
risch | \(-\frac {\left (-b x +a \right ) \sqrt {b x +a}\, c}{2 x^{2} \sqrt {-c \left (b x -a \right )}}+\frac {b^{2} \ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {-b^{2} c \,x^{2}+a^{2} c}}{x}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}\, c}{2 \sqrt {a^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\) | \(120\) |
Input:
int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^3,x,method=_RETURNVERBOSE)
Output:
1/2*(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)*(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 = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.22 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^3} \, dx=\left [\frac {b^{2} \sqrt {c} x^{2} \log \left (-\frac {b^{2} c x^{2} - 2 \, a^{2} c - 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {c}}{x^{2}}\right ) - 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} a}{4 \, a x^{2}}, \frac {b^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {-c}}{b^{2} c x^{2} - a^{2} c}\right ) - \sqrt {-b c x + a c} \sqrt {b x + a} a}{2 \, a x^{2}}\right ] \] Input:
integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^3,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*x^2) , 1/2*(b^2*sqrt(-c)*x^2*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*x^2)]
\[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^3} \, dx=\int \frac {\sqrt {- c \left (- a + b x\right )} \sqrt {a + b x}}{x^{3}}\, dx \] Input:
integrate((b*x+a)**(1/2)*(-b*c*x+a*c)**(1/2)/x**3,x)
Output:
Integral(sqrt(-c*(-a + b*x))*sqrt(a + b*x)/x**3, x)
Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.35 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^3} \, dx=\frac {b^{2} \sqrt {c} \log \left (\frac {2 \, a^{2} c}{{\left | x \right |}} + \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a \sqrt {c}}{{\left | x \right |}}\right )}{2 \, a} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} b^{2}}{2 \, a^{2}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}}}{2 \, a^{2} c x^{2}} \] Input:
integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^3,x, algorithm="maxima")
Output:
1/2*b^2*sqrt(c)*log(2*a^2*c/abs(x) + 2*sqrt(-b^2*c*x^2 + a^2*c)*a*sqrt(c)/ abs(x))/a - 1/2*sqrt(-b^2*c*x^2 + a^2*c)*b^2/a^2 - 1/2*(-b^2*c*x^2 + a^2*c )^(3/2)/(a^2*c*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (61) = 122\).
Time = 0.17 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.57 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^3} \, dx=\frac {\frac {b^{3} \sqrt {-c} \arctan \left (\frac {{\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2}}{2 \, a c}\right )}{a} - \frac {2 \, {\left (b^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{6} \sqrt {-c} c - 4 \, a^{2} b^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} \sqrt {-c} c^{3}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} + 4 \, a^{2} c^{2}\right )}^{2}}}{b} \] Input:
integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^3,x, algorithm="giac")
Output:
(b^3*sqrt(-c)*arctan(1/2*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a *c))^2/(a*c))/a - 2*(b^3*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a *c))^6*sqrt(-c)*c - 4*a^2*b^3*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*sqrt(-c)*c^3)/((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2 *a*c))^4 + 4*a^2*c^2)^2)/b
Time = 2.75 (sec) , antiderivative size = 395, normalized size of antiderivative = 5.13 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^3} \, dx=\frac {b^2\,\ln \left (\frac {\sqrt {c\,\left (a-b\,x\right )}-\sqrt {a\,c}}{\sqrt {a+b\,x}-\sqrt {a}}\right )\,\sqrt {a\,c}}{2\,a^{3/2}}-\frac {b^2\,\ln \left (\frac {{\left (\sqrt {c\,\left (a-b\,x\right )}-\sqrt {a\,c}\right )}^2}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}-c\right )\,\sqrt {a\,c}}{2\,a^{3/2}}-\frac {\frac {15\,b^2\,c\,\sqrt {a\,c}\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^4}{32\,a^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}-\frac {b^2\,c^3\,\sqrt {a\,c}}{32\,a^{3/2}}+\frac {b^2\,c^2\,\sqrt {a\,c}\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^2}{16\,a^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}}{\frac {{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^6}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}-\frac {2\,c\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^4}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}+\frac {c^2\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^2}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}}+\frac {b^2\,\sqrt {a\,c}\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^2}{32\,a^{3/2}\,c\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2} \] Input:
int(((a*c - b*c*x)^(1/2)*(a + b*x)^(1/2))/x^3,x)
Output:
(b^2*log(((c*(a - b*x))^(1/2) - (a*c)^(1/2))/((a + b*x)^(1/2) - a^(1/2)))* (a*c)^(1/2))/(2*a^(3/2)) - (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)^(1/2))/(2*a^(3/2)) - ((15*b^2*c*( a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/(32*a^(3/2)*((a + b*x)^( 1/2) - a^(1/2))^4) - (b^2*c^3*(a*c)^(1/2))/(32*a^(3/2)) + (b^2*c^2*(a*c)^( 1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(16*a^(3/2)*((a + b*x)^(1/2) - a^(1/2))^2))/(((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*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(3 2*a^(3/2)*c*((a + b*x)^(1/2) - a^(1/2))^2)
Time = 0.16 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.09 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^3} \, dx=\frac {\sqrt {c}\, \left (-\sqrt {b x +a}\, \sqrt {-b x +a}\, a +\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right ) b^{2} x^{2}-\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right ) b^{2} x^{2}+\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right ) b^{2} x^{2}-\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right ) b^{2} x^{2}\right )}{2 a \,x^{2}} \] Input:
int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^3,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*x**2)