Integrand size = 26, antiderivative size = 65 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^2} \, dx=-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x}-2 b \sqrt {c} \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a c-b c x}}\right ) \] Output:
-(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x-2*b*c^(1/2)*arctan(c^(1/2)*(b*x+a)^(1/ 2)/(-b*c*x+a*c)^(1/2))
Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^2} \, dx=-\frac {\sqrt {c (a-b x)} \left (\sqrt {a-b x} \sqrt {a+b x}+2 b x \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )\right )}{x \sqrt {a-b x}} \] Input:
Integrate[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/x^2,x]
Output:
-((Sqrt[c*(a - b*x)]*(Sqrt[a - b*x]*Sqrt[a + b*x] + 2*b*x*ArcTan[Sqrt[a + b*x]/Sqrt[a - b*x]]))/(x*Sqrt[a - b*x]))
Time = 0.17 (sec) , antiderivative size = 65, 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 = {108, 25, 27, 45, 218}
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.
\(\displaystyle \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^2} \, dx\) |
\(\Big \downarrow \) 108 |
\(\displaystyle \int -\frac {b^2 c}{\sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {b^2 c}{\sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle b^2 (-c) \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x}\) |
\(\Big \downarrow \) 45 |
\(\displaystyle -2 b^2 c \int \frac {1}{\frac {c (a+b x) b}{a c-b c x}+b}d\frac {\sqrt {a+b x}}{\sqrt {a c-b c x}}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle -2 b \sqrt {c} \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a c-b c x}}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x}\) |
Input:
Int[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/x^2,x]
Output:
-((Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/x) - 2*b*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[ a + b*x])/Sqrt[a*c - b*c*x]]
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_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && !GtQ[c, 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/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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]
Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.60
method | result | size |
default | \(\frac {\left (-\arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) b^{2} c x -\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\right ) \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, x \sqrt {b^{2} c}}\) | \(104\) |
risch | \(-\frac {\left (-b x +a \right ) \sqrt {b x +a}\, c}{x \sqrt {-c \left (b x -a \right )}}-\frac {b^{2} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}\, c}{\sqrt {b^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\) | \(109\) |
Input:
int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^2,x,method=_RETURNVERBOSE)
Output:
(-arctan((b^2*c)^(1/2)*x/(c*(-b^2*x^2+a^2))^(1/2))*b^2*c*x-(c*(-b^2*x^2+a^ 2))^(1/2)*(b^2*c)^(1/2))*(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)/(c*(-b^2*x^2+a^2 ))^(1/2)/x/(b^2*c)^(1/2)
Time = 0.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 2.34 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^2} \, dx=\left [\frac {b \sqrt {-c} x \log \left (2 \, b^{2} c x^{2} - 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {-c} x - a^{2} c\right ) - 2 \, \sqrt {-b c x + a c} \sqrt {b x + a}}{2 \, x}, \frac {b \sqrt {c} x \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {c} x}{b^{2} c x^{2} - a^{2} c}\right ) - \sqrt {-b c x + a c} \sqrt {b x + a}}{x}\right ] \] Input:
integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^2,x, algorithm="fricas")
Output:
[1/2*(b*sqrt(-c)*x*log(2*b^2*c*x^2 - 2*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b* sqrt(-c)*x - a^2*c) - 2*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/x, (b*sqrt(c)*x* arctan(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*sqrt(c)*x/(b^2*c*x^2 - a^2*c)) - sqrt(-b*c*x + a*c)*sqrt(b*x + a))/x]
\[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^2} \, dx=\int \frac {\sqrt {- c \left (- a + b x\right )} \sqrt {a + b x}}{x^{2}}\, dx \] Input:
integrate((b*x+a)**(1/2)*(-b*c*x+a*c)**(1/2)/x**2,x)
Output:
Integral(sqrt(-c*(-a + b*x))*sqrt(a + b*x)/x**2, x)
Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^2} \, dx=-b \sqrt {c} \arcsin \left (\frac {b x}{a}\right ) - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c}}{x} \] Input:
integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^2,x, algorithm="maxima")
Output:
-b*sqrt(c)*arcsin(b*x/a) - sqrt(-b^2*c*x^2 + a^2*c)/x
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (53) = 106\).
Time = 0.17 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.68 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^2} \, dx=-\frac {\frac {16 \, a^{2} b^{2} \sqrt {-c} c^{2}}{{\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} + 4 \, a^{2} c^{2}} + b^{2} \sqrt {-c} \log \left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4}\right )}{2 \, b} \] Input:
integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^2,x, algorithm="giac")
Output:
-1/2*(16*a^2*b^2*sqrt(-c)*c^2/((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4 + 4*a^2*c^2) + b^2*sqrt(-c)*log((sqrt(b*x + a)*sqrt(-c) - sqr t(-(b*x + a)*c + 2*a*c))^4))/b
Time = 1.34 (sec) , antiderivative size = 211, normalized size of antiderivative = 3.25 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^2} \, dx=4\,b\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}}{\sqrt {c}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}\right )-\frac {b\,\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}{4\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}-\frac {b\,c^2-\frac {5\,b\,c\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^2}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}}{\frac {4\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^3}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}-\frac {4\,c\,\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}{\sqrt {a+b\,x}-\sqrt {a}}} \] Input:
int(((a*c - b*c*x)^(1/2)*(a + b*x)^(1/2))/x^2,x)
Output:
4*b*c^(1/2)*atan(((a*c - b*c*x)^(1/2) - (a*c)^(1/2))/(c^(1/2)*((a + b*x)^( 1/2) - a^(1/2)))) - (b*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(4*((a + b*x)^ (1/2) - a^(1/2))) - (b*c^2 - (5*b*c*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2) /((a + b*x)^(1/2) - a^(1/2))^2)/((4*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3) /((a + b*x)^(1/2) - a^(1/2))^3 - (4*c*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))) /((a + b*x)^(1/2) - a^(1/2)))
Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^2} \, dx=\frac {\sqrt {c}\, \left (2 \mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right ) b x -\sqrt {b x +a}\, \sqrt {-b x +a}\right )}{x} \] Input:
int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^2,x)
Output:
(sqrt(c)*(2*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*b*x - sqrt(a + b*x)*sqrt (a - b*x)))/x