Integrand size = 26, antiderivative size = 64 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x} \, dx=\sqrt {a+b x} \sqrt {a c-b c x}-a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right ) \] Output:
(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)-a*c^(1/2)*arctanh((b*x+a)^(1/2)*(-b*c*x+a *c)^(1/2)/a/c^(1/2))
Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x} \, dx=\frac {\sqrt {c (a-b x)} \left (\sqrt {a-b x} \sqrt {a+b x}-2 a \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )\right )}{\sqrt {a-b x}} \] Input:
Integrate[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/x,x]
Output:
(Sqrt[c*(a - b*x)]*(Sqrt[a - b*x]*Sqrt[a + b*x] - 2*a*ArcTanh[Sqrt[a + b*x ]/Sqrt[a - b*x]]))/Sqrt[a - b*x]
Time = 0.19 (sec) , antiderivative size = 64, 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 = {112, 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.
\(\displaystyle \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x} \, dx\) |
\(\Big \downarrow \) 112 |
\(\displaystyle \sqrt {a+b x} \sqrt {a c-b c x}-\int -\frac {a^2 c}{x \sqrt {a+b x} \sqrt {a c-b c x}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {a^2 c}{x \sqrt {a+b x} \sqrt {a c-b c x}}dx+\sqrt {a+b x} \sqrt {a c-b c x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle a^2 c \int \frac {1}{x \sqrt {a+b x} \sqrt {a c-b c x}}dx+\sqrt {a+b x} \sqrt {a c-b c x}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle a^2 b c \int \frac {1}{b (a+b x) (a c-b c x)-a^2 b c}d\left (\sqrt {a+b x} \sqrt {a c-b c x}\right )+\sqrt {a+b x} \sqrt {a c-b c x}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \sqrt {a+b x} \sqrt {a c-b c x}-a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )\) |
Input:
Int[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/x,x]
Output:
Sqrt[a + b*x]*Sqrt[a*c - b*c*x] - a*Sqrt[c]*ArcTanh[(Sqrt[a + b*x]*Sqrt[a* c - b*c*x])/(a*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[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 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)*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. \(108\) vs. \(2(52)=104\).
Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.70
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (-a^{2} c \ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}{x}\right )+\sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\right )}{\sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\) | \(109\) |
Input:
int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x,x,method=_RETURNVERBOSE)
Output:
(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)*(-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))/(a^2*c)^(1/2 )/(c*(-b^2*x^2+a^2))^(1/2)
Time = 0.09 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.23 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x} \, dx=\left [\frac {1}{2} \, a \sqrt {c} \log \left (-\frac {b^{2} c x^{2} - 2 \, a^{2} c + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {c}}{x^{2}}\right ) + \sqrt {-b c x + a c} \sqrt {b x + a}, -a \sqrt {-c} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {-c}}{b^{2} c x^{2} - a^{2} c}\right ) + \sqrt {-b c x + a c} \sqrt {b x + a}\right ] \] Input:
integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x,x, algorithm="fricas")
Output:
[1/2*a*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) + sqrt(-b*c*x + a*c)*sqrt(b*x + a), -a*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)]
\[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x} \, dx=\int \frac {\sqrt {- c \left (- a + b x\right )} \sqrt {a + b x}}{x}\, dx \] Input:
integrate((b*x+a)**(1/2)*(-b*c*x+a*c)**(1/2)/x,x)
Output:
Integral(sqrt(-c*(-a + b*x))*sqrt(a + b*x)/x, x)
Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x} \, dx=-a \sqrt {c} \log \left (\frac {2 \, a^{2} c}{{\left | x \right |}} + \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a \sqrt {c}}{{\left | x \right |}}\right ) + \sqrt {-b^{2} c x^{2} + a^{2} c} \] Input:
integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x,x, algorithm="maxima")
Output:
-a*sqrt(c)*log(2*a^2*c/abs(x) + 2*sqrt(-b^2*c*x^2 + a^2*c)*a*sqrt(c)/abs(x )) + sqrt(-b^2*c*x^2 + a^2*c)
Time = 0.15 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x} \, dx=-\frac {2 \, a b \sqrt {-c} \arctan \left (\frac {{\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2}}{2 \, a c}\right ) - \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a} b}{b} \] Input:
integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x,x, algorithm="giac")
Output:
-(2*a*b*sqrt(-c)*arctan(1/2*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2/(a*c)) - sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a)*b)/b
Time = 1.77 (sec) , antiderivative size = 225, normalized size of antiderivative = 3.52 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x} \, dx=\sqrt {a}\,\ln \left (\frac {{\left (\sqrt {c\,\left (a-b\,x\right )}-\sqrt {a\,c}\right )}^2}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}-c\right )\,\sqrt {a\,c}-\sqrt {a}\,\ln \left (\frac {\sqrt {c\,\left (a-b\,x\right )}-\sqrt {a\,c}}{\sqrt {a+b\,x}-\sqrt {a}}\right )\,\sqrt {a\,c}+\frac {8\,\sqrt {a}\,c\,\sqrt {a\,c}\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^2}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2\,\left (\frac {{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^4}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}+c^2+\frac {2\,c\,{\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}^2}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}\right )} \] Input:
int(((a*c - b*c*x)^(1/2)*(a + b*x)^(1/2))/x,x)
Output:
a^(1/2)*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) - a^(1/2)*log(((c*(a - b*x))^(1/2) - (a*c)^(1/2))/( (a + b*x)^(1/2) - a^(1/2)))*(a*c)^(1/2) + (8*a^(1/2)*c*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(((a + b*x)^(1/2) - a^(1/2))^2*(((a*c - b* c*x)^(1/2) - (a*c)^(1/2))^4/((a + b*x)^(1/2) - a^(1/2))^4 + c^2 + (2*c*((a *c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/2))^2))
Time = 0.16 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.06 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x} \, dx=\sqrt {c}\, \left (\sqrt {b x +a}\, \sqrt {-b x +a}-\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right ) a +\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right ) a -\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right ) a +\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right ) a \right ) \] Input:
int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x,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)*a + log( - sqrt(2) + tan(asin(sqrt(a - b*x )/(sqrt(a)*sqrt(2)))/2) + 1)*a - log(sqrt(2) + tan(asin(sqrt(a - b*x)/(sqr t(a)*sqrt(2)))/2) - 1)*a + log(sqrt(2) + tan(asin(sqrt(a - b*x)/(sqrt(a)*s qrt(2)))/2) + 1)*a)