Integrand size = 23, antiderivative size = 123 \[ \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx=\sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} x+\frac {(b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a} \sqrt {c}}-2 \sqrt {b} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b} \sqrt {c+\frac {d}{x}}}\right ) \]
(a*d+b*c)*arctanh(c^(1/2)*(a+b/x)^(1/2)/a^(1/2)/(c+d/x)^(1/2))/a^(1/2)/c^( 1/2)-2*arctanh(d^(1/2)*(a+b/x)^(1/2)/b^(1/2)/(c+d/x)^(1/2))*b^(1/2)*d^(1/2 )+x*(a+b/x)^(1/2)*(c+d/x)^(1/2)
Time = 0.52 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.43 \[ \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} x \left ((b c+a d) \sqrt {b+a x} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d+c x}}{\sqrt {c} \sqrt {b+a x}}\right )+\sqrt {a} \sqrt {c} \left ((b+a x) \sqrt {d+c x}-2 \sqrt {b} \sqrt {d} \sqrt {b+a x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+c x}}{\sqrt {d} \sqrt {b+a x}}\right )\right )\right )}{\sqrt {a} \sqrt {c} (b+a x) \sqrt {d+c x}} \]
(Sqrt[a + b/x]*Sqrt[c + d/x]*x*((b*c + a*d)*Sqrt[b + a*x]*ArcTanh[(Sqrt[a] *Sqrt[d + c*x])/(Sqrt[c]*Sqrt[b + a*x])] + Sqrt[a]*Sqrt[c]*((b + a*x)*Sqrt [d + c*x] - 2*Sqrt[b]*Sqrt[d]*Sqrt[b + a*x]*ArcTanh[(Sqrt[b]*Sqrt[d + c*x] )/(Sqrt[d]*Sqrt[b + a*x])])))/(Sqrt[a]*Sqrt[c]*(b + a*x)*Sqrt[d + c*x])
Time = 0.25 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {899, 108, 27, 175, 66, 104, 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.
| \(\displaystyle \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} x^2d\frac {1}{x}\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle x \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}}-\int \frac {\left (b c+a d+\frac {2 b d}{x}\right ) x}{2 \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle x \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}}-\frac {1}{2} \int \frac {\left (b c+a d+\frac {2 b d}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{2} \left (-2 b d \int \frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}}}d\frac {1}{x}-(a d+b c) \int \frac {x}{\sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}}}d\frac {1}{x}\right )+x \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {1}{2} \left (-4 b d \int \frac {1}{b-\frac {d}{x^2}}d\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}}-(a d+b c) \int \frac {x}{\sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}}}d\frac {1}{x}\right )+x \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{2} \left (-2 (a d+b c) \int \frac {1}{\frac {c}{x^2}-a}d\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}}-4 b d \int \frac {1}{b-\frac {d}{x^2}}d\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}}\right )+x \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 (a d+b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a} \sqrt {c}}-4 \sqrt {b} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b} \sqrt {c+\frac {d}{x}}}\right )\right )+x \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}}\) |
Sqrt[a + b/x]*Sqrt[c + d/x]*x + ((2*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b/x])/(Sqrt[a]*Sqrt[c + d/x])])/(Sqrt[a]*Sqrt[c]) - 4*Sqrt[b]*Sqrt[d]*ArcT anh[(Sqrt[d]*Sqrt[a + b/x])/(Sqrt[b]*Sqrt[c + d/x])])/2
3.3.66.3.1 Defintions of rubi rules used
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] && !GtQ[c - a*(d/b), 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
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[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
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]
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(217\) vs. \(2(95)=190\).
Time = 0.10 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.77
| method | result | size |
| default | \(\frac {\sqrt {\frac {a x +b}{x}}\, x \sqrt {\frac {c x +d}{x}}\, \left (\sqrt {b d}\, \ln \left (\frac {2 a c x +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) a d +\sqrt {b d}\, \ln \left (\frac {2 a c x +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) b c -2 b d \ln \left (\frac {a d x +b c x +2 \sqrt {b d}\, \sqrt {\left (a x +b \right ) \left (c x +d \right )}+2 b d}{x}\right ) \sqrt {a c}+2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}\, \sqrt {b d}\right )}{2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}\, \sqrt {b d}}\) | \(218\) |
1/2*((a*x+b)/x)^(1/2)*x*((c*x+d)/x)^(1/2)*((b*d)^(1/2)*ln(1/2*(2*a*c*x+2*( (a*x+b)*(c*x+d))^(1/2)*(a*c)^(1/2)+a*d+b*c)/(a*c)^(1/2))*a*d+(b*d)^(1/2)*l n(1/2*(2*a*c*x+2*((a*x+b)*(c*x+d))^(1/2)*(a*c)^(1/2)+a*d+b*c)/(a*c)^(1/2)) *b*c-2*b*d*ln((a*d*x+b*c*x+2*(b*d)^(1/2)*((a*x+b)*(c*x+d))^(1/2)+2*b*d)/x) *(a*c)^(1/2)+2*((a*x+b)*(c*x+d))^(1/2)*(a*c)^(1/2)*(b*d)^(1/2))/((a*x+b)*( c*x+d))^(1/2)/(a*c)^(1/2)/(b*d)^(1/2)
Time = 0.60 (sec) , antiderivative size = 890, normalized size of antiderivative = 7.24 \[ \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx=\left [\frac {4 \, a c x \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} + 2 \, \sqrt {b d} a c \log \left (-\frac {8 \, b^{2} d^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, b d x + {\left (b c + a d\right )} x^{2}\right )} \sqrt {b d} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x}{x^{2}}\right ) + \sqrt {a c} {\left (b c + a d\right )} \log \left (-8 \, a^{2} c^{2} x^{2} - b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2} - 4 \, {\left (2 \, a c x^{2} + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} - 8 \, {\left (a b c^{2} + a^{2} c d\right )} x\right )}{4 \, a c}, \frac {4 \, a c x \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} + 4 \, \sqrt {-b d} a c \arctan \left (\frac {{\left (2 \, b d x + {\left (b c + a d\right )} x^{2}\right )} \sqrt {-b d} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{2 \, {\left (a b c d x^{2} + b^{2} d^{2} + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + \sqrt {a c} {\left (b c + a d\right )} \log \left (-8 \, a^{2} c^{2} x^{2} - b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2} - 4 \, {\left (2 \, a c x^{2} + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} - 8 \, {\left (a b c^{2} + a^{2} c d\right )} x\right )}{4 \, a c}, \frac {2 \, a c x \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} + \sqrt {b d} a c \log \left (-\frac {8 \, b^{2} d^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, b d x + {\left (b c + a d\right )} x^{2}\right )} \sqrt {b d} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x}{x^{2}}\right ) - \sqrt {-a c} {\left (b c + a d\right )} \arctan \left (\frac {2 \, \sqrt {-a c} x \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{2 \, a c x + b c + a d}\right )}{2 \, a c}, \frac {2 \, a c x \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} + 2 \, \sqrt {-b d} a c \arctan \left (\frac {{\left (2 \, b d x + {\left (b c + a d\right )} x^{2}\right )} \sqrt {-b d} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{2 \, {\left (a b c d x^{2} + b^{2} d^{2} + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - \sqrt {-a c} {\left (b c + a d\right )} \arctan \left (\frac {2 \, \sqrt {-a c} x \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{2 \, a c x + b c + a d}\right )}{2 \, a c}\right ] \]
[1/4*(4*a*c*x*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) + 2*sqrt(b*d)*a*c*log(-( 8*b^2*d^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*b*d*x + (b*c + a*d) *x^2)*sqrt(b*d)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) + 8*(b^2*c*d + a*b*d^2 )*x)/x^2) + sqrt(a*c)*(b*c + a*d)*log(-8*a^2*c^2*x^2 - b^2*c^2 - 6*a*b*c*d - a^2*d^2 - 4*(2*a*c*x^2 + (b*c + a*d)*x)*sqrt(a*c)*sqrt((a*x + b)/x)*sqr t((c*x + d)/x) - 8*(a*b*c^2 + a^2*c*d)*x))/(a*c), 1/4*(4*a*c*x*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) + 4*sqrt(-b*d)*a*c*arctan(1/2*(2*b*d*x + (b*c + a *d)*x^2)*sqrt(-b*d)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x)/(a*b*c*d*x^2 + b^2 *d^2 + (b^2*c*d + a*b*d^2)*x)) + sqrt(a*c)*(b*c + a*d)*log(-8*a^2*c^2*x^2 - b^2*c^2 - 6*a*b*c*d - a^2*d^2 - 4*(2*a*c*x^2 + (b*c + a*d)*x)*sqrt(a*c)* sqrt((a*x + b)/x)*sqrt((c*x + d)/x) - 8*(a*b*c^2 + a^2*c*d)*x))/(a*c), 1/2 *(2*a*c*x*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) + sqrt(b*d)*a*c*log(-(8*b^2* d^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*b*d*x + (b*c + a*d)*x^2)* sqrt(b*d)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) + 8*(b^2*c*d + a*b*d^2)*x)/x ^2) - sqrt(-a*c)*(b*c + a*d)*arctan(2*sqrt(-a*c)*x*sqrt((a*x + b)/x)*sqrt( (c*x + d)/x)/(2*a*c*x + b*c + a*d)))/(a*c), 1/2*(2*a*c*x*sqrt((a*x + b)/x) *sqrt((c*x + d)/x) + 2*sqrt(-b*d)*a*c*arctan(1/2*(2*b*d*x + (b*c + a*d)*x^ 2)*sqrt(-b*d)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x)/(a*b*c*d*x^2 + b^2*d^2 + (b^2*c*d + a*b*d^2)*x)) - sqrt(-a*c)*(b*c + a*d)*arctan(2*sqrt(-a*c)*x*sq rt((a*x + b)/x)*sqrt((c*x + d)/x)/(2*a*c*x + b*c + a*d)))/(a*c)]
\[ \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx=\int \sqrt {a + \frac {b}{x}} \sqrt {c + \frac {d}{x}}\, dx \]
\[ \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx=\int { \sqrt {a + \frac {b}{x}} \sqrt {c + \frac {d}{x}} \,d x } \]
\[ \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx=\int { \sqrt {a + \frac {b}{x}} \sqrt {c + \frac {d}{x}} \,d x } \]
Time = 26.27 (sec) , antiderivative size = 4674, normalized size of antiderivative = 38.00 \[ \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx=\text {Too large to display} \]
atan(((b*d)^(1/2)*(2*(b*d)^(1/2)*(2*(b*d)^(1/2)*(2*(b*d)^(1/2)*((2*(4*a^(9 /2)*b^9*c^(19/2) - 4*a^(13/2)*b^7*c^(15/2)*d^2 - 4*a^(15/2)*b^6*c^(13/2)*d ^3 + 4*a^(19/2)*b^4*c^(9/2)*d^5))/(a^7*c^7*d^9) - (((a + b/x)^(1/2) - a^(1 /2))*(32*a^4*b^9*c^10 - 120*a^5*b^8*c^9*d + 288*a^6*b^7*c^8*d^2 - 400*a^7* b^6*c^7*d^3 + 288*a^8*b^5*c^6*d^4 - 120*a^9*b^4*c^5*d^5 + 32*a^10*b^3*c^4* d^6))/(2*a^7*c^7*d^9*((c + d/x)^(1/2) - c^(1/2)))) - (2*(8*a^5*b^9*c^9*d + 16*a^6*b^8*c^8*d^2 - 48*a^7*b^7*c^7*d^3 + 16*a^8*b^6*c^6*d^4 + 8*a^9*b^5* c^5*d^5))/(a^7*c^7*d^9) + (((a + b/x)^(1/2) - a^(1/2))*(16*a^(7/2)*b^10*c^ (21/2) - 76*a^(9/2)*b^9*c^(19/2)*d + 228*a^(11/2)*b^8*c^(17/2)*d^2 - 168*a ^(13/2)*b^7*c^(15/2)*d^3 - 168*a^(15/2)*b^6*c^(13/2)*d^4 + 228*a^(17/2)*b^ 5*c^(11/2)*d^5 - 76*a^(19/2)*b^4*c^(9/2)*d^6 + 16*a^(21/2)*b^3*c^(7/2)*d^7 ))/(2*a^7*c^7*d^9*((c + d/x)^(1/2) - c^(1/2)))) - (2*(a^(7/2)*b^11*c^(21/2 ) + 16*a^(9/2)*b^10*c^(19/2)*d - 42*a^(11/2)*b^9*c^(17/2)*d^2 + 25*a^(13/2 )*b^8*c^(15/2)*d^3 + 25*a^(15/2)*b^7*c^(13/2)*d^4 - 42*a^(17/2)*b^6*c^(11/ 2)*d^5 + 16*a^(19/2)*b^5*c^(9/2)*d^6 + a^(21/2)*b^4*c^(7/2)*d^7))/(a^7*c^7 *d^9) + (((a + b/x)^(1/2) - a^(1/2))*(146*a^4*b^10*c^10*d - 556*a^5*b^9*c^ 9*d^2 + 1006*a^6*b^8*c^8*d^3 - 1192*a^7*b^7*c^7*d^4 + 1006*a^8*b^6*c^6*d^5 - 556*a^9*b^5*c^5*d^6 + 146*a^10*b^4*c^4*d^7))/(2*a^7*c^7*d^9*((c + d/x)^ (1/2) - c^(1/2)))) + (2*(2*a^4*b^11*c^10*d + 8*a^5*b^10*c^9*d^2 - 2*a^6*b^ 9*c^8*d^3 - 16*a^7*b^8*c^7*d^4 - 2*a^8*b^7*c^6*d^5 + 8*a^9*b^6*c^5*d^6 ...