Integrand size = 22, antiderivative size = 85 \[ \int \frac {\sqrt {c+d x}}{x \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}} \]
-2*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))*c^(1/2)/a^(1/2)+2* arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))*d^(1/2)/b^(1/2)
Result contains complex when optimal does not.
Time = 1.50 (sec) , antiderivative size = 392, normalized size of antiderivative = 4.61 \[ \int \frac {\sqrt {c+d x}}{x \sqrt {a+b x}} \, dx=-\frac {2 \left (\left (\sqrt {a} \sqrt {d}-i \sqrt {b c-a d}\right ) \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )+\left (\sqrt {a} \sqrt {d}+i \sqrt {b c-a d}\right ) \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )+2 \sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )\right )}{\sqrt {a} b \sqrt {c}} \]
(-2*((Sqrt[a]*Sqrt[d] - I*Sqrt[b*c - a*d])*Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a ]*Sqrt[d]*Sqrt[b*c - a*d]]*ArcTan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d ]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sq rt[a + b*x]))] + (Sqrt[a]*Sqrt[d] + I*Sqrt[b*c - a*d])*Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*ArcTan[(Sqrt[b*c - 2*a*d + (2*I)*Sq rt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - ( b*c)/d] - Sqrt[a + b*x]))] + 2*Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]*ArcTanh[(Sq rt[b]*Sqrt[c + d*x])/(Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))]))/(Sqr t[a]*b*Sqrt[c])
Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {140, 27, 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.
\(\displaystyle \int \frac {\sqrt {c+d x}}{x \sqrt {a+b x}} \, dx\) |
\(\Big \downarrow \) 140 |
\(\displaystyle d \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx+\int \frac {c}{x \sqrt {a+b x} \sqrt {c+d x}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle d \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx+c \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx\) |
\(\Big \downarrow \) 66 |
\(\displaystyle c \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx+2 d \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle 2 c \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+2 d \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}}-\frac {2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}\) |
(-2*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt [a] + (2*Sqrt[d]*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]) /Sqrt[b]
3.8.2.3.1 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[b*d^(m + n)*f^p Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] , x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x )*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n, -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]
Leaf count of result is larger than twice the leaf count of optimal. \(132\) vs. \(2(61)=122\).
Time = 0.57 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.56
method | result | size |
default | \(\frac {\left (-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) \sqrt {b d}\, c +\sqrt {a c}\, \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) d \right ) \sqrt {d x +c}\, \sqrt {b x +a}}{\sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}}\) | \(133\) |
(-ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*(b*d)^(1 /2)*c+(a*c)^(1/2)*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a* d+b*c)/(b*d)^(1/2))*d)*(d*x+c)^(1/2)*(b*x+a)^(1/2)/((b*x+a)*(d*x+c))^(1/2) /(b*d)^(1/2)/(a*c)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (61) = 122\).
Time = 0.31 (sec) , antiderivative size = 711, normalized size of antiderivative = 8.36 \[ \int \frac {\sqrt {c+d x}}{x \sqrt {a+b x}} \, dx=\left [\frac {1}{2} \, \sqrt {\frac {d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + \frac {1}{2} \, \sqrt {\frac {c}{a}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ), -\sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) + \frac {1}{2} \, \sqrt {\frac {c}{a}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ), \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) + \frac {1}{2} \, \sqrt {\frac {d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ), \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) - \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right )\right ] \]
[1/2*sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^ 2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x) + 1/2*sqrt(c/a)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2* d^2)*x^2 - 4*(2*a^2*c + (a*b*c + a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqr t(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2), -sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-d/b)/(b*d^2*x^2 + a*c*d + (b* c*d + a*d^2)*x)) + 1/2*sqrt(c/a)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a ^2*d^2)*x^2 - 4*(2*a^2*c + (a*b*c + a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)* sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2), sqrt(-c/a)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-c/a)/(b*c*d*x^2 + a*c^2 + (b*c^2 + a*c*d)*x)) + 1/2*sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c *d + a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*s qrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x), sqrt(-c/a)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-c/a)/(b*c*d*x^2 + a*c^2 + (b*c^ 2 + a*c*d)*x)) - sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a) *sqrt(d*x + c)*sqrt(-d/b)/(b*d^2*x^2 + a*c*d + (b*c*d + a*d^2)*x))]
\[ \int \frac {\sqrt {c+d x}}{x \sqrt {a+b x}} \, dx=\int \frac {\sqrt {c + d x}}{x \sqrt {a + b x}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {c+d x}}{x \sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Exception generated. \[ \int \frac {\sqrt {c+d x}}{x \sqrt {a+b x}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Time = 17.92 (sec) , antiderivative size = 4311, normalized size of antiderivative = 50.72 \[ \int \frac {\sqrt {c+d x}}{x \sqrt {a+b x}} \, dx=\text {Too large to display} \]
(c^(1/2)*log(((c^(1/2)*(a + b*x)^(1/2) - a^(1/2)*(c + d*x)^(1/2))*(b*c^(1/ 2) - (a^(1/2)*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)))) /((c + d*x)^(1/2) - c^(1/2))) - c^(1/2)*log(((a + b*x)^(1/2) - a^(1/2))/(( c + d*x)^(1/2) - c^(1/2))))/a^(1/2) + (4*atanh((64*a^2*b^2*(b*d)^(1/2))/(( 200*a^(1/2)*b^4*c^(3/2))/d - 288*a^(3/2)*b^3*c^(1/2) + (544*b^5*c^(5/2))/( a^(1/2)*d^2) - (528*b^6*c^(7/2))/(a^(3/2)*d^3) + (64*b^7*c^(9/2))/(a^(5/2) *d^4) + (8*b^8*c^(11/2))/(a^(7/2)*d^5) + (64*a^2*b^2*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) - (864*b^4*c^2*((a + b*x)^(1/2) - a^ (1/2)))/(d*((c + d*x)^(1/2) - c^(1/2))) + (368*a*b^3*c*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (320*b^5*c^3*((a + b*x)^(1/2) - a^ (1/2)))/(a*d^2*((c + d*x)^(1/2) - c^(1/2))) + (160*b^6*c^4*((a + b*x)^(1/2 ) - a^(1/2)))/(a^2*d^3*((c + d*x)^(1/2) - c^(1/2))) - (48*b^7*c^5*((a + b* x)^(1/2) - a^(1/2)))/(a^3*d^4*((c + d*x)^(1/2) - c^(1/2)))) - (864*b^4*(b* d)^(1/2))/((544*b^5*c^(1/2))/a^(1/2) + (200*a^(1/2)*b^4*d)/c^(1/2) - (288* a^(3/2)*b^3*d^2)/c^(3/2) - (528*b^6*c^(3/2))/(a^(3/2)*d) + (64*b^7*c^(5/2) )/(a^(5/2)*d^2) + (8*b^8*c^(7/2))/(a^(7/2)*d^3) - (864*b^4*d*((a + b*x)^(1 /2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (320*b^5*c*((a + b*x)^(1/2) - a^(1/2)))/(a*((c + d*x)^(1/2) - c^(1/2))) + (368*a*b^3*d^2*((a + b*x)^(1 /2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))) + (64*a^2*b^2*d^3*((a + b* x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) + (160*b^6*c^2*(...