Integrand size = 21, antiderivative size = 80 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x} \, dx=\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )-\frac {\sqrt {b+a c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {b+a c}}\right )}{\sqrt {c}} \] Output:
a^(1/2)*arctanh((a+b/(d*x^2+c))^(1/2)/a^(1/2))-(a*c+b)^(1/2)*arctanh(c^(1/ 2)*(a+b/(d*x^2+c))^(1/2)/(a*c+b)^(1/2))/c^(1/2)
Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x} \, dx=-\frac {\sqrt {-b-a c} \arctan \left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {-b-a c}}\right )}{\sqrt {c}}+\sqrt {a} \text {arctanh}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right ) \] Input:
Integrate[Sqrt[a + b/(c + d*x^2)]/x,x]
Output:
-((Sqrt[-b - a*c]*ArcTan[(Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/S qrt[-b - a*c]])/Sqrt[c]) + Sqrt[a]*ArcTanh[Sqrt[(b + a*c + a*d*x^2)/(c + d *x^2)]/Sqrt[a]]
Time = 0.62 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.30, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {2057, 2053, 2052, 25, 27, 383, 219, 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.
\(\displaystyle \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x} \, dx\) |
\(\Big \downarrow \) 2057 |
\(\displaystyle \int \frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{x}dx\) |
\(\Big \downarrow \) 2053 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{x^2}dx^2\) |
\(\Big \downarrow \) 2052 |
\(\displaystyle -b d \int -\frac {x^4}{d \left (a-x^4\right ) \left (-c x^4+b+a c\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle b d \int \frac {x^4}{d \left (a-x^4\right ) \left (-c x^4+b+a c\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle b \int \frac {x^4}{\left (a-x^4\right ) \left (-c x^4+b+a c\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
\(\Big \downarrow \) 383 |
\(\displaystyle b \left (\frac {a \int \frac {1}{a-x^4}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{b}-\frac {(a c+b) \int \frac {1}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{b}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle b \left (\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a}}\right )}{b}-\frac {(a c+b) \int \frac {1}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{b}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle b \left (\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a}}\right )}{b}-\frac {\sqrt {a c+b} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a c+b}}\right )}{b \sqrt {c}}\right )\) |
Input:
Int[Sqrt[a + b/(c + d*x^2)]/x,x]
Output:
b*((Sqrt[a]*ArcTanh[Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]/Sqrt[a]])/b - (S qrt[b + a*c]*ArcTanh[(Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[ b + a*c]])/(b*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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_Sym bol] :> Simp[(-a)*(e^2/(b*c - a*d)) Int[(e*x)^(m - 2)/(a + b*x^2), x], x] + Simp[c*(e^2/(b*c - a*d)) Int[(e*x)^(m - 2)/(c + d*x^2), x], x] /; Free Q[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LeQ[2, m, 3]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d) Subst[Int[x^(q* (p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) ))^(p_), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( (a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* ((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(234\) vs. \(2(64)=128\).
Time = 0.10 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.94
method | result | size |
default | \(\frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (\ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a c d -\sqrt {a \,c^{2}+b c}\, \ln \left (\frac {2 a d \,x^{2} c +b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) \sqrt {a \,d^{2}}\right )}{2 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, c \sqrt {a \,d^{2}}}\) | \(235\) |
Input:
int((a+b/(d*x^2+c))^(1/2)/x,x,method=_RETURNVERBOSE)
Output:
1/2*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)*(d*x^2+c)*(ln(1/2*(2*a*d^2*x^2+2*a*c *d+2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*d^2)^(1/2)+b*d)/(a *d^2)^(1/2))*a*c*d-(a*c^2+b*c)^(1/2)*ln((2*a*d*x^2*c+b*d*x^2+2*a*c^2+2*(a* c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)+2*b*c)/x^2) *(a*d^2)^(1/2))/((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)/c/(a*d^2)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (64) = 128\).
Time = 0.14 (sec) , antiderivative size = 927, normalized size of antiderivative = 11.59 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x} \, dx =\text {Too large to display} \] Input:
integrate((a+b/(d*x^2+c))^(1/2)/x,x, algorithm="fricas")
Output:
[1/4*sqrt(a)*log(8*a^2*d^2*x^4 + 8*a^2*c^2 + 8*(2*a^2*c + a*b)*d*x^2 + 8*a *b*c + b^2 + 4*(2*a*d^2*x^4 + (4*a*c + b)*d*x^2 + 2*a*c^2 + b*c)*sqrt(a)*s qrt((a*d*x^2 + a*c + b)/(d*x^2 + c))) + 1/4*sqrt((a*c + b)/c)*log(((8*a^2* c^2 + 8*a*b*c + b^2)*d^2*x^4 + 8*a^2*c^4 + 16*a*b*c^3 + 8*b^2*c^2 + 8*(2*a ^2*c^3 + 3*a*b*c^2 + b^2*c)*d*x^2 - 4*((2*a*c^2 + b*c)*d^2*x^4 + 2*a*c^4 + 2*b*c^3 + (4*a*c^3 + 3*b*c^2)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c) )*sqrt((a*c + b)/c))/x^4), -1/2*sqrt(-a)*arctan(1/2*(2*a*d*x^2 + 2*a*c + b )*sqrt(-a)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/(a^2*d*x^2 + a^2*c + a*b) ) + 1/4*sqrt((a*c + b)/c)*log(((8*a^2*c^2 + 8*a*b*c + b^2)*d^2*x^4 + 8*a^2 *c^4 + 16*a*b*c^3 + 8*b^2*c^2 + 8*(2*a^2*c^3 + 3*a*b*c^2 + b^2*c)*d*x^2 - 4*((2*a*c^2 + b*c)*d^2*x^4 + 2*a*c^4 + 2*b*c^3 + (4*a*c^3 + 3*b*c^2)*d*x^2 )*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))*sqrt((a*c + b)/c))/x^4), 1/2*sqrt( -(a*c + b)/c)*arctan(1/2*((2*a*c + b)*d*x^2 + 2*a*c^2 + 2*b*c)*sqrt((a*d*x ^2 + a*c + b)/(d*x^2 + c))*sqrt(-(a*c + b)/c)/(a^2*c^2 + (a^2*c + a*b)*d*x ^2 + 2*a*b*c + b^2)) + 1/4*sqrt(a)*log(8*a^2*d^2*x^4 + 8*a^2*c^2 + 8*(2*a^ 2*c + a*b)*d*x^2 + 8*a*b*c + b^2 + 4*(2*a*d^2*x^4 + (4*a*c + b)*d*x^2 + 2* a*c^2 + b*c)*sqrt(a)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))), -1/2*sqrt(-a) *arctan(1/2*(2*a*d*x^2 + 2*a*c + b)*sqrt(-a)*sqrt((a*d*x^2 + a*c + b)/(d*x ^2 + c))/(a^2*d*x^2 + a^2*c + a*b)) + 1/2*sqrt(-(a*c + b)/c)*arctan(1/2*(( 2*a*c + b)*d*x^2 + 2*a*c^2 + 2*b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c...
\[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x} \, dx=\int \frac {\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}{x}\, dx \] Input:
integrate((a+b/(d*x**2+c))**(1/2)/x,x)
Output:
Integral(sqrt((a*c + a*d*x**2 + b)/(c + d*x**2))/x, x)
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (64) = 128\).
Time = 0.13 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.99 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x} \, dx=\frac {{\left (a c + b\right )} \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{2 \, \sqrt {{\left (a c + b\right )} c}} - \frac {1}{2} \, \sqrt {a} \log \left (-\frac {\sqrt {a} - \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{\sqrt {a} + \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}\right ) \] Input:
integrate((a+b/(d*x^2+c))^(1/2)/x,x, algorithm="maxima")
Output:
1/2*(a*c + b)*log((c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) - sqrt((a*c + b )*c))/(c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) + sqrt((a*c + b)*c)))/sqrt( (a*c + b)*c) - 1/2*sqrt(a)*log(-(sqrt(a) - sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(sqrt(a) + sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))))
Exception generated. \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b/(d*x^2+c))^(1/2)/x,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x} \, dx=\int \frac {\sqrt {a+\frac {b}{d\,x^2+c}}}{x} \,d x \] Input:
int((a + b/(c + d*x^2))^(1/2)/x,x)
Output:
int((a + b/(c + d*x^2))^(1/2)/x, x)
Time = 0.22 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.35 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x} \, dx=\frac {\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right )-\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right )+\sqrt {a}\, \mathrm {log}\left (-\sqrt {a}\, \sqrt {a d \,x^{2}+a c +b}-\sqrt {d \,x^{2}+c}\, a \right ) c}{c} \] Input:
int((a+b/(d*x^2+c))^(1/2)/x,x)
Output:
(sqrt(c)*sqrt(a*c + b)*log(sqrt(a*c + b)*sqrt(a*c + a*d*x**2 + b)*c - sqrt (c)*sqrt(c + d*x**2)*a*c - sqrt(c)*sqrt(c + d*x**2)*b) - sqrt(c)*sqrt(a*c + b)*log(x) + sqrt(a)*log( - sqrt(a)*sqrt(a*c + a*d*x**2 + b) - sqrt(c + d *x**2)*a)*c)/c