Integrand size = 21, antiderivative size = 67 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^2} \, dx=-\frac {\sqrt {a+b \sqrt {c x^2}}}{x}-\frac {b \sqrt {c x^2} \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{\sqrt {a} x} \] Output:
-(a+b*(c*x^2)^(1/2))^(1/2)/x-b*(c*x^2)^(1/2)*arctanh((a+b*(c*x^2)^(1/2))^( 1/2)/a^(1/2))/a^(1/2)/x
Time = 1.18 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^2} \, dx=-\frac {a+b \sqrt {c x^2}+b \sqrt {c x^2} \sqrt {1+\frac {b \sqrt {c x^2}}{a}} \text {arctanh}\left (\sqrt {1+\frac {b \sqrt {c x^2}}{a}}\right )}{x \sqrt {a+b \sqrt {c x^2}}} \] Input:
Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x^2,x]
Output:
-((a + b*Sqrt[c*x^2] + b*Sqrt[c*x^2]*Sqrt[1 + (b*Sqrt[c*x^2])/a]*ArcTanh[S qrt[1 + (b*Sqrt[c*x^2])/a]])/(x*Sqrt[a + b*Sqrt[c*x^2]]))
Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {892, 51, 73, 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.
\(\displaystyle \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^2} \, dx\) |
\(\Big \downarrow \) 892 |
\(\displaystyle \frac {\sqrt {c x^2} \int \frac {\sqrt {a+b \sqrt {c x^2}}}{c x^2}d\sqrt {c x^2}}{x}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {\sqrt {c x^2} \left (\frac {1}{2} b \int \frac {1}{\sqrt {c x^2} \sqrt {a+b \sqrt {c x^2}}}d\sqrt {c x^2}-\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {c x^2}}\right )}{x}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\sqrt {c x^2} \left (\int \frac {1}{\frac {c x^2}{b}-\frac {a}{b}}d\sqrt {a+b \sqrt {c x^2}}-\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {c x^2}}\right )}{x}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\sqrt {c x^2} \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {c x^2}}\right )}{x}\) |
Input:
Int[Sqrt[a + b*Sqrt[c*x^2]]/x^2,x]
Output:
(Sqrt[c*x^2]*(-(Sqrt[a + b*Sqrt[c*x^2]]/Sqrt[c*x^2]) - (b*ArcTanh[Sqrt[a + b*Sqrt[c*x^2]]/Sqrt[a]])/Sqrt[a]))/x
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo l] :> Simp[(d*x)^(m + 1)/(d*((c*x^q)^(1/q))^(m + 1)) Subst[Int[x^m*(a + b *x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x ] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]
Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\frac {\operatorname {arctanh}\left (\frac {\sqrt {a +b \sqrt {c \,x^{2}}}}{\sqrt {a}}\right ) b \sqrt {c \,x^{2}}+\sqrt {a +b \sqrt {c \,x^{2}}}\, \sqrt {a}}{x \sqrt {a}}\) | \(54\) |
Input:
int((a+b*(c*x^2)^(1/2))^(1/2)/x^2,x,method=_RETURNVERBOSE)
Output:
-(arctanh((a+b*(c*x^2)^(1/2))^(1/2)/a^(1/2))*b*(c*x^2)^(1/2)+(a+b*(c*x^2)^ (1/2))^(1/2)*a^(1/2))/x/a^(1/2)
Time = 0.12 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.69 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^2} \, dx=\left [\frac {b x \sqrt {\frac {c}{a}} \log \left (\frac {b c x^{2} - 2 \, \sqrt {\sqrt {c x^{2}} b + a} a x \sqrt {\frac {c}{a}} + 2 \, \sqrt {c x^{2}} a}{x^{2}}\right ) - 2 \, \sqrt {\sqrt {c x^{2}} b + a}}{2 \, x}, -\frac {b x \sqrt {-\frac {c}{a}} \arctan \left (-\frac {{\left (a b c x^{2} \sqrt {-\frac {c}{a}} - \sqrt {c x^{2}} a^{2} \sqrt {-\frac {c}{a}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{b^{2} c^{2} x^{3} - a^{2} c x}\right ) + \sqrt {\sqrt {c x^{2}} b + a}}{x}\right ] \] Input:
integrate((a+b*(c*x^2)^(1/2))^(1/2)/x^2,x, algorithm="fricas")
Output:
[1/2*(b*x*sqrt(c/a)*log((b*c*x^2 - 2*sqrt(sqrt(c*x^2)*b + a)*a*x*sqrt(c/a) + 2*sqrt(c*x^2)*a)/x^2) - 2*sqrt(sqrt(c*x^2)*b + a))/x, -(b*x*sqrt(-c/a)* arctan(-(a*b*c*x^2*sqrt(-c/a) - sqrt(c*x^2)*a^2*sqrt(-c/a))*sqrt(sqrt(c*x^ 2)*b + a)/(b^2*c^2*x^3 - a^2*c*x)) + sqrt(sqrt(c*x^2)*b + a))/x]
\[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^2} \, dx=\int \frac {\sqrt {a + b \sqrt {c x^{2}}}}{x^{2}}\, dx \] Input:
integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**2,x)
Output:
Integral(sqrt(a + b*sqrt(c*x**2))/x**2, x)
\[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^2} \, dx=\int { \frac {\sqrt {\sqrt {c x^{2}} b + a}}{x^{2}} \,d x } \] Input:
integrate((a+b*(c*x^2)^(1/2))^(1/2)/x^2,x, algorithm="maxima")
Output:
integrate(sqrt(sqrt(c*x^2)*b + a)/x^2, x)
Time = 0.14 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^2} \, dx=b \sqrt {c} {\left (\frac {\arctan \left (\frac {\sqrt {b \sqrt {c} x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\sqrt {b \sqrt {c} x + a}}{b \sqrt {c} x}\right )} \] Input:
integrate((a+b*(c*x^2)^(1/2))^(1/2)/x^2,x, algorithm="giac")
Output:
b*sqrt(c)*(arctan(sqrt(b*sqrt(c)*x + a)/sqrt(-a))/sqrt(-a) - sqrt(b*sqrt(c )*x + a)/(b*sqrt(c)*x))
Timed out. \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^2} \, dx=\int \frac {\sqrt {a+b\,\sqrt {c\,x^2}}}{x^2} \,d x \] Input:
int((a + b*(c*x^2)^(1/2))^(1/2)/x^2,x)
Output:
int((a + b*(c*x^2)^(1/2))^(1/2)/x^2, x)
Time = 0.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^2} \, dx=\frac {-2 \sqrt {\sqrt {c}\, b x +a}\, a +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, b x +a}-\sqrt {a}\right ) b x -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, b x +a}+\sqrt {a}\right ) b x}{2 a x} \] Input:
int((a+b*(c*x^2)^(1/2))^(1/2)/x^2,x)
Output:
( - 2*sqrt(sqrt(c)*b*x + a)*a + sqrt(c)*sqrt(a)*log(sqrt(sqrt(c)*b*x + a) - sqrt(a))*b*x - sqrt(c)*sqrt(a)*log(sqrt(sqrt(c)*b*x + a) + sqrt(a))*b*x) /(2*a*x)