Integrand size = 21, antiderivative size = 97 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx=-\frac {\sqrt {a+b \sqrt {c x^2}}}{2 x^2}-\frac {b c \sqrt {a+b \sqrt {c x^2}}}{4 a \sqrt {c x^2}}+\frac {b^2 c \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{4 a^{3/2}} \] Output:
-1/2*(a+b*(c*x^2)^(1/2))^(1/2)/x^2-1/4*b*c*(a+b*(c*x^2)^(1/2))^(1/2)/a/(c* x^2)^(1/2)+1/4*b^2*c*arctanh((a+b*(c*x^2)^(1/2))^(1/2)/a^(1/2))/a^(3/2)
Time = 1.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx=-\frac {\sqrt {a+b \sqrt {c x^2}} \left (2 a+b \sqrt {c x^2}\right )}{4 a x^2}+\frac {b^2 c \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{4 a^{3/2}} \] Input:
Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x^3,x]
Output:
-1/4*(Sqrt[a + b*Sqrt[c*x^2]]*(2*a + b*Sqrt[c*x^2]))/(a*x^2) + (b^2*c*ArcT anh[Sqrt[a + b*Sqrt[c*x^2]]/Sqrt[a]])/(4*a^(3/2))
Time = 0.31 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {892, 51, 52, 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.
\(\displaystyle \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx\) |
\(\Big \downarrow \) 892 |
\(\displaystyle c \int \frac {\sqrt {a+b \sqrt {c x^2}}}{\left (c x^2\right )^{3/2}}d\sqrt {c x^2}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle c \left (\frac {1}{4} b \int \frac {1}{c x^2 \sqrt {a+b \sqrt {c x^2}}}d\sqrt {c x^2}-\frac {\sqrt {a+b \sqrt {c x^2}}}{2 c x^2}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle c \left (\frac {1}{4} b \left (-\frac {b \int \frac {1}{\sqrt {c x^2} \sqrt {a+b \sqrt {c x^2}}}d\sqrt {c x^2}}{2 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{a \sqrt {c x^2}}\right )-\frac {\sqrt {a+b \sqrt {c x^2}}}{2 c x^2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle c \left (\frac {1}{4} b \left (-\frac {\int \frac {1}{\frac {c x^2}{b}-\frac {a}{b}}d\sqrt {a+b \sqrt {c x^2}}}{a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{a \sqrt {c x^2}}\right )-\frac {\sqrt {a+b \sqrt {c x^2}}}{2 c x^2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle c \left (\frac {1}{4} b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b \sqrt {c x^2}}}{a \sqrt {c x^2}}\right )-\frac {\sqrt {a+b \sqrt {c x^2}}}{2 c x^2}\right )\) |
Input:
Int[Sqrt[a + b*Sqrt[c*x^2]]/x^3,x]
Output:
c*(-1/2*Sqrt[a + b*Sqrt[c*x^2]]/(c*x^2) + (b*(-(Sqrt[a + b*Sqrt[c*x^2]]/(a *Sqrt[c*x^2])) + (b*ArcTanh[Sqrt[a + b*Sqrt[c*x^2]]/Sqrt[a]])/a^(3/2)))/4)
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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[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 = 72, normalized size of antiderivative = 0.74
method | result | size |
default | \(-\frac {-\operatorname {arctanh}\left (\frac {\sqrt {a +b \sqrt {c \,x^{2}}}}{\sqrt {a}}\right ) a \,b^{2} c \,x^{2}+\left (a +b \sqrt {c \,x^{2}}\right )^{\frac {3}{2}} a^{\frac {3}{2}}+\sqrt {a +b \sqrt {c \,x^{2}}}\, a^{\frac {5}{2}}}{4 a^{\frac {5}{2}} x^{2}}\) | \(72\) |
Input:
int((a+b*(c*x^2)^(1/2))^(1/2)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/4*(-arctanh((a+b*(c*x^2)^(1/2))^(1/2)/a^(1/2))*a*b^2*c*x^2+(a+b*(c*x^2) ^(1/2))^(3/2)*a^(3/2)+(a+b*(c*x^2)^(1/2))^(1/2)*a^(5/2))/a^(5/2)/x^2
Time = 0.12 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.11 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx=\left [\frac {\sqrt {a} b^{2} c x^{2} \log \left (\frac {b c x^{2} + 2 \, \sqrt {c x^{2}} \sqrt {\sqrt {c x^{2}} b + a} \sqrt {a} + 2 \, \sqrt {c x^{2}} a}{x^{2}}\right ) - 2 \, {\left (\sqrt {c x^{2}} a b + 2 \, a^{2}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{8 \, a^{2} x^{2}}, -\frac {\sqrt {-a} b^{2} c x^{2} \arctan \left (\frac {{\left (\sqrt {c x^{2}} \sqrt {-a} b - \sqrt {-a} a\right )} \sqrt {\sqrt {c x^{2}} b + a}}{b^{2} c x^{2} - a^{2}}\right ) + {\left (\sqrt {c x^{2}} a b + 2 \, a^{2}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{4 \, a^{2} x^{2}}\right ] \] Input:
integrate((a+b*(c*x^2)^(1/2))^(1/2)/x^3,x, algorithm="fricas")
Output:
[1/8*(sqrt(a)*b^2*c*x^2*log((b*c*x^2 + 2*sqrt(c*x^2)*sqrt(sqrt(c*x^2)*b + a)*sqrt(a) + 2*sqrt(c*x^2)*a)/x^2) - 2*(sqrt(c*x^2)*a*b + 2*a^2)*sqrt(sqrt (c*x^2)*b + a))/(a^2*x^2), -1/4*(sqrt(-a)*b^2*c*x^2*arctan((sqrt(c*x^2)*sq rt(-a)*b - sqrt(-a)*a)*sqrt(sqrt(c*x^2)*b + a)/(b^2*c*x^2 - a^2)) + (sqrt( c*x^2)*a*b + 2*a^2)*sqrt(sqrt(c*x^2)*b + a))/(a^2*x^2)]
\[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx=\int \frac {\sqrt {a + b \sqrt {c x^{2}}}}{x^{3}}\, dx \] Input:
integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**3,x)
Output:
Integral(sqrt(a + b*sqrt(c*x**2))/x**3, x)
Time = 0.11 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx=-\frac {1}{8} \, {\left (\frac {b^{2} \log \left (\frac {\sqrt {\sqrt {c x^{2}} b + a} - \sqrt {a}}{\sqrt {\sqrt {c x^{2}} b + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (\sqrt {c x^{2}} b + a\right )}^{\frac {3}{2}} b^{2} + \sqrt {\sqrt {c x^{2}} b + a} a b^{2}\right )}}{{\left (\sqrt {c x^{2}} b + a\right )}^{2} a - 2 \, {\left (\sqrt {c x^{2}} b + a\right )} a^{2} + a^{3}}\right )} c \] Input:
integrate((a+b*(c*x^2)^(1/2))^(1/2)/x^3,x, algorithm="maxima")
Output:
-1/8*(b^2*log((sqrt(sqrt(c*x^2)*b + a) - sqrt(a))/(sqrt(sqrt(c*x^2)*b + a) + sqrt(a)))/a^(3/2) + 2*((sqrt(c*x^2)*b + a)^(3/2)*b^2 + sqrt(sqrt(c*x^2) *b + a)*a*b^2)/((sqrt(c*x^2)*b + a)^2*a - 2*(sqrt(c*x^2)*b + a)*a^2 + a^3) )*c
Exception generated. \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*(c*x^2)^(1/2))^(1/2)/x^3,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx=\int \frac {\sqrt {a+b\,\sqrt {c\,x^2}}}{x^3} \,d x \] Input:
int((a + b*(c*x^2)^(1/2))^(1/2)/x^3,x)
Output:
int((a + b*(c*x^2)^(1/2))^(1/2)/x^3, x)
Time = 0.14 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx=\frac {-2 \sqrt {c}\, \sqrt {\sqrt {c}\, b x +a}\, a b x -4 \sqrt {\sqrt {c}\, b x +a}\, a^{2}-\sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, b x +a}-\sqrt {a}\right ) b^{2} c \,x^{2}+\sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, b x +a}+\sqrt {a}\right ) b^{2} c \,x^{2}}{8 a^{2} x^{2}} \] Input:
int((a+b*(c*x^2)^(1/2))^(1/2)/x^3,x)
Output:
( - 2*sqrt(c)*sqrt(sqrt(c)*b*x + a)*a*b*x - 4*sqrt(sqrt(c)*b*x + a)*a**2 - sqrt(a)*log(sqrt(sqrt(c)*b*x + a) - sqrt(a))*b**2*c*x**2 + sqrt(a)*log(sq rt(sqrt(c)*b*x + a) + sqrt(a))*b**2*c*x**2)/(8*a**2*x**2)