Integrand size = 15, antiderivative size = 61 \[ \int \sqrt {\frac {a+b x^n}{x^2}} \, dx=\frac {2 x \sqrt {\frac {a}{x^2}+b x^{-2+n}}}{n}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^{-2+n}}}\right )}{n} \] Output:
2*x*(a/x^2+b*x^(-2+n))^(1/2)/n-2*a^(1/2)*arctanh(a^(1/2)/x/(a/x^2+b*x^(-2+ n))^(1/2))/n
Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.13 \[ \int \sqrt {\frac {a+b x^n}{x^2}} \, dx=\frac {2 x \sqrt {\frac {a+b x^n}{x^2}} \left (\sqrt {a+b x^n}-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )\right )}{n \sqrt {a+b x^n}} \] Input:
Integrate[Sqrt[(a + b*x^n)/x^2],x]
Output:
(2*x*Sqrt[(a + b*x^n)/x^2]*(Sqrt[a + b*x^n] - Sqrt[a]*ArcTanh[Sqrt[a + b*x ^n]/Sqrt[a]]))/(n*Sqrt[a + b*x^n])
Time = 0.36 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2078, 1913, 1935, 219}
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.
\(\displaystyle \int \sqrt {\frac {a+b x^n}{x^2}} \, dx\) |
\(\Big \downarrow \) 2078 |
\(\displaystyle \int \sqrt {\frac {a}{x^2}+b x^{n-2}}dx\) |
\(\Big \downarrow \) 1913 |
\(\displaystyle a \int \frac {1}{x^2 \sqrt {b x^{n-2}+\frac {a}{x^2}}}dx+\frac {2 x \sqrt {\frac {a}{x^2}+b x^{n-2}}}{n}\) |
\(\Big \downarrow \) 1935 |
\(\displaystyle \frac {2 x \sqrt {\frac {a}{x^2}+b x^{n-2}}}{n}-\frac {2 a \int \frac {1}{1-\frac {a}{x^2 \left (b x^{n-2}+\frac {a}{x^2}\right )}}d\frac {1}{x \sqrt {b x^{n-2}+\frac {a}{x^2}}}}{n}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 x \sqrt {\frac {a}{x^2}+b x^{n-2}}}{n}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^{n-2}}}\right )}{n}\) |
Input:
Int[Sqrt[(a + b*x^n)/x^2],x]
Output:
(2*x*Sqrt[a/x^2 + b*x^(-2 + n)])/n - (2*Sqrt[a]*ArcTanh[Sqrt[a]/(x*Sqrt[a/ x^2 + b*x^(-2 + n)])])/n
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_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[x*((a*x^j + b*x^n)^p/(p*(n - j))), x] + Simp[a Int[x^j*(a*x^j + b*x^n)^(p - 1), x] , x] /; FreeQ[{a, b, j, n}, x] && IGtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[Simp lify[j*p + 1], 0]
Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp [-2/(n - j) Subst[Int[1/(1 - a*x^2), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]
Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && G eneralizedBinomialQ[u, x] && !GeneralizedBinomialMatchQ[u, x]
Time = 0.89 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.21
method | result | size |
risch | \(\frac {2 \sqrt {\frac {a +b \,{\mathrm e}^{n \ln \left (x \right )}}{x^{2}}}\, x}{n}-\frac {2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}}}{\sqrt {a}}\right ) \sqrt {\frac {a +b \,{\mathrm e}^{n \ln \left (x \right )}}{x^{2}}}\, x}{n \sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}}}\) | \(74\) |
Input:
int(((a+b*x^n)/x^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/n*((a+b*exp(n*ln(x)))/x^2)^(1/2)*x-2*a^(1/2)/n*arctanh((a+b*exp(n*ln(x)) )^(1/2)/a^(1/2))*((a+b*exp(n*ln(x)))/x^2)^(1/2)/(a+b*exp(n*ln(x)))^(1/2)*x
Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.93 \[ \int \sqrt {\frac {a+b x^n}{x^2}} \, dx=\left [\frac {2 \, x \sqrt {\frac {b x^{n} + a}{x^{2}}} + \sqrt {a} \log \left (\frac {b x^{n} - 2 \, \sqrt {a} x \sqrt {\frac {b x^{n} + a}{x^{2}}} + 2 \, a}{x^{n}}\right )}{n}, \frac {2 \, {\left (x \sqrt {\frac {b x^{n} + a}{x^{2}}} + \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {b x^{n} + a}{x^{2}}}}{b x^{n} + a}\right )\right )}}{n}\right ] \] Input:
integrate(((a+b*x^n)/x^2)^(1/2),x, algorithm="fricas")
Output:
[(2*x*sqrt((b*x^n + a)/x^2) + sqrt(a)*log((b*x^n - 2*sqrt(a)*x*sqrt((b*x^n + a)/x^2) + 2*a)/x^n))/n, 2*(x*sqrt((b*x^n + a)/x^2) + sqrt(-a)*arctan(sq rt(-a)*x*sqrt((b*x^n + a)/x^2)/(b*x^n + a)))/n]
\[ \int \sqrt {\frac {a+b x^n}{x^2}} \, dx=\int \sqrt {\frac {a + b x^{n}}{x^{2}}}\, dx \] Input:
integrate(((a+b*x**n)/x**2)**(1/2),x)
Output:
Integral(sqrt((a + b*x**n)/x**2), x)
\[ \int \sqrt {\frac {a+b x^n}{x^2}} \, dx=\int { \sqrt {\frac {b x^{n} + a}{x^{2}}} \,d x } \] Input:
integrate(((a+b*x^n)/x^2)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt((b*x^n + a)/x^2), x)
\[ \int \sqrt {\frac {a+b x^n}{x^2}} \, dx=\int { \sqrt {\frac {b x^{n} + a}{x^{2}}} \,d x } \] Input:
integrate(((a+b*x^n)/x^2)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt((b*x^n + a)/x^2), x)
Timed out. \[ \int \sqrt {\frac {a+b x^n}{x^2}} \, dx=\int \sqrt {\frac {a+b\,x^n}{x^2}} \,d x \] Input:
int(((a + b*x^n)/x^2)^(1/2),x)
Output:
int(((a + b*x^n)/x^2)^(1/2), x)
\[ \int \sqrt {\frac {a+b x^n}{x^2}} \, dx=\frac {2 \sqrt {x^{n} b +a}+\left (\int \frac {\sqrt {x^{n} b +a}}{x^{n} b x +a x}d x \right ) a n}{n} \] Input:
int(((a+b*x^n)/x^2)^(1/2),x)
Output:
(2*sqrt(x**n*b + a) + int(sqrt(x**n*b + a)/(x**n*b*x + a*x),x)*a*n)/n