Integrand size = 17, antiderivative size = 63 \[ \int \sqrt {\frac {-a+b x^n}{x^2}} \, dx=\frac {2 x \sqrt {-\frac {a}{x^2}+b x^{-2+n}}}{n}+\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a}}{x \sqrt {-\frac {a}{x^2}+b x^{-2+n}}}\right )}{n} \] Output:
2*x*(-a/x^2+b*x^(-2+n))^(1/2)/n+2*a^(1/2)*arctan(a^(1/2)/x/(-a/x^2+b*x^(-2 +n))^(1/2))/n
Time = 0.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.22 \[ \int \sqrt {\frac {-a+b x^n}{x^2}} \, dx=\frac {2 x \sqrt {\frac {-a+b x^n}{x^2}} \left (\sqrt {-a+b x^n}-\sqrt {a} \arctan \left (\frac {\sqrt {-a+b x^n}}{\sqrt {a}}\right )\right )}{n \sqrt {-a+b x^n}} \] 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]*ArcTan[Sqrt[-a + b *x^n]/Sqrt[a]]))/(n*Sqrt[-a + b*x^n])
Time = 0.36 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2078, 1913, 1935, 216}
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.
\(\displaystyle \int \sqrt {\frac {b x^n-a}{x^2}} \, dx\) |
\(\Big \downarrow \) 2078 |
\(\displaystyle \int \sqrt {b x^{n-2}-\frac {a}{x^2}}dx\) |
\(\Big \downarrow \) 1913 |
\(\displaystyle \frac {2 x \sqrt {b x^{n-2}-\frac {a}{x^2}}}{n}-a \int \frac {1}{x^2 \sqrt {b x^{n-2}-\frac {a}{x^2}}}dx\) |
\(\Big \downarrow \) 1935 |
\(\displaystyle \frac {2 a \int \frac {1}{\frac {a}{x^2 \left (b x^{n-2}-\frac {a}{x^2}\right )}+1}d\frac {1}{x \sqrt {b x^{n-2}-\frac {a}{x^2}}}}{n}+\frac {2 x \sqrt {b x^{n-2}-\frac {a}{x^2}}}{n}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a}}{x \sqrt {b x^{n-2}-\frac {a}{x^2}}}\right )}{n}+\frac {2 x \sqrt {b x^{n-2}-\frac {a}{x^2}}}{n}\) |
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]*ArcTan[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]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[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 = 0.92 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.67
method | result | size |
risch | \(-\frac {2 \left (a -b \,{\mathrm e}^{n \ln \left (x \right )}\right ) \sqrt {\frac {-a +b \,{\mathrm e}^{n \ln \left (x \right )}}{x^{2}}}\, x}{n \left (-a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}-\frac {2 \sqrt {a}\, \arctan \left (\frac {\sqrt {-a +b \,{\mathrm e}^{n \ln \left (x \right )}}}{\sqrt {a}}\right ) \sqrt {\frac {-a +b \,{\mathrm e}^{n \ln \left (x \right )}}{x^{2}}}\, x}{n \sqrt {-a +b \,{\mathrm e}^{n \ln \left (x \right )}}}\) | \(105\) |
Input:
int(((-a+b*x^n)/x^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2*(a-b*exp(n*ln(x)))/n/(-a+b*exp(n*ln(x)))*((-a+b*exp(n*ln(x)))/x^2)^(1/2 )*x-2*a^(1/2)/n*arctan((-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 = 128, normalized size of antiderivative = 2.03 \[ \int \sqrt {\frac {-a+b x^n}{x^2}} \, dx=\left [\frac {2 \, x \sqrt {\frac {b x^{n} - a}{x^{2}}} + \sqrt {-a} \log \left (\frac {b x^{n} - 2 \, \sqrt {-a} x \sqrt {\frac {b x^{n} - a}{x^{2}}} - 2 \, a}{x^{n}}\right )}{n}, \frac {2 \, {\left (x \sqrt {\frac {b x^{n} - a}{x^{2}}} + \sqrt {a} \arctan \left (\frac {\sqrt {a} x \sqrt {\frac {b x^{n} - a}{x^{2}}}}{b x^{n} - a}\right )\right )}}{n}\right ] \] 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(s qrt(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