Integrand size = 19, antiderivative size = 53 \[ \int \frac {\sqrt {-a+b (c x)^n}}{x} \, dx=\frac {2 \sqrt {-a+b (c x)^n}}{n}-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {-a+b (c x)^n}}{\sqrt {a}}\right )}{n} \] Output:
2*(-a+b*(c*x)^n)^(1/2)/n-2*a^(1/2)*arctan((-a+b*(c*x)^n)^(1/2)/a^(1/2))/n
Time = 0.20 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {-a+b (c x)^n}}{x} \, dx=\frac {2 \left (\sqrt {-a+b (c x)^n}-\sqrt {a} \arctan \left (\frac {\sqrt {-a+b (c x)^n}}{\sqrt {a}}\right )\right )}{n} \] Input:
Integrate[Sqrt[-a + b*(c*x)^n]/x,x]
Output:
(2*(Sqrt[-a + b*(c*x)^n] - Sqrt[a]*ArcTan[Sqrt[-a + b*(c*x)^n]/Sqrt[a]]))/ n
Time = 0.30 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {891, 27, 798, 60, 73, 218}
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.
\(\displaystyle \int \frac {\sqrt {b (c x)^n-a}}{x} \, dx\) |
\(\Big \downarrow \) 891 |
\(\displaystyle \frac {\int \frac {\sqrt {b (c x)^n-a}}{x}d(c x)}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {\sqrt {b (c x)^n-a}}{c x}d(c x)\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int \frac {\sqrt {b (c x)^n-a}}{c x}d(c x)^n}{n}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {2 \sqrt {b (c x)^n-a}-a \int \frac {1}{c x \sqrt {b (c x)^n-a}}d(c x)^n}{n}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2 \sqrt {b (c x)^n-a}-\frac {2 a \int \frac {1}{\frac {c^2 x^2}{b}+\frac {a}{b}}d\sqrt {b (c x)^n-a}}{b}}{n}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2 \sqrt {b (c x)^n-a}-2 \sqrt {a} \arctan \left (\frac {\sqrt {b (c x)^n-a}}{\sqrt {a}}\right )}{n}\) |
Input:
Int[Sqrt[-a + b*(c*x)^n]/x,x]
Output:
(2*Sqrt[-a + b*(c*x)^n] - 2*Sqrt[a]*ArcTan[Sqrt[-a + b*(c*x)^n]/Sqrt[a]])/ n
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_)*(x_))^(n_))^(p_.), x_Symbol] :> Simp[1/c Subst[Int[(d*(x/c))^m*(a + b*x^n)^p, x], x, c*x], x] /; FreeQ[{a , b, c, d, m, n, p}, x]
Time = 0.43 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {2 \sqrt {-a +b \left (c x \right )^{n}}-2 \sqrt {a}\, \arctan \left (\frac {\sqrt {-a +b \left (c x \right )^{n}}}{\sqrt {a}}\right )}{n}\) | \(44\) |
default | \(\frac {2 \sqrt {-a +b \left (c x \right )^{n}}-2 \sqrt {a}\, \arctan \left (\frac {\sqrt {-a +b \left (c x \right )^{n}}}{\sqrt {a}}\right )}{n}\) | \(44\) |
risch | \(-\frac {2 \left (a -b \,{\mathrm e}^{n \ln \left (c x \right )}\right )}{n \sqrt {-a +b \,{\mathrm e}^{n \ln \left (c x \right )}}}-\frac {2 \sqrt {a}\, \arctan \left (\frac {\sqrt {-a +b \,{\mathrm e}^{n \ln \left (c x \right )}}}{\sqrt {a}}\right )}{n}\) | \(62\) |
Input:
int((-a+b*(c*x)^n)^(1/2)/x,x,method=_RETURNVERBOSE)
Output:
1/n*(2*(-a+b*(c*x)^n)^(1/2)-2*a^(1/2)*arctan((-a+b*(c*x)^n)^(1/2)/a^(1/2)) )
Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.04 \[ \int \frac {\sqrt {-a+b (c x)^n}}{x} \, dx=\left [\frac {\sqrt {-a} \log \left (\frac {\left (c x\right )^{n} b - 2 \, \sqrt {\left (c x\right )^{n} b - a} \sqrt {-a} - 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \, \sqrt {\left (c x\right )^{n} b - a}}{n}, \frac {2 \, {\left (\sqrt {a} \arctan \left (\frac {\sqrt {a}}{\sqrt {\left (c x\right )^{n} b - a}}\right ) + \sqrt {\left (c x\right )^{n} b - a}\right )}}{n}\right ] \] Input:
integrate((-a+b*(c*x)^n)^(1/2)/x,x, algorithm="fricas")
Output:
[(sqrt(-a)*log(((c*x)^n*b - 2*sqrt((c*x)^n*b - a)*sqrt(-a) - 2*a)/(c*x)^n) + 2*sqrt((c*x)^n*b - a))/n, 2*(sqrt(a)*arctan(sqrt(a)/sqrt((c*x)^n*b - a) ) + sqrt((c*x)^n*b - a))/n]
\[ \int \frac {\sqrt {-a+b (c x)^n}}{x} \, dx=\int \frac {\sqrt {- a + b \left (c x\right )^{n}}}{x}\, dx \] Input:
integrate((-a+b*(c*x)**n)**(1/2)/x,x)
Output:
Integral(sqrt(-a + b*(c*x)**n)/x, x)
\[ \int \frac {\sqrt {-a+b (c x)^n}}{x} \, dx=\int { \frac {\sqrt {\left (c x\right )^{n} b - a}}{x} \,d x } \] Input:
integrate((-a+b*(c*x)^n)^(1/2)/x,x, algorithm="maxima")
Output:
integrate(sqrt((c*x)^n*b - a)/x, x)
\[ \int \frac {\sqrt {-a+b (c x)^n}}{x} \, dx=\int { \frac {\sqrt {\left (c x\right )^{n} b - a}}{x} \,d x } \] Input:
integrate((-a+b*(c*x)^n)^(1/2)/x,x, algorithm="giac")
Output:
integrate(sqrt((c*x)^n*b - a)/x, x)
Timed out. \[ \int \frac {\sqrt {-a+b (c x)^n}}{x} \, dx=\int \frac {\sqrt {b\,{\left (c\,x\right )}^n-a}}{x} \,d x \] Input:
int((b*(c*x)^n - a)^(1/2)/x,x)
Output:
int((b*(c*x)^n - a)^(1/2)/x, x)
\[ \int \frac {\sqrt {-a+b (c x)^n}}{x} \, dx=\frac {2 \sqrt {x^{n} c^{n} b -a}-\left (\int \frac {\sqrt {x^{n} c^{n} b -a}}{x^{n} c^{n} b x -a x}d x \right ) a n}{n} \] Input:
int((-a+b*(c*x)^n)^(1/2)/x,x)
Output:
(2*sqrt(x**n*c**n*b - a) - int(sqrt(x**n*c**n*b - a)/(x**n*c**n*b*x - a*x) ,x)*a*n)/n