Integrand size = 19, antiderivative size = 113 \[ \int x^{-1-3 n} \sqrt {a+b x^n} \, dx=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n}+\frac {b^2 x^{-n} \sqrt {a+b x^n}}{8 a^2 n}-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{8 a^{5/2} n} \] Output:
-1/3*(a+b*x^n)^(1/2)/n/(x^(3*n))-1/12*b*(a+b*x^n)^(1/2)/a/n/(x^(2*n))+1/8* b^2*(a+b*x^n)^(1/2)/a^2/n/(x^n)-1/8*b^3*arctanh((a+b*x^n)^(1/2)/a^(1/2))/a ^(5/2)/n
Time = 0.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.72 \[ \int x^{-1-3 n} \sqrt {a+b x^n} \, dx=\frac {\sqrt {a} x^{-3 n} \sqrt {a+b x^n} \left (-8 a^2-2 a b x^n+3 b^2 x^{2 n}\right )-3 b^3 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{24 a^{5/2} n} \] Input:
Integrate[x^(-1 - 3*n)*Sqrt[a + b*x^n],x]
Output:
((Sqrt[a]*Sqrt[a + b*x^n]*(-8*a^2 - 2*a*b*x^n + 3*b^2*x^(2*n)))/x^(3*n) - 3*b^3*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(24*a^(5/2)*n)
Time = 0.33 (sec) , antiderivative size = 109, 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 = {798, 51, 52, 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.
\(\displaystyle \int x^{-3 n-1} \sqrt {a+b x^n} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int x^{-4 n} \sqrt {b x^n+a}dx^n}{n}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {\frac {1}{6} b \int \frac {x^{-3 n}}{\sqrt {b x^n+a}}dx^n-\frac {1}{3} x^{-3 n} \sqrt {a+b x^n}}{n}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {\frac {1}{6} b \left (-\frac {3 b \int \frac {x^{-2 n}}{\sqrt {b x^n+a}}dx^n}{4 a}-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 a}\right )-\frac {1}{3} x^{-3 n} \sqrt {a+b x^n}}{n}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {\frac {1}{6} b \left (-\frac {3 b \left (-\frac {b \int \frac {x^{-n}}{\sqrt {b x^n+a}}dx^n}{2 a}-\frac {x^{-n} \sqrt {a+b x^n}}{a}\right )}{4 a}-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 a}\right )-\frac {1}{3} x^{-3 n} \sqrt {a+b x^n}}{n}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {1}{6} b \left (-\frac {3 b \left (-\frac {\int \frac {1}{\frac {x^{2 n}}{b}-\frac {a}{b}}d\sqrt {b x^n+a}}{a}-\frac {x^{-n} \sqrt {a+b x^n}}{a}\right )}{4 a}-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 a}\right )-\frac {1}{3} x^{-3 n} \sqrt {a+b x^n}}{n}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {1}{6} b \left (-\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {x^{-n} \sqrt {a+b x^n}}{a}\right )}{4 a}-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 a}\right )-\frac {1}{3} x^{-3 n} \sqrt {a+b x^n}}{n}\) |
Input:
Int[x^(-1 - 3*n)*Sqrt[a + b*x^n],x]
Output:
(-1/3*Sqrt[a + b*x^n]/x^(3*n) + (b*(-1/2*Sqrt[a + b*x^n]/(a*x^(2*n)) - (3* b*(-(Sqrt[a + b*x^n]/(a*x^n)) + (b*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/a^(3/ 2)))/(4*a)))/6)/n
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[(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]]
\[\int x^{-1-3 n} \sqrt {a +b \,x^{n}}d x\]
Input:
int(x^(-1-3*n)*(a+b*x^n)^(1/2),x)
Output:
int(x^(-1-3*n)*(a+b*x^n)^(1/2),x)
Time = 0.10 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.59 \[ \int x^{-1-3 n} \sqrt {a+b x^n} \, dx=\left [\frac {3 \, \sqrt {a} b^{3} x^{3 \, n} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, {\left (3 \, a b^{2} x^{2 \, n} - 2 \, a^{2} b x^{n} - 8 \, a^{3}\right )} \sqrt {b x^{n} + a}}{48 \, a^{3} n x^{3 \, n}}, \frac {3 \, \sqrt {-a} b^{3} x^{3 \, n} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{n} + a}}\right ) + {\left (3 \, a b^{2} x^{2 \, n} - 2 \, a^{2} b x^{n} - 8 \, a^{3}\right )} \sqrt {b x^{n} + a}}{24 \, a^{3} n x^{3 \, n}}\right ] \] Input:
integrate(x^(-1-3*n)*(a+b*x^n)^(1/2),x, algorithm="fricas")
Output:
[1/48*(3*sqrt(a)*b^3*x^(3*n)*log((b*x^n - 2*sqrt(b*x^n + a)*sqrt(a) + 2*a) /x^n) + 2*(3*a*b^2*x^(2*n) - 2*a^2*b*x^n - 8*a^3)*sqrt(b*x^n + a))/(a^3*n* x^(3*n)), 1/24*(3*sqrt(-a)*b^3*x^(3*n)*arctan(sqrt(-a)/sqrt(b*x^n + a)) + (3*a*b^2*x^(2*n) - 2*a^2*b*x^n - 8*a^3)*sqrt(b*x^n + a))/(a^3*n*x^(3*n))]
Time = 17.30 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.27 \[ \int x^{-1-3 n} \sqrt {a+b x^n} \, dx=- \frac {a x^{- \frac {7 n}{2}}}{3 \sqrt {b} n \sqrt {\frac {a x^{- n}}{b} + 1}} - \frac {5 \sqrt {b} x^{- \frac {5 n}{2}}}{12 n \sqrt {\frac {a x^{- n}}{b} + 1}} + \frac {b^{\frac {3}{2}} x^{- \frac {3 n}{2}}}{24 a n \sqrt {\frac {a x^{- n}}{b} + 1}} + \frac {b^{\frac {5}{2}} x^{- \frac {n}{2}}}{8 a^{2} n \sqrt {\frac {a x^{- n}}{b} + 1}} - \frac {b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a} x^{- \frac {n}{2}}}{\sqrt {b}} \right )}}{8 a^{\frac {5}{2}} n} \] Input:
integrate(x**(-1-3*n)*(a+b*x**n)**(1/2),x)
Output:
-a/(3*sqrt(b)*n*x**(7*n/2)*sqrt(a/(b*x**n) + 1)) - 5*sqrt(b)/(12*n*x**(5*n /2)*sqrt(a/(b*x**n) + 1)) + b**(3/2)/(24*a*n*x**(3*n/2)*sqrt(a/(b*x**n) + 1)) + b**(5/2)/(8*a**2*n*x**(n/2)*sqrt(a/(b*x**n) + 1)) - b**3*asinh(sqrt( a)/(sqrt(b)*x**(n/2)))/(8*a**(5/2)*n)
\[ \int x^{-1-3 n} \sqrt {a+b x^n} \, dx=\int { \sqrt {b x^{n} + a} x^{-3 \, n - 1} \,d x } \] Input:
integrate(x^(-1-3*n)*(a+b*x^n)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^n + a)*x^(-3*n - 1), x)
\[ \int x^{-1-3 n} \sqrt {a+b x^n} \, dx=\int { \sqrt {b x^{n} + a} x^{-3 \, n - 1} \,d x } \] Input:
integrate(x^(-1-3*n)*(a+b*x^n)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*x^n + a)*x^(-3*n - 1), x)
Timed out. \[ \int x^{-1-3 n} \sqrt {a+b x^n} \, dx=\int \frac {\sqrt {a+b\,x^n}}{x^{3\,n+1}} \,d x \] Input:
int((a + b*x^n)^(1/2)/x^(3*n + 1),x)
Output:
int((a + b*x^n)^(1/2)/x^(3*n + 1), x)
\[ \int x^{-1-3 n} \sqrt {a+b x^n} \, dx=\frac {6 x^{2 n} \sqrt {x^{n} b +a}\, b^{2}-4 x^{n} \sqrt {x^{n} b +a}\, a b -16 \sqrt {x^{n} b +a}\, a^{2}+3 x^{3 n} \left (\int \frac {\sqrt {x^{n} b +a}}{x^{n} b x +a x}d x \right ) b^{3} n}{48 x^{3 n} a^{2} n} \] Input:
int(x^(-1-3*n)*(a+b*x^n)^(1/2),x)
Output:
(6*x**(2*n)*sqrt(x**n*b + a)*b**2 - 4*x**n*sqrt(x**n*b + a)*a*b - 16*sqrt( x**n*b + a)*a**2 + 3*x**(3*n)*int(sqrt(x**n*b + a)/(x**n*b*x + a*x),x)*b** 3*n)/(48*x**(3*n)*a**2*n)