Integrand size = 21, antiderivative size = 98 \[ \int \frac {x^{-1+\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx=-\frac {3 a x^{n/2} \sqrt {a+b x^n}}{4 b^2 n}+\frac {x^{3 n/2} \sqrt {a+b x^n}}{2 b n}+\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{4 b^{5/2} n} \] Output:
-3/4*a*x^(1/2*n)*(a+b*x^n)^(1/2)/b^2/n+1/2*x^(3/2*n)*(a+b*x^n)^(1/2)/b/n+3 /4*a^2*arctanh(b^(1/2)*x^(1/2*n)/(a+b*x^n)^(1/2))/b^(5/2)/n
Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.02 \[ \int \frac {x^{-1+\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {\sqrt {b} x^{n/2} \left (-3 a^2-a b x^n+2 b^2 x^{2 n}\right )+3 a^{5/2} \sqrt {1+\frac {b x^n}{a}} \text {arcsinh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a}}\right )}{4 b^{5/2} n \sqrt {a+b x^n}} \] Input:
Integrate[x^(-1 + (5*n)/2)/Sqrt[a + b*x^n],x]
Output:
(Sqrt[b]*x^(n/2)*(-3*a^2 - a*b*x^n + 2*b^2*x^(2*n)) + 3*a^(5/2)*Sqrt[1 + ( b*x^n)/a]*ArcSinh[(Sqrt[b]*x^(n/2))/Sqrt[a]])/(4*b^(5/2)*n*Sqrt[a + b*x^n] )
Time = 0.33 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.42, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {880, 252, 252, 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.
\(\displaystyle \int \frac {x^{\frac {5 n}{2}-1}}{\sqrt {a+b x^n}} \, dx\) |
\(\Big \downarrow \) 880 |
\(\displaystyle \frac {2 a^2 \int \frac {x^{2 n}}{\left (b x^n+a\right )^2 \left (1-\frac {b x^n}{b x^n+a}\right )^3}d\frac {x^{n/2}}{\sqrt {b x^n+a}}}{n}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {2 a^2 \left (\frac {x^{3 n/2}}{4 b \left (a+b x^n\right )^{3/2} \left (1-\frac {b x^n}{a+b x^n}\right )^2}-\frac {3 \int \frac {x^n}{\left (b x^n+a\right ) \left (1-\frac {b x^n}{b x^n+a}\right )^2}d\frac {x^{n/2}}{\sqrt {b x^n+a}}}{4 b}\right )}{n}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {2 a^2 \left (\frac {x^{3 n/2}}{4 b \left (a+b x^n\right )^{3/2} \left (1-\frac {b x^n}{a+b x^n}\right )^2}-\frac {3 \left (\frac {x^{n/2}}{2 b \sqrt {a+b x^n} \left (1-\frac {b x^n}{a+b x^n}\right )}-\frac {\int \frac {1}{1-\frac {b x^n}{b x^n+a}}d\frac {x^{n/2}}{\sqrt {b x^n+a}}}{2 b}\right )}{4 b}\right )}{n}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 a^2 \left (\frac {x^{3 n/2}}{4 b \left (a+b x^n\right )^{3/2} \left (1-\frac {b x^n}{a+b x^n}\right )^2}-\frac {3 \left (\frac {x^{n/2}}{2 b \sqrt {a+b x^n} \left (1-\frac {b x^n}{a+b x^n}\right )}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{2 b^{3/2}}\right )}{4 b}\right )}{n}\) |
Input:
Int[x^(-1 + (5*n)/2)/Sqrt[a + b*x^n],x]
Output:
(2*a^2*(x^((3*n)/2)/(4*b*(a + b*x^n)^(3/2)*(1 - (b*x^n)/(a + b*x^n))^2) - (3*(x^(n/2)/(2*b*Sqrt[a + b*x^n]*(1 - (b*x^n)/(a + b*x^n))) - ArcTanh[(Sqr t[b]*x^(n/2))/Sqrt[a + b*x^n]]/(2*b^(3/2))))/(4*b)))/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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi nator[p]}, Simp[k*(a^(p + Simplify[(m + 1)/n])/n) Subst[Int[x^(k*Simplify [(m + 1)/n] - 1)/(1 - b*x^k)^(p + Simplify[(m + 1)/n] + 1), x], x, x^(n/k)/ (a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[p + Simpli fy[(m + 1)/n]] && LtQ[-1, p, 0]
Time = 0.77 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.84
method | result | size |
risch | \(-\frac {{\mathrm e}^{\frac {n \ln \left (x \right )}{2}} \left (-2 b \,{\mathrm e}^{n \ln \left (x \right )}+3 a \right ) \sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}}}{4 b^{2} n}+\frac {3 a^{2} \ln \left (\sqrt {b}\, {\mathrm e}^{\frac {n \ln \left (x \right )}{2}}+\sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}}\right )}{4 b^{\frac {5}{2}} n}\) | \(82\) |
Input:
int(x^(-1+5/2*n)/(a+b*x^n)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/4*exp(1/2*n*ln(x))*(-2*b*exp(1/2*n*ln(x))^2+3*a)*(a+b*exp(1/2*n*ln(x))^ 2)^(1/2)/b^2/n+3/4*a^2/b^(5/2)/n*ln(b^(1/2)*exp(1/2*n*ln(x))+(a+b*exp(1/2* n*ln(x))^2)^(1/2))
Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.58 \[ \int \frac {x^{-1+\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx=\left [\frac {3 \, a^{2} \sqrt {b} \log \left (-2 \, \sqrt {b x^{n} + a} \sqrt {b} x^{\frac {1}{2} \, n} - 2 \, b x^{n} - a\right ) + 2 \, {\left (2 \, b^{2} x^{\frac {3}{2} \, n} - 3 \, a b x^{\frac {1}{2} \, n}\right )} \sqrt {b x^{n} + a}}{8 \, b^{3} n}, -\frac {3 \, a^{2} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-b}}{b x^{\frac {1}{2} \, n}}\right ) - {\left (2 \, b^{2} x^{\frac {3}{2} \, n} - 3 \, a b x^{\frac {1}{2} \, n}\right )} \sqrt {b x^{n} + a}}{4 \, b^{3} n}\right ] \] Input:
integrate(x^(-1+5/2*n)/(a+b*x^n)^(1/2),x, algorithm="fricas")
Output:
[1/8*(3*a^2*sqrt(b)*log(-2*sqrt(b*x^n + a)*sqrt(b)*x^(1/2*n) - 2*b*x^n - a ) + 2*(2*b^2*x^(3/2*n) - 3*a*b*x^(1/2*n))*sqrt(b*x^n + a))/(b^3*n), -1/4*( 3*a^2*sqrt(-b)*arctan(sqrt(b*x^n + a)*sqrt(-b)/(b*x^(1/2*n))) - (2*b^2*x^( 3/2*n) - 3*a*b*x^(1/2*n))*sqrt(b*x^n + a))/(b^3*n)]
Time = 3.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.18 \[ \int \frac {x^{-1+\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx=- \frac {3 a^{\frac {3}{2}} x^{\frac {n}{2}}}{4 b^{2} n \sqrt {1 + \frac {b x^{n}}{a}}} - \frac {\sqrt {a} x^{\frac {3 n}{2}}}{4 b n \sqrt {1 + \frac {b x^{n}}{a}}} + \frac {3 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{4 b^{\frac {5}{2}} n} + \frac {x^{\frac {5 n}{2}}}{2 \sqrt {a} n \sqrt {1 + \frac {b x^{n}}{a}}} \] Input:
integrate(x**(-1+5/2*n)/(a+b*x**n)**(1/2),x)
Output:
-3*a**(3/2)*x**(n/2)/(4*b**2*n*sqrt(1 + b*x**n/a)) - sqrt(a)*x**(3*n/2)/(4 *b*n*sqrt(1 + b*x**n/a)) + 3*a**2*asinh(sqrt(b)*x**(n/2)/sqrt(a))/(4*b**(5 /2)*n) + x**(5*n/2)/(2*sqrt(a)*n*sqrt(1 + b*x**n/a))
\[ \int \frac {x^{-1+\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int { \frac {x^{\frac {5}{2} \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \] Input:
integrate(x^(-1+5/2*n)/(a+b*x^n)^(1/2),x, algorithm="maxima")
Output:
integrate(x^(5/2*n - 1)/sqrt(b*x^n + a), x)
\[ \int \frac {x^{-1+\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int { \frac {x^{\frac {5}{2} \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \] Input:
integrate(x^(-1+5/2*n)/(a+b*x^n)^(1/2),x, algorithm="giac")
Output:
integrate(x^(5/2*n - 1)/sqrt(b*x^n + a), x)
Timed out. \[ \int \frac {x^{-1+\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int \frac {x^{\frac {5\,n}{2}-1}}{\sqrt {a+b\,x^n}} \,d x \] Input:
int(x^((5*n)/2 - 1)/(a + b*x^n)^(1/2),x)
Output:
int(x^((5*n)/2 - 1)/(a + b*x^n)^(1/2), x)
\[ \int \frac {x^{-1+\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int \frac {x^{\frac {5 n}{2}} \sqrt {x^{n} b +a}}{x^{n} b x +a x}d x \] Input:
int(x^(-1+5/2*n)/(a+b*x^n)^(1/2),x)
Output:
int((x**((5*n)/2)*sqrt(x**n*b + a))/(x**n*b*x + a*x),x)