Integrand size = 15, antiderivative size = 69 \[ \int \frac {1}{x \left (a+b x^n\right )^{5/2}} \, dx=\frac {2}{3 a n \left (a+b x^n\right )^{3/2}}+\frac {2}{a^2 n \sqrt {a+b x^n}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{5/2} n} \] Output:
2/3/a/n/(a+b*x^n)^(3/2)+2/a^2/n/(a+b*x^n)^(1/2)-2*arctanh((a+b*x^n)^(1/2)/ a^(1/2))/a^(5/2)/n
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x \left (a+b x^n\right )^{5/2}} \, dx=\frac {2 \left (a+3 \left (a+b x^n\right )\right )}{3 a^2 n \left (a+b x^n\right )^{3/2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{5/2} n} \] Input:
Integrate[1/(x*(a + b*x^n)^(5/2)),x]
Output:
(2*(a + 3*(a + b*x^n)))/(3*a^2*n*(a + b*x^n)^(3/2)) - (2*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(a^(5/2)*n)
Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {798, 61, 61, 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.
\(\displaystyle \int \frac {1}{x \left (a+b x^n\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int \frac {x^{-n}}{\left (b x^n+a\right )^{5/2}}dx^n}{n}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {\frac {\int \frac {x^{-n}}{\left (b x^n+a\right )^{3/2}}dx^n}{a}+\frac {2}{3 a \left (a+b x^n\right )^{3/2}}}{n}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {\frac {\frac {\int \frac {x^{-n}}{\sqrt {b x^n+a}}dx^n}{a}+\frac {2}{a \sqrt {a+b x^n}}}{a}+\frac {2}{3 a \left (a+b x^n\right )^{3/2}}}{n}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {\frac {2 \int \frac {1}{\frac {x^{2 n}}{b}-\frac {a}{b}}d\sqrt {b x^n+a}}{a b}+\frac {2}{a \sqrt {a+b x^n}}}{a}+\frac {2}{3 a \left (a+b x^n\right )^{3/2}}}{n}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {\frac {2}{a \sqrt {a+b x^n}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{3/2}}}{a}+\frac {2}{3 a \left (a+b x^n\right )^{3/2}}}{n}\) |
Input:
Int[1/(x*(a + b*x^n)^(5/2)),x]
Output:
(2/(3*a*(a + b*x^n)^(3/2)) + (2/(a*Sqrt[a + b*x^n]) - (2*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/a^(3/2))/a)/n
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] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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)*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]]
Time = 0.54 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(\frac {-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \,x^{n}}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}+\frac {2}{a^{2} \sqrt {a +b \,x^{n}}}+\frac {2}{3 a \left (a +b \,x^{n}\right )^{\frac {3}{2}}}}{n}\) | \(53\) |
default | \(\frac {-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \,x^{n}}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}+\frac {2}{a^{2} \sqrt {a +b \,x^{n}}}+\frac {2}{3 a \left (a +b \,x^{n}\right )^{\frac {3}{2}}}}{n}\) | \(53\) |
Input:
int(1/x/(a+b*x^n)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/n*(-2/a^(5/2)*arctanh((a+b*x^n)^(1/2)/a^(1/2))+2/a^2/(a+b*x^n)^(1/2)+2/3 /a/(a+b*x^n)^(3/2))
Time = 0.09 (sec) , antiderivative size = 227, normalized size of antiderivative = 3.29 \[ \int \frac {1}{x \left (a+b x^n\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (\sqrt {a} b^{2} x^{2 \, n} + 2 \, a^{\frac {3}{2}} b x^{n} + a^{\frac {5}{2}}\right )} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, {\left (3 \, a b x^{n} + 4 \, a^{2}\right )} \sqrt {b x^{n} + a}}{3 \, {\left (a^{3} b^{2} n x^{2 \, n} + 2 \, a^{4} b n x^{n} + a^{5} n\right )}}, \frac {2 \, {\left (3 \, {\left (\sqrt {-a} b^{2} x^{2 \, n} + 2 \, \sqrt {-a} a b x^{n} + \sqrt {-a} a^{2}\right )} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{n} + a}}\right ) + {\left (3 \, a b x^{n} + 4 \, a^{2}\right )} \sqrt {b x^{n} + a}\right )}}{3 \, {\left (a^{3} b^{2} n x^{2 \, n} + 2 \, a^{4} b n x^{n} + a^{5} n\right )}}\right ] \] Input:
integrate(1/x/(a+b*x^n)^(5/2),x, algorithm="fricas")
Output:
[1/3*(3*(sqrt(a)*b^2*x^(2*n) + 2*a^(3/2)*b*x^n + a^(5/2))*log((b*x^n - 2*s qrt(b*x^n + a)*sqrt(a) + 2*a)/x^n) + 2*(3*a*b*x^n + 4*a^2)*sqrt(b*x^n + a) )/(a^3*b^2*n*x^(2*n) + 2*a^4*b*n*x^n + a^5*n), 2/3*(3*(sqrt(-a)*b^2*x^(2*n ) + 2*sqrt(-a)*a*b*x^n + sqrt(-a)*a^2)*arctan(sqrt(-a)/sqrt(b*x^n + a)) + (3*a*b*x^n + 4*a^2)*sqrt(b*x^n + a))/(a^3*b^2*n*x^(2*n) + 2*a^4*b*n*x^n + a^5*n)]
Leaf count of result is larger than twice the leaf count of optimal. 860 vs. \(2 (58) = 116\).
Time = 2.86 (sec) , antiderivative size = 860, normalized size of antiderivative = 12.46 \[ \int \frac {1}{x \left (a+b x^n\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate(1/x/(a+b*x**n)**(5/2),x)
Output:
8*a**7*sqrt(1 + b*x**n/a)/(3*a**(19/2)*n + 9*a**(17/2)*b*n*x**n + 9*a**(15 /2)*b**2*n*x**(2*n) + 3*a**(13/2)*b**3*n*x**(3*n)) + 3*a**7*log(b*x**n/a)/ (3*a**(19/2)*n + 9*a**(17/2)*b*n*x**n + 9*a**(15/2)*b**2*n*x**(2*n) + 3*a* *(13/2)*b**3*n*x**(3*n)) - 6*a**7*log(sqrt(1 + b*x**n/a) + 1)/(3*a**(19/2) *n + 9*a**(17/2)*b*n*x**n + 9*a**(15/2)*b**2*n*x**(2*n) + 3*a**(13/2)*b**3 *n*x**(3*n)) + 14*a**6*b*x**n*sqrt(1 + b*x**n/a)/(3*a**(19/2)*n + 9*a**(17 /2)*b*n*x**n + 9*a**(15/2)*b**2*n*x**(2*n) + 3*a**(13/2)*b**3*n*x**(3*n)) + 9*a**6*b*x**n*log(b*x**n/a)/(3*a**(19/2)*n + 9*a**(17/2)*b*n*x**n + 9*a* *(15/2)*b**2*n*x**(2*n) + 3*a**(13/2)*b**3*n*x**(3*n)) - 18*a**6*b*x**n*lo g(sqrt(1 + b*x**n/a) + 1)/(3*a**(19/2)*n + 9*a**(17/2)*b*n*x**n + 9*a**(15 /2)*b**2*n*x**(2*n) + 3*a**(13/2)*b**3*n*x**(3*n)) + 6*a**5*b**2*x**(2*n)* sqrt(1 + b*x**n/a)/(3*a**(19/2)*n + 9*a**(17/2)*b*n*x**n + 9*a**(15/2)*b** 2*n*x**(2*n) + 3*a**(13/2)*b**3*n*x**(3*n)) + 9*a**5*b**2*x**(2*n)*log(b*x **n/a)/(3*a**(19/2)*n + 9*a**(17/2)*b*n*x**n + 9*a**(15/2)*b**2*n*x**(2*n) + 3*a**(13/2)*b**3*n*x**(3*n)) - 18*a**5*b**2*x**(2*n)*log(sqrt(1 + b*x** n/a) + 1)/(3*a**(19/2)*n + 9*a**(17/2)*b*n*x**n + 9*a**(15/2)*b**2*n*x**(2 *n) + 3*a**(13/2)*b**3*n*x**(3*n)) + 3*a**4*b**3*x**(3*n)*log(b*x**n/a)/(3 *a**(19/2)*n + 9*a**(17/2)*b*n*x**n + 9*a**(15/2)*b**2*n*x**(2*n) + 3*a**( 13/2)*b**3*n*x**(3*n)) - 6*a**4*b**3*x**(3*n)*log(sqrt(1 + b*x**n/a) + 1)/ (3*a**(19/2)*n + 9*a**(17/2)*b*n*x**n + 9*a**(15/2)*b**2*n*x**(2*n) + 3...
Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x \left (a+b x^n\right )^{5/2}} \, dx=\frac {\log \left (\frac {\sqrt {b x^{n} + a} - \sqrt {a}}{\sqrt {b x^{n} + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}} n} + \frac {2 \, {\left (3 \, b x^{n} + 4 \, a\right )}}{3 \, {\left (b x^{n} + a\right )}^{\frac {3}{2}} a^{2} n} \] Input:
integrate(1/x/(a+b*x^n)^(5/2),x, algorithm="maxima")
Output:
log((sqrt(b*x^n + a) - sqrt(a))/(sqrt(b*x^n + a) + sqrt(a)))/(a^(5/2)*n) + 2/3*(3*b*x^n + 4*a)/((b*x^n + a)^(3/2)*a^2*n)
\[ \int \frac {1}{x \left (a+b x^n\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{\frac {5}{2}} x} \,d x } \] Input:
integrate(1/x/(a+b*x^n)^(5/2),x, algorithm="giac")
Output:
integrate(1/((b*x^n + a)^(5/2)*x), x)
Timed out. \[ \int \frac {1}{x \left (a+b x^n\right )^{5/2}} \, dx=\int \frac {1}{x\,{\left (a+b\,x^n\right )}^{5/2}} \,d x \] Input:
int(1/(x*(a + b*x^n)^(5/2)),x)
Output:
int(1/(x*(a + b*x^n)^(5/2)), x)
\[ \int \frac {1}{x \left (a+b x^n\right )^{5/2}} \, dx=\int \frac {\sqrt {x^{n} b +a}}{x^{3 n} b^{3} x +3 x^{2 n} a \,b^{2} x +3 x^{n} a^{2} b x +a^{3} x}d x \] Input:
int(1/x/(a+b*x^n)^(5/2),x)
Output:
int(sqrt(x**n*b + a)/(x**(3*n)*b**3*x + 3*x**(2*n)*a*b**2*x + 3*x**n*a**2* b*x + a**3*x),x)