Integrand size = 22, antiderivative size = 82 \[ \int \frac {A+B x^n}{x \left (a+b x^n\right )^{5/2}} \, dx=\frac {2 (A b-a B)}{3 a b n \left (a+b x^n\right )^{3/2}}+\frac {2 A}{a^2 n \sqrt {a+b x^n}}-\frac {2 A \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{5/2} n} \] Output:
2/3*(A*b-B*a)/a/b/n/(a+b*x^n)^(3/2)+2*A/a^2/n/(a+b*x^n)^(1/2)-2*A*arctanh( (a+b*x^n)^(1/2)/a^(1/2))/a^(5/2)/n
Time = 0.15 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x^n}{x \left (a+b x^n\right )^{5/2}} \, dx=-\frac {2 \left (-a A b+a^2 B-3 A b \left (a+b x^n\right )\right )}{3 a^2 b n \left (a+b x^n\right )^{3/2}}-\frac {2 A \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{5/2} n} \] Input:
Integrate[(A + B*x^n)/(x*(a + b*x^n)^(5/2)),x]
Output:
(-2*(-(a*A*b) + a^2*B - 3*A*b*(a + b*x^n)))/(3*a^2*b*n*(a + b*x^n)^(3/2)) - (2*A*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(a^(5/2)*n)
Time = 0.33 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {948, 87, 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.
\(\displaystyle \int \frac {A+B x^n}{x \left (a+b x^n\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {\int \frac {x^{-n} \left (B x^n+A\right )}{\left (b x^n+a\right )^{5/2}}dx^n}{n}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {\frac {A \int \frac {x^{-n}}{\left (b x^n+a\right )^{3/2}}dx^n}{a}+\frac {2 (A b-a B)}{3 a b \left (a+b x^n\right )^{3/2}}}{n}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {\frac {A \left (\frac {\int \frac {x^{-n}}{\sqrt {b x^n+a}}dx^n}{a}+\frac {2}{a \sqrt {a+b x^n}}\right )}{a}+\frac {2 (A b-a B)}{3 a b \left (a+b x^n\right )^{3/2}}}{n}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {A \left (\frac {2 \int \frac {1}{\frac {x^{2 n}}{b}-\frac {a}{b}}d\sqrt {b x^n+a}}{a b}+\frac {2}{a \sqrt {a+b x^n}}\right )}{a}+\frac {2 (A b-a B)}{3 a b \left (a+b x^n\right )^{3/2}}}{n}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {A \left (\frac {2}{a \sqrt {a+b x^n}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{3/2}}\right )}{a}+\frac {2 (A b-a B)}{3 a b \left (a+b x^n\right )^{3/2}}}{n}\) |
Input:
Int[(A + B*x^n)/(x*(a + b*x^n)^(5/2)),x]
Output:
((2*(A*b - a*B))/(3*a*b*(a + b*x^n)^(3/2)) + (A*(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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (-A b +B a \right )}{3 a \left (a +b \,x^{n}\right )^{\frac {3}{2}}}+\frac {2 A b}{a^{2} \sqrt {a +b \,x^{n}}}-\frac {2 A b \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \,x^{n}}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}}{b n}\) | \(68\) |
default | \(\frac {-\frac {2 \left (-A b +B a \right )}{3 a \left (a +b \,x^{n}\right )^{\frac {3}{2}}}+\frac {2 A b}{a^{2} \sqrt {a +b \,x^{n}}}-\frac {2 A b \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \,x^{n}}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}}{b n}\) | \(68\) |
Input:
int((A+B*x^n)/x/(a+b*x^n)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/n/b*(-1/3*(-A*b+B*a)/a/(a+b*x^n)^(3/2)+A*b/a^2/(a+b*x^n)^(1/2)-A*b/a^(5/ 2)*arctanh((a+b*x^n)^(1/2)/a^(1/2)))
Time = 0.11 (sec) , antiderivative size = 268, normalized size of antiderivative = 3.27 \[ \int \frac {A+B x^n}{x \left (a+b x^n\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (A \sqrt {a} b^{3} x^{2 \, n} + 2 \, A a^{\frac {3}{2}} b^{2} x^{n} + A a^{\frac {5}{2}} b\right )} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, {\left (3 \, A a b^{2} x^{n} - B a^{3} + 4 \, A a^{2} b\right )} \sqrt {b x^{n} + a}}{3 \, {\left (a^{3} b^{3} n x^{2 \, n} + 2 \, a^{4} b^{2} n x^{n} + a^{5} b n\right )}}, \frac {2 \, {\left (3 \, {\left (A \sqrt {-a} b^{3} x^{2 \, n} + 2 \, A \sqrt {-a} a b^{2} x^{n} + A \sqrt {-a} a^{2} b\right )} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{n} + a}}\right ) + {\left (3 \, A a b^{2} x^{n} - B a^{3} + 4 \, A a^{2} b\right )} \sqrt {b x^{n} + a}\right )}}{3 \, {\left (a^{3} b^{3} n x^{2 \, n} + 2 \, a^{4} b^{2} n x^{n} + a^{5} b n\right )}}\right ] \] Input:
integrate((A+B*x^n)/x/(a+b*x^n)^(5/2),x, algorithm="fricas")
Output:
[1/3*(3*(A*sqrt(a)*b^3*x^(2*n) + 2*A*a^(3/2)*b^2*x^n + A*a^(5/2)*b)*log((b *x^n - 2*sqrt(b*x^n + a)*sqrt(a) + 2*a)/x^n) + 2*(3*A*a*b^2*x^n - B*a^3 + 4*A*a^2*b)*sqrt(b*x^n + a))/(a^3*b^3*n*x^(2*n) + 2*a^4*b^2*n*x^n + a^5*b*n ), 2/3*(3*(A*sqrt(-a)*b^3*x^(2*n) + 2*A*sqrt(-a)*a*b^2*x^n + A*sqrt(-a)*a^ 2*b)*arctan(sqrt(-a)/sqrt(b*x^n + a)) + (3*A*a*b^2*x^n - B*a^3 + 4*A*a^2*b )*sqrt(b*x^n + a))/(a^3*b^3*n*x^(2*n) + 2*a^4*b^2*n*x^n + a^5*b*n)]
Time = 29.03 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.24 \[ \int \frac {A+B x^n}{x \left (a+b x^n\right )^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {A b}{a^{2} n \sqrt {a + b x^{n}}} + \frac {A b \operatorname {atan}{\left (\frac {\sqrt {a + b x^{n}}}{\sqrt {- a}} \right )}}{a^{2} n \sqrt {- a}} - \frac {- A b + B a}{3 a n \left (a + b x^{n}\right )^{\frac {3}{2}}}\right )}{b} & \text {for}\: b \neq 0 \\\frac {A \log {\left (B x^{n} \right )} + B x^{n}}{a^{\frac {5}{2}} n} & \text {otherwise} \end {cases} \] Input:
integrate((A+B*x**n)/x/(a+b*x**n)**(5/2),x)
Output:
Piecewise((2*(A*b/(a**2*n*sqrt(a + b*x**n)) + A*b*atan(sqrt(a + b*x**n)/sq rt(-a))/(a**2*n*sqrt(-a)) - (-A*b + B*a)/(3*a*n*(a + b*x**n)**(3/2)))/b, N e(b, 0)), ((A*log(B*x**n) + B*x**n)/(a**(5/2)*n), True))
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x^n}{x \left (a+b x^n\right )^{5/2}} \, dx=\frac {1}{3} \, A {\left (\frac {3 \, \log \left (\frac {\sqrt {b x^{n} + a} - \sqrt {a}}{\sqrt {b x^{n} + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}} n} + \frac {2 \, {\left (3 \, b x^{n} + 4 \, a\right )}}{{\left (b x^{n} + a\right )}^{\frac {3}{2}} a^{2} n}\right )} - \frac {2 \, B}{3 \, {\left (b x^{n} + a\right )}^{\frac {3}{2}} b n} \] Input:
integrate((A+B*x^n)/x/(a+b*x^n)^(5/2),x, algorithm="maxima")
Output:
1/3*A*(3*log((sqrt(b*x^n + a) - sqrt(a))/(sqrt(b*x^n + a) + sqrt(a)))/(a^( 5/2)*n) + 2*(3*b*x^n + 4*a)/((b*x^n + a)^(3/2)*a^2*n)) - 2/3*B/((b*x^n + a )^(3/2)*b*n)
\[ \int \frac {A+B x^n}{x \left (a+b x^n\right )^{5/2}} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{\frac {5}{2}} x} \,d x } \] Input:
integrate((A+B*x^n)/x/(a+b*x^n)^(5/2),x, algorithm="giac")
Output:
integrate((B*x^n + A)/((b*x^n + a)^(5/2)*x), x)
Timed out. \[ \int \frac {A+B x^n}{x \left (a+b x^n\right )^{5/2}} \, dx=\int \frac {A+B\,x^n}{x\,{\left (a+b\,x^n\right )}^{5/2}} \,d x \] Input:
int((A + B*x^n)/(x*(a + b*x^n)^(5/2)),x)
Output:
int((A + B*x^n)/(x*(a + b*x^n)^(5/2)), x)
\[ \int \frac {A+B x^n}{x \left (a+b x^n\right )^{5/2}} \, dx=\int \frac {\sqrt {x^{n} b +a}}{x^{2 n} b^{2} x +2 x^{n} a b x +a^{2} x}d x \] Input:
int((A+B*x^n)/x/(a+b*x^n)^(5/2),x)
Output:
int(sqrt(x**n*b + a)/(x**(2*n)*b**2*x + 2*x**n*a*b*x + a**2*x),x)