Integrand size = 19, antiderivative size = 89 \[ \int \frac {x^{-1+\frac {n}{2}}}{\left (a+b x^n\right )^3} \, dx=\frac {x^{n/2}}{2 a n \left (a+b x^n\right )^2}+\frac {3 x^{n/2}}{4 a^2 n \left (a+b x^n\right )}+\frac {3 \arctan \left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b} n} \] Output:
1/2*x^(1/2*n)/a/n/(a+b*x^n)^2+3/4*x^(1/2*n)/a^2/n/(a+b*x^n)+3/4*arctan(1/a ^(1/2)*b^(1/2)*x^(1/2*n))/a^(5/2)/b^(1/2)/n
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.36 \[ \int \frac {x^{-1+\frac {n}{2}}}{\left (a+b x^n\right )^3} \, dx=\frac {2 x^{n/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3,\frac {3}{2},-\frac {b x^n}{a}\right )}{a^3 n} \] Input:
Integrate[x^(-1 + n/2)/(a + b*x^n)^3,x]
Output:
(2*x^(n/2)*Hypergeometric2F1[1/2, 3, 3/2, -((b*x^n)/a)])/(a^3*n)
Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {868, 215, 215, 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.
\(\displaystyle \int \frac {x^{\frac {n}{2}-1}}{\left (a+b x^n\right )^3} \, dx\) |
\(\Big \downarrow \) 868 |
\(\displaystyle \frac {2 \int \frac {1}{\left (b x^n+a\right )^3}dx^{n/2}}{n}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {2 \left (\frac {3 \int \frac {1}{\left (b x^n+a\right )^2}dx^{n/2}}{4 a}+\frac {x^{n/2}}{4 a \left (a+b x^n\right )^2}\right )}{n}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {2 \left (\frac {3 \left (\frac {\int \frac {1}{b x^n+a}dx^{n/2}}{2 a}+\frac {x^{n/2}}{2 a \left (a+b x^n\right )}\right )}{4 a}+\frac {x^{n/2}}{4 a \left (a+b x^n\right )^2}\right )}{n}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2 \left (\frac {3 \left (\frac {\arctan \left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {x^{n/2}}{2 a \left (a+b x^n\right )}\right )}{4 a}+\frac {x^{n/2}}{4 a \left (a+b x^n\right )^2}\right )}{n}\) |
Input:
Int[x^(-1 + n/2)/(a + b*x^n)^3,x]
Output:
(2*(x^(n/2)/(4*a*(a + b*x^n)^2) + (3*(x^(n/2)/(2*a*(a + b*x^n)) + ArcTan[( Sqrt[b]*x^(n/2))/Sqrt[a]]/(2*a^(3/2)*Sqrt[b])))/(4*a)))/n
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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/(m + 1) Subst[Int[(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[ {a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
Time = 0.89 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.13
method | result | size |
risch | \(\frac {x^{\frac {n}{2}} \left (3 b \,x^{n}+5 a \right )}{4 a^{2} n \left (a +b \,x^{n}\right )^{2}}-\frac {3 \ln \left (x^{\frac {n}{2}}-\frac {a}{\sqrt {-a b}}\right )}{8 \sqrt {-a b}\, n \,a^{2}}+\frac {3 \ln \left (x^{\frac {n}{2}}+\frac {a}{\sqrt {-a b}}\right )}{8 \sqrt {-a b}\, n \,a^{2}}\) | \(101\) |
Input:
int(x^(-1+1/2*n)/(a+b*x^n)^3,x,method=_RETURNVERBOSE)
Output:
1/4*x^(1/2*n)*(3*b*(x^(1/2*n))^2+5*a)/a^2/n/(a+b*(x^(1/2*n))^2)^2-3/8/(-a* b)^(1/2)/n/a^2*ln(x^(1/2*n)-1/(-a*b)^(1/2)*a)+3/8/(-a*b)^(1/2)/n/a^2*ln(x^ (1/2*n)+1/(-a*b)^(1/2)*a)
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (69) = 138\).
Time = 0.09 (sec) , antiderivative size = 318, normalized size of antiderivative = 3.57 \[ \int \frac {x^{-1+\frac {n}{2}}}{\left (a+b x^n\right )^3} \, dx=\left [\frac {6 \, a b^{2} x^{3} x^{\frac {3}{2} \, n - 3} + 10 \, a^{2} b x x^{\frac {1}{2} \, n - 1} - 3 \, {\left (\sqrt {-a b} b^{2} x^{4} x^{2 \, n - 4} + 2 \, \sqrt {-a b} a b x^{2} x^{n - 2} + \sqrt {-a b} a^{2}\right )} \log \left (\frac {b x^{2} x^{n - 2} - 2 \, \sqrt {-a b} x x^{\frac {1}{2} \, n - 1} - a}{b x^{2} x^{n - 2} + a}\right )}{8 \, {\left (a^{3} b^{3} n x^{4} x^{2 \, n - 4} + 2 \, a^{4} b^{2} n x^{2} x^{n - 2} + a^{5} b n\right )}}, \frac {3 \, a b^{2} x^{3} x^{\frac {3}{2} \, n - 3} + 5 \, a^{2} b x x^{\frac {1}{2} \, n - 1} + 3 \, {\left (\sqrt {a b} b^{2} x^{4} x^{2 \, n - 4} + 2 \, \sqrt {a b} a b x^{2} x^{n - 2} + \sqrt {a b} a^{2}\right )} \arctan \left (\frac {\sqrt {a b} x x^{\frac {1}{2} \, n - 1}}{a}\right )}{4 \, {\left (a^{3} b^{3} n x^{4} x^{2 \, n - 4} + 2 \, a^{4} b^{2} n x^{2} x^{n - 2} + a^{5} b n\right )}}\right ] \] Input:
integrate(x^(-1+1/2*n)/(a+b*x^n)^3,x, algorithm="fricas")
Output:
[1/8*(6*a*b^2*x^3*x^(3/2*n - 3) + 10*a^2*b*x*x^(1/2*n - 1) - 3*(sqrt(-a*b) *b^2*x^4*x^(2*n - 4) + 2*sqrt(-a*b)*a*b*x^2*x^(n - 2) + sqrt(-a*b)*a^2)*lo g((b*x^2*x^(n - 2) - 2*sqrt(-a*b)*x*x^(1/2*n - 1) - a)/(b*x^2*x^(n - 2) + a)))/(a^3*b^3*n*x^4*x^(2*n - 4) + 2*a^4*b^2*n*x^2*x^(n - 2) + a^5*b*n), 1/ 4*(3*a*b^2*x^3*x^(3/2*n - 3) + 5*a^2*b*x*x^(1/2*n - 1) + 3*(sqrt(a*b)*b^2* x^4*x^(2*n - 4) + 2*sqrt(a*b)*a*b*x^2*x^(n - 2) + sqrt(a*b)*a^2)*arctan(sq rt(a*b)*x*x^(1/2*n - 1)/a))/(a^3*b^3*n*x^4*x^(2*n - 4) + 2*a^4*b^2*n*x^2*x ^(n - 2) + a^5*b*n)]
Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (70) = 140\).
Time = 3.18 (sec) , antiderivative size = 683, normalized size of antiderivative = 7.67 \[ \int \frac {x^{-1+\frac {n}{2}}}{\left (a+b x^n\right )^3} \, dx=\frac {3 a^{\frac {11}{2}} x^{\frac {n}{2}} \operatorname {atan}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{4 a^{8} \sqrt {b} n x^{\frac {n}{2}} + 12 a^{7} b^{\frac {3}{2}} n x^{\frac {3 n}{2}} + 12 a^{6} b^{\frac {5}{2}} n x^{\frac {5 n}{2}} + 4 a^{5} b^{\frac {7}{2}} n x^{\frac {7 n}{2}}} + \frac {9 a^{\frac {9}{2}} b x^{\frac {3 n}{2}} \operatorname {atan}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{4 a^{8} \sqrt {b} n x^{\frac {n}{2}} + 12 a^{7} b^{\frac {3}{2}} n x^{\frac {3 n}{2}} + 12 a^{6} b^{\frac {5}{2}} n x^{\frac {5 n}{2}} + 4 a^{5} b^{\frac {7}{2}} n x^{\frac {7 n}{2}}} + \frac {9 a^{\frac {7}{2}} b^{2} x^{\frac {5 n}{2}} \operatorname {atan}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{4 a^{8} \sqrt {b} n x^{\frac {n}{2}} + 12 a^{7} b^{\frac {3}{2}} n x^{\frac {3 n}{2}} + 12 a^{6} b^{\frac {5}{2}} n x^{\frac {5 n}{2}} + 4 a^{5} b^{\frac {7}{2}} n x^{\frac {7 n}{2}}} + \frac {3 a^{\frac {5}{2}} b^{3} x^{\frac {7 n}{2}} \operatorname {atan}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{4 a^{8} \sqrt {b} n x^{\frac {n}{2}} + 12 a^{7} b^{\frac {3}{2}} n x^{\frac {3 n}{2}} + 12 a^{6} b^{\frac {5}{2}} n x^{\frac {5 n}{2}} + 4 a^{5} b^{\frac {7}{2}} n x^{\frac {7 n}{2}}} + \frac {5 a^{5} \sqrt {b} x^{n}}{4 a^{8} \sqrt {b} n x^{\frac {n}{2}} + 12 a^{7} b^{\frac {3}{2}} n x^{\frac {3 n}{2}} + 12 a^{6} b^{\frac {5}{2}} n x^{\frac {5 n}{2}} + 4 a^{5} b^{\frac {7}{2}} n x^{\frac {7 n}{2}}} + \frac {8 a^{4} b^{\frac {3}{2}} x^{2 n}}{4 a^{8} \sqrt {b} n x^{\frac {n}{2}} + 12 a^{7} b^{\frac {3}{2}} n x^{\frac {3 n}{2}} + 12 a^{6} b^{\frac {5}{2}} n x^{\frac {5 n}{2}} + 4 a^{5} b^{\frac {7}{2}} n x^{\frac {7 n}{2}}} + \frac {3 a^{3} b^{\frac {5}{2}} x^{3 n}}{4 a^{8} \sqrt {b} n x^{\frac {n}{2}} + 12 a^{7} b^{\frac {3}{2}} n x^{\frac {3 n}{2}} + 12 a^{6} b^{\frac {5}{2}} n x^{\frac {5 n}{2}} + 4 a^{5} b^{\frac {7}{2}} n x^{\frac {7 n}{2}}} \] Input:
integrate(x**(-1+1/2*n)/(a+b*x**n)**3,x)
Output:
3*a**(11/2)*x**(n/2)*atan(sqrt(b)*x**(n/2)/sqrt(a))/(4*a**8*sqrt(b)*n*x**( n/2) + 12*a**7*b**(3/2)*n*x**(3*n/2) + 12*a**6*b**(5/2)*n*x**(5*n/2) + 4*a **5*b**(7/2)*n*x**(7*n/2)) + 9*a**(9/2)*b*x**(3*n/2)*atan(sqrt(b)*x**(n/2) /sqrt(a))/(4*a**8*sqrt(b)*n*x**(n/2) + 12*a**7*b**(3/2)*n*x**(3*n/2) + 12* a**6*b**(5/2)*n*x**(5*n/2) + 4*a**5*b**(7/2)*n*x**(7*n/2)) + 9*a**(7/2)*b* *2*x**(5*n/2)*atan(sqrt(b)*x**(n/2)/sqrt(a))/(4*a**8*sqrt(b)*n*x**(n/2) + 12*a**7*b**(3/2)*n*x**(3*n/2) + 12*a**6*b**(5/2)*n*x**(5*n/2) + 4*a**5*b** (7/2)*n*x**(7*n/2)) + 3*a**(5/2)*b**3*x**(7*n/2)*atan(sqrt(b)*x**(n/2)/sqr t(a))/(4*a**8*sqrt(b)*n*x**(n/2) + 12*a**7*b**(3/2)*n*x**(3*n/2) + 12*a**6 *b**(5/2)*n*x**(5*n/2) + 4*a**5*b**(7/2)*n*x**(7*n/2)) + 5*a**5*sqrt(b)*x* *n/(4*a**8*sqrt(b)*n*x**(n/2) + 12*a**7*b**(3/2)*n*x**(3*n/2) + 12*a**6*b* *(5/2)*n*x**(5*n/2) + 4*a**5*b**(7/2)*n*x**(7*n/2)) + 8*a**4*b**(3/2)*x**( 2*n)/(4*a**8*sqrt(b)*n*x**(n/2) + 12*a**7*b**(3/2)*n*x**(3*n/2) + 12*a**6* b**(5/2)*n*x**(5*n/2) + 4*a**5*b**(7/2)*n*x**(7*n/2)) + 3*a**3*b**(5/2)*x* *(3*n)/(4*a**8*sqrt(b)*n*x**(n/2) + 12*a**7*b**(3/2)*n*x**(3*n/2) + 12*a** 6*b**(5/2)*n*x**(5*n/2) + 4*a**5*b**(7/2)*n*x**(7*n/2))
\[ \int \frac {x^{-1+\frac {n}{2}}}{\left (a+b x^n\right )^3} \, dx=\int { \frac {x^{\frac {1}{2} \, n - 1}}{{\left (b x^{n} + a\right )}^{3}} \,d x } \] Input:
integrate(x^(-1+1/2*n)/(a+b*x^n)^3,x, algorithm="maxima")
Output:
1/4*(3*b*x^(3/2*n) + 5*a*x^(1/2*n))/(a^2*b^2*n*x^(2*n) + 2*a^3*b*n*x^n + a ^4*n) + 3*integrate(1/8*x^(1/2*n)/(a^2*b*x*x^n + a^3*x), x)
Time = 0.14 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.70 \[ \int \frac {x^{-1+\frac {n}{2}}}{\left (a+b x^n\right )^3} \, dx=\frac {\frac {3 \, \arctan \left (\frac {b \sqrt {x^{n}}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, b x^{\frac {1}{2} \, n} x^{n} + 5 \, a \sqrt {x^{n}}}{{\left (b x^{n} + a\right )}^{2} a^{2}}}{4 \, n} \] Input:
integrate(x^(-1+1/2*n)/(a+b*x^n)^3,x, algorithm="giac")
Output:
1/4*(3*arctan(b*sqrt(x^n)/sqrt(a*b))/(sqrt(a*b)*a^2) + (3*b*x^(1/2*n)*x^n + 5*a*sqrt(x^n))/((b*x^n + a)^2*a^2))/n
Timed out. \[ \int \frac {x^{-1+\frac {n}{2}}}{\left (a+b x^n\right )^3} \, dx=\int \frac {x^{\frac {n}{2}-1}}{{\left (a+b\,x^n\right )}^3} \,d x \] Input:
int(x^(n/2 - 1)/(a + b*x^n)^3,x)
Output:
int(x^(n/2 - 1)/(a + b*x^n)^3, x)
\[ \int \frac {x^{-1+\frac {n}{2}}}{\left (a+b x^n\right )^3} \, dx=\int \frac {x^{\frac {n}{2}}}{x^{3 n} b^{3} x +3 x^{2 n} a \,b^{2} x +3 x^{n} a^{2} b x +a^{3} x}d x \] Input:
int(x^(-1+1/2*n)/(a+b*x^n)^3,x)
Output:
int(x**(n/2)/(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)