Integrand size = 15, antiderivative size = 96 \[ \int \left (a+b x^n\right )^{-3-\frac {1}{n}} \, dx=\frac {x \left (a+b x^n\right )^{-2-\frac {1}{n}}}{a (1+2 n)}+\frac {2 n x \left (a+b x^n\right )^{-1-\frac {1}{n}}}{a^2 (1+n) (1+2 n)}+\frac {2 n^2 x \left (a+b x^n\right )^{-1/n}}{a^3 (1+n) (1+2 n)} \]
x*(a+b*x^n)^(-2-1/n)/a/(1+2*n)+2*n*x*(a+b*x^n)^(-1-1/n)/a^2/(2*n^2+3*n+1)+ 2*n^2*x/a^3/(2*n^2+3*n+1)/((a+b*x^n)^(1/n))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.57 \[ \int \left (a+b x^n\right )^{-3-\frac {1}{n}} \, dx=\frac {x \left (a+b x^n\right )^{-1/n} \left (1+\frac {b x^n}{a}\right )^{\frac {1}{n}} \operatorname {Hypergeometric2F1}\left (3+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^3} \]
(x*(1 + (b*x^n)/a)^n^(-1)*Hypergeometric2F1[3 + n^(-1), n^(-1), 1 + n^(-1) , -((b*x^n)/a)])/(a^3*(a + b*x^n)^n^(-1))
Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {777, 777, 746}
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.
\(\displaystyle \int \left (a+b x^n\right )^{-\frac {1}{n}-3} \, dx\) |
\(\Big \downarrow \) 777 |
\(\displaystyle \frac {2 n \int \left (b x^n+a\right )^{-2-\frac {1}{n}}dx}{a (2 n+1)}+\frac {x \left (a+b x^n\right )^{-\frac {1}{n}-2}}{a (2 n+1)}\) |
\(\Big \downarrow \) 777 |
\(\displaystyle \frac {2 n \left (\frac {n \int \left (b x^n+a\right )^{-1-\frac {1}{n}}dx}{a (n+1)}+\frac {x \left (a+b x^n\right )^{-\frac {1}{n}-1}}{a (n+1)}\right )}{a (2 n+1)}+\frac {x \left (a+b x^n\right )^{-\frac {1}{n}-2}}{a (2 n+1)}\) |
\(\Big \downarrow \) 746 |
\(\displaystyle \frac {2 n \left (\frac {n x \left (a+b x^n\right )^{-1/n}}{a^2 (n+1)}+\frac {x \left (a+b x^n\right )^{-\frac {1}{n}-1}}{a (n+1)}\right )}{a (2 n+1)}+\frac {x \left (a+b x^n\right )^{-\frac {1}{n}-2}}{a (2 n+1)}\) |
(x*(a + b*x^n)^(-2 - n^(-1)))/(a*(1 + 2*n)) + (2*n*((x*(a + b*x^n)^(-1 - n ^(-1)))/(a*(1 + n)) + (n*x)/(a^2*(1 + n)*(a + b*x^n)^n^(-1))))/(a*(1 + 2*n ))
3.28.6.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1) /a), x] /; FreeQ[{a, b, n, p}, x] && EqQ[1/n + p + 1, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1)) Int[(a + b*x^ n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1], 0] && NeQ[p, -1]
\[\int \left (a +b \,x^{n}\right )^{-3-\frac {1}{n}}d x\]
Time = 0.31 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.31 \[ \int \left (a+b x^n\right )^{-3-\frac {1}{n}} \, dx=\frac {2 \, b^{3} n^{2} x x^{3 \, n} + 2 \, {\left (3 \, a b^{2} n^{2} + a b^{2} n\right )} x x^{2 \, n} + {\left (6 \, a^{2} b n^{2} + 5 \, a^{2} b n + a^{2} b\right )} x x^{n} + {\left (2 \, a^{3} n^{2} + 3 \, a^{3} n + a^{3}\right )} x}{{\left (2 \, a^{3} n^{2} + 3 \, a^{3} n + a^{3}\right )} {\left (b x^{n} + a\right )}^{\frac {3 \, n + 1}{n}}} \]
(2*b^3*n^2*x*x^(3*n) + 2*(3*a*b^2*n^2 + a*b^2*n)*x*x^(2*n) + (6*a^2*b*n^2 + 5*a^2*b*n + a^2*b)*x*x^n + (2*a^3*n^2 + 3*a^3*n + a^3)*x)/((2*a^3*n^2 + 3*a^3*n + a^3)*(b*x^n + a)^((3*n + 1)/n))
Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (83) = 166\).
Time = 0.93 (sec) , antiderivative size = 695, normalized size of antiderivative = 7.24 \[ \int \left (a+b x^n\right )^{-3-\frac {1}{n}} \, dx=\frac {2 a^{2} a^{\frac {1}{n}} a^{-3 - \frac {2}{n}} n^{2} x \Gamma \left (\frac {1}{n}\right )}{a^{2} n^{3} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right ) + 2 a b n^{3} x^{n} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right ) + b^{2} n^{3} x^{2 n} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right )} + \frac {3 a^{2} a^{\frac {1}{n}} a^{-3 - \frac {2}{n}} n x \Gamma \left (\frac {1}{n}\right )}{a^{2} n^{3} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right ) + 2 a b n^{3} x^{n} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right ) + b^{2} n^{3} x^{2 n} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right )} + \frac {a^{2} a^{\frac {1}{n}} a^{-3 - \frac {2}{n}} x \Gamma \left (\frac {1}{n}\right )}{a^{2} n^{3} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right ) + 2 a b n^{3} x^{n} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right ) + b^{2} n^{3} x^{2 n} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right )} + \frac {4 a a^{\frac {1}{n}} a^{-3 - \frac {2}{n}} b n^{2} x x^{n} \Gamma \left (\frac {1}{n}\right )}{a^{2} n^{3} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right ) + 2 a b n^{3} x^{n} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right ) + b^{2} n^{3} x^{2 n} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right )} + \frac {2 a a^{\frac {1}{n}} a^{-3 - \frac {2}{n}} b n x x^{n} \Gamma \left (\frac {1}{n}\right )}{a^{2} n^{3} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right ) + 2 a b n^{3} x^{n} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right ) + b^{2} n^{3} x^{2 n} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right )} + \frac {2 a^{\frac {1}{n}} a^{-3 - \frac {2}{n}} b^{2} n^{2} x x^{2 n} \Gamma \left (\frac {1}{n}\right )}{a^{2} n^{3} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right ) + 2 a b n^{3} x^{n} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right ) + b^{2} n^{3} x^{2 n} \left (1 + \frac {b x^{n}}{a}\right )^{\frac {1}{n}} \Gamma \left (3 + \frac {1}{n}\right )} \]
2*a**2*a**(1/n)*a**(-3 - 2/n)*n**2*x*gamma(1/n)/(a**2*n**3*(1 + b*x**n/a)* *(1/n)*gamma(3 + 1/n) + 2*a*b*n**3*x**n*(1 + b*x**n/a)**(1/n)*gamma(3 + 1/ n) + b**2*n**3*x**(2*n)*(1 + b*x**n/a)**(1/n)*gamma(3 + 1/n)) + 3*a**2*a** (1/n)*a**(-3 - 2/n)*n*x*gamma(1/n)/(a**2*n**3*(1 + b*x**n/a)**(1/n)*gamma( 3 + 1/n) + 2*a*b*n**3*x**n*(1 + b*x**n/a)**(1/n)*gamma(3 + 1/n) + b**2*n** 3*x**(2*n)*(1 + b*x**n/a)**(1/n)*gamma(3 + 1/n)) + a**2*a**(1/n)*a**(-3 - 2/n)*x*gamma(1/n)/(a**2*n**3*(1 + b*x**n/a)**(1/n)*gamma(3 + 1/n) + 2*a*b* n**3*x**n*(1 + b*x**n/a)**(1/n)*gamma(3 + 1/n) + b**2*n**3*x**(2*n)*(1 + b *x**n/a)**(1/n)*gamma(3 + 1/n)) + 4*a*a**(1/n)*a**(-3 - 2/n)*b*n**2*x*x**n *gamma(1/n)/(a**2*n**3*(1 + b*x**n/a)**(1/n)*gamma(3 + 1/n) + 2*a*b*n**3*x **n*(1 + b*x**n/a)**(1/n)*gamma(3 + 1/n) + b**2*n**3*x**(2*n)*(1 + b*x**n/ a)**(1/n)*gamma(3 + 1/n)) + 2*a*a**(1/n)*a**(-3 - 2/n)*b*n*x*x**n*gamma(1/ n)/(a**2*n**3*(1 + b*x**n/a)**(1/n)*gamma(3 + 1/n) + 2*a*b*n**3*x**n*(1 + b*x**n/a)**(1/n)*gamma(3 + 1/n) + b**2*n**3*x**(2*n)*(1 + b*x**n/a)**(1/n) *gamma(3 + 1/n)) + 2*a**(1/n)*a**(-3 - 2/n)*b**2*n**2*x*x**(2*n)*gamma(1/n )/(a**2*n**3*(1 + b*x**n/a)**(1/n)*gamma(3 + 1/n) + 2*a*b*n**3*x**n*(1 + b *x**n/a)**(1/n)*gamma(3 + 1/n) + b**2*n**3*x**(2*n)*(1 + b*x**n/a)**(1/n)* gamma(3 + 1/n))
\[ \int \left (a+b x^n\right )^{-3-\frac {1}{n}} \, dx=\int { {\left (b x^{n} + a\right )}^{-\frac {1}{n} - 3} \,d x } \]
\[ \int \left (a+b x^n\right )^{-3-\frac {1}{n}} \, dx=\int { {\left (b x^{n} + a\right )}^{-\frac {1}{n} - 3} \,d x } \]
Time = 5.93 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.67 \[ \int \left (a+b x^n\right )^{-3-\frac {1}{n}} \, dx=-\frac {x^{1-3\,n}\,{\left (\frac {a}{b\,x^n}+1\right )}^{1/n}\,{{}}_2{\mathrm {F}}_1\left (3,\frac {1}{n}+3;\ 4;\ -\frac {a}{b\,x^n}\right )}{3\,b^3\,n\,{\left (a+b\,x^n\right )}^{1/n}} \]