Integrand size = 18, antiderivative size = 147 \[ \int \left (a+b x^n\right )^{-\frac {1+4 n}{n}} \, dx=\frac {x \left (a+b x^n\right )^{-3-\frac {1}{n}}}{a (1+3 n)}+\frac {3 n x \left (a+b x^n\right )^{-2-\frac {1}{n}}}{a^2 \left (1+5 n+6 n^2\right )}+\frac {6 n^3 x \left (a+b x^n\right )^{-1/n}}{a^4 (1+n) (1+2 n) (1+3 n)}+\frac {6 n^2 x \left (a+b x^n\right )^{-\frac {1+n}{n}}}{a^3 (1+n) (1+2 n) (1+3 n)} \] Output:
x*(a+b*x^n)^(-3-1/n)/a/(1+3*n)+3*n*x*(a+b*x^n)^(-2-1/n)/a^2/(6*n^2+5*n+1)+ 6*n^3*x/a^4/(1+n)/(1+2*n)/(1+3*n)/((a+b*x^n)^(1/n))+6*n^2*x/a^3/(1+n)/(1+2 *n)/(1+3*n)/((a+b*x^n)^((1+n)/n))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.37 \[ \int \left (a+b x^n\right )^{-\frac {1+4 n}{n}} \, dx=\frac {x \left (a+b x^n\right )^{-1/n} \left (1+\frac {b x^n}{a}\right )^{\frac {1}{n}} \operatorname {Hypergeometric2F1}\left (4+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^4} \] Input:
Integrate[(a + b*x^n)^(-((1 + 4*n)/n)),x]
Output:
(x*(1 + (b*x^n)/a)^n^(-1)*Hypergeometric2F1[4 + n^(-1), n^(-1), 1 + n^(-1) , -((b*x^n)/a)])/(a^4*(a + b*x^n)^n^(-1))
Time = 0.40 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {777, 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.
\(\displaystyle \int \left (a+b x^n\right )^{-\frac {4 n+1}{n}} \, dx\) |
\(\Big \downarrow \) 777 |
\(\displaystyle \frac {3 n \int \left (b x^n+a\right )^{-3-\frac {1}{n}}dx}{a (3 n+1)}+\frac {x \left (a+b x^n\right )^{-\frac {1}{n}-3}}{a (3 n+1)}\) |
\(\Big \downarrow \) 777 |
\(\displaystyle \frac {3 n \left (\frac {2 n \int \left (b x^n+a\right )^{-2-\frac {1}{n}}dx}{a (2 n+1)}+\frac {x \left (a+b x^n\right )^{-\frac {1}{n}-2}}{a (2 n+1)}\right )}{a (3 n+1)}+\frac {x \left (a+b x^n\right )^{-\frac {1}{n}-3}}{a (3 n+1)}\) |
\(\Big \downarrow \) 777 |
\(\displaystyle \frac {3 n \left (\frac {2 n \left (\frac {n \int \left (b x^n+a\right )^{-1-\frac {1}{n}}dx}{a (n+1)}+\frac {x \left (a+b x^n\right )^{-\frac {1}{n}-1}}{a (n+1)}\right )}{a (2 n+1)}+\frac {x \left (a+b x^n\right )^{-\frac {1}{n}-2}}{a (2 n+1)}\right )}{a (3 n+1)}+\frac {x \left (a+b x^n\right )^{-\frac {1}{n}-3}}{a (3 n+1)}\) |
\(\Big \downarrow \) 746 |
\(\displaystyle \frac {3 n \left (\frac {2 n \left (\frac {n x \left (a+b x^n\right )^{-1/n}}{a^2 (n+1)}+\frac {x \left (a+b x^n\right )^{-\frac {1}{n}-1}}{a (n+1)}\right )}{a (2 n+1)}+\frac {x \left (a+b x^n\right )^{-\frac {1}{n}-2}}{a (2 n+1)}\right )}{a (3 n+1)}+\frac {x \left (a+b x^n\right )^{-\frac {1}{n}-3}}{a (3 n+1)}\) |
Input:
Int[(a + b*x^n)^(-((1 + 4*n)/n)),x]
Output:
(x*(a + b*x^n)^(-3 - n^(-1)))/(a*(1 + 3*n)) + (3*n*((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))))/(a*(1 + 3*n))
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]
\[\int \left (a +b \,x^{n}\right )^{-\frac {1+4 n}{n}}d x\]
Input:
int((a+b*x^n)^(-(1+4*n)/n),x)
Output:
int((a+b*x^n)^(-(1+4*n)/n),x)
Time = 0.08 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.31 \[ \int \left (a+b x^n\right )^{-\frac {1+4 n}{n}} \, dx=\frac {6 \, b^{4} n^{3} x x^{4 \, n} + 6 \, {\left (4 \, a b^{3} n^{3} + a b^{3} n^{2}\right )} x x^{3 \, n} + 3 \, {\left (12 \, a^{2} b^{2} n^{3} + 7 \, a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x x^{2 \, n} + {\left (24 \, a^{3} b n^{3} + 26 \, a^{3} b n^{2} + 9 \, a^{3} b n + a^{3} b\right )} x x^{n} + {\left (6 \, a^{4} n^{3} + 11 \, a^{4} n^{2} + 6 \, a^{4} n + a^{4}\right )} x}{{\left (6 \, a^{4} n^{3} + 11 \, a^{4} n^{2} + 6 \, a^{4} n + a^{4}\right )} {\left (b x^{n} + a\right )}^{\frac {4 \, n + 1}{n}}} \] Input:
integrate((a+b*x^n)^(-(1+4*n)/n),x, algorithm="fricas")
Output:
(6*b^4*n^3*x*x^(4*n) + 6*(4*a*b^3*n^3 + a*b^3*n^2)*x*x^(3*n) + 3*(12*a^2*b ^2*n^3 + 7*a^2*b^2*n^2 + a^2*b^2*n)*x*x^(2*n) + (24*a^3*b*n^3 + 26*a^3*b*n ^2 + 9*a^3*b*n + a^3*b)*x*x^n + (6*a^4*n^3 + 11*a^4*n^2 + 6*a^4*n + a^4)*x )/((6*a^4*n^3 + 11*a^4*n^2 + 6*a^4*n + a^4)*(b*x^n + a)^((4*n + 1)/n))
Leaf count of result is larger than twice the leaf count of optimal. 1911 vs. \(2 (122) = 244\).
Time = 4.26 (sec) , antiderivative size = 1911, normalized size of antiderivative = 13.00 \[ \int \left (a+b x^n\right )^{-\frac {1+4 n}{n}} \, dx=\text {Too large to display} \] Input:
integrate((a+b*x**n)**(-(1+4*n)/n),x)
Output:
6*a**3*a**(1/n)*n**3*gamma(1/n)/(a**3*a**(4 + 1/n)*b**(1/n)*n**4*(a/(b*x** n) + 1)**(1/n)*gamma(4 + 1/n) + 3*a**2*a**(4 + 1/n)*b*b**(1/n)*n**4*x**n*( a/(b*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*a*a**(4 + 1/n)*b**2*b**(1/n)*n** 4*x**(2*n)*(a/(b*x**n) + 1)**(1/n)*gamma(4 + 1/n) + a**(4 + 1/n)*b**3*b**( 1/n)*n**4*x**(3*n)*(a/(b*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + 11*a**3*a**(1 /n)*n**2*gamma(1/n)/(a**3*a**(4 + 1/n)*b**(1/n)*n**4*(a/(b*x**n) + 1)**(1/ n)*gamma(4 + 1/n) + 3*a**2*a**(4 + 1/n)*b*b**(1/n)*n**4*x**n*(a/(b*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*a*a**(4 + 1/n)*b**2*b**(1/n)*n**4*x**(2*n)*( a/(b*x**n) + 1)**(1/n)*gamma(4 + 1/n) + a**(4 + 1/n)*b**3*b**(1/n)*n**4*x* *(3*n)*(a/(b*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + 6*a**3*a**(1/n)*n*gamma(1 /n)/(a**3*a**(4 + 1/n)*b**(1/n)*n**4*(a/(b*x**n) + 1)**(1/n)*gamma(4 + 1/n ) + 3*a**2*a**(4 + 1/n)*b*b**(1/n)*n**4*x**n*(a/(b*x**n) + 1)**(1/n)*gamma (4 + 1/n) + 3*a*a**(4 + 1/n)*b**2*b**(1/n)*n**4*x**(2*n)*(a/(b*x**n) + 1)* *(1/n)*gamma(4 + 1/n) + a**(4 + 1/n)*b**3*b**(1/n)*n**4*x**(3*n)*(a/(b*x** n) + 1)**(1/n)*gamma(4 + 1/n)) + a**3*a**(1/n)*gamma(1/n)/(a**3*a**(4 + 1/ n)*b**(1/n)*n**4*(a/(b*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*a**2*a**(4 + 1 /n)*b*b**(1/n)*n**4*x**n*(a/(b*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*a*a**( 4 + 1/n)*b**2*b**(1/n)*n**4*x**(2*n)*(a/(b*x**n) + 1)**(1/n)*gamma(4 + 1/n ) + a**(4 + 1/n)*b**3*b**(1/n)*n**4*x**(3*n)*(a/(b*x**n) + 1)**(1/n)*gamma (4 + 1/n)) + 18*a**2*a**(1/n)*b*n**3*x**n*gamma(1/n)/(a**3*a**(4 + 1/n)...
\[ \int \left (a+b x^n\right )^{-\frac {1+4 n}{n}} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{\frac {4 \, n + 1}{n}}} \,d x } \] Input:
integrate((a+b*x^n)^(-(1+4*n)/n),x, algorithm="maxima")
Output:
integrate(1/((b*x^n + a)^((4*n + 1)/n)), x)
\[ \int \left (a+b x^n\right )^{-\frac {1+4 n}{n}} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{\frac {4 \, n + 1}{n}}} \,d x } \] Input:
integrate((a+b*x^n)^(-(1+4*n)/n),x, algorithm="giac")
Output:
integrate(1/((b*x^n + a)^((4*n + 1)/n)), x)
Time = 0.34 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.44 \[ \int \left (a+b x^n\right )^{-\frac {1+4 n}{n}} \, dx=-\frac {x^{1-4\,n}\,{\left (\frac {a}{b\,x^n}+1\right )}^{1/n}\,{{}}_2{\mathrm {F}}_1\left (4,\frac {1}{n}+4;\ 5;\ -\frac {a}{b\,x^n}\right )}{4\,b^4\,n\,{\left (a+b\,x^n\right )}^{1/n}} \] Input:
int(1/(a + b*x^n)^((4*n + 1)/n),x)
Output:
-(x^(1 - 4*n)*(a/(b*x^n) + 1)^(1/n)*hypergeom([4, 1/n + 4], 5, -a/(b*x^n)) )/(4*b^4*n*(a + b*x^n)^(1/n))
\[ \int \left (a+b x^n\right )^{-\frac {1+4 n}{n}} \, dx=\int \frac {1}{x^{4 n} \left (x^{n} b +a \right )^{\frac {1}{n}} b^{4}+4 x^{3 n} \left (x^{n} b +a \right )^{\frac {1}{n}} a \,b^{3}+6 x^{2 n} \left (x^{n} b +a \right )^{\frac {1}{n}} a^{2} b^{2}+4 x^{n} \left (x^{n} b +a \right )^{\frac {1}{n}} a^{3} b +\left (x^{n} b +a \right )^{\frac {1}{n}} a^{4}}d x \] Input:
int((a+b*x^n)^(-(1+4*n)/n),x)
Output:
int(1/(x**(4*n)*(x**n*b + a)**(1/n)*b**4 + 4*x**(3*n)*(x**n*b + a)**(1/n)* a*b**3 + 6*x**(2*n)*(x**n*b + a)**(1/n)*a**2*b**2 + 4*x**n*(x**n*b + a)**( 1/n)*a**3*b + (x**n*b + a)**(1/n)*a**4),x)