Integrand size = 20, antiderivative size = 66 \[ \int x^{-1-n (-3+p)} \left (a+b x^n\right )^p \, dx=\frac {x^{n (3-p)} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (3-p,-p,4-p,-\frac {b x^n}{a}\right )}{n (3-p)} \] Output:
x^(n*(3-p))*(a+b*x^n)^p*hypergeom([-p, 3-p],[4-p],-b*x^n/a)/n/(3-p)/((1+b* x^n/a)^p)
Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97 \[ \int x^{-1-n (-3+p)} \left (a+b x^n\right )^p \, dx=-\frac {x^{-n (-3+p)} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (3-p,-p,4-p,-\frac {b x^n}{a}\right )}{n (-3+p)} \] Input:
Integrate[x^(-1 - n*(-3 + p))*(a + b*x^n)^p,x]
Output:
-(((a + b*x^n)^p*Hypergeometric2F1[3 - p, -p, 4 - p, -((b*x^n)/a)])/(n*(-3 + p)*x^(n*(-3 + p))*(1 + (b*x^n)/a)^p))
Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {882, 74}
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.
\(\displaystyle \int x^{-n (p-3)-1} \left (a+b x^n\right )^p \, dx\) |
\(\Big \downarrow \) 882 |
\(\displaystyle \frac {a^3 x^{-n p} \left (\frac {x^n}{a+b x^n}\right )^p \left (a+b x^n\right )^p \int \frac {\left (\frac {x^n}{b x^n+a}\right )^{2-p}}{\left (1-\frac {b x^n}{b x^n+a}\right )^4}d\frac {x^n}{b x^n+a}}{n}\) |
\(\Big \downarrow \) 74 |
\(\displaystyle \frac {a^3 x^{3 n-n p} \left (a+b x^n\right )^{p-3} \operatorname {Hypergeometric2F1}\left (4,3-p,4-p,\frac {b x^n}{b x^n+a}\right )}{n (3-p)}\) |
Input:
Int[x^(-1 - n*(-3 + p))*(a + b*x^n)^p,x]
Output:
(a^3*x^(3*n - n*p)*(a + b*x^n)^(-3 + p)*Hypergeometric2F1[4, 3 - p, 4 - p, (b*x^n)/(a + b*x^n)])/(n*(3 - p))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^Simplify[ (m + 1)/n + p]*x^m*(a + b*x^n)^p*((x^n/(a + b*x^n))^p/(n*x^Simplify[m + n*p ])) Subst[Int[x^((m + 1)/n - 1)/(1 - b*x)^(Simplify[(m + 1)/n + p] + 1), x], x, x^n/(a + b*x^n)], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simpli fy[(m + 1)/n + p]]
\[\int x^{-1-n \left (-3+p \right )} \left (a +b \,x^{n}\right )^{p}d x\]
Input:
int(x^(-1-n*(-3+p))*(a+b*x^n)^p,x)
Output:
int(x^(-1-n*(-3+p))*(a+b*x^n)^p,x)
\[ \int x^{-1-n (-3+p)} \left (a+b x^n\right )^p \, dx=\int { {\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 3\right )} - 1} \,d x } \] Input:
integrate(x^(-1-n*(-3+p))*(a+b*x^n)^p,x, algorithm="fricas")
Output:
integral((b*x^n + a)^p*x^(-n*p + 3*n - 1), x)
Result contains complex when optimal does not.
Time = 60.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.62 \[ \int x^{-1-n (-3+p)} \left (a+b x^n\right )^p \, dx=\frac {a^{p} x^{- n p + 3 n} \Gamma \left (3 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 3 - p \\ 4 - p \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (4 - p\right )} \] Input:
integrate(x**(-1-n*(-3+p))*(a+b*x**n)**p,x)
Output:
a**p*x**(-n*p + 3*n)*gamma(3 - p)*hyper((-p, 3 - p), (4 - p,), b*x**n*exp_ polar(I*pi)/a)/(n*gamma(4 - p))
\[ \int x^{-1-n (-3+p)} \left (a+b x^n\right )^p \, dx=\int { {\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 3\right )} - 1} \,d x } \] Input:
integrate(x^(-1-n*(-3+p))*(a+b*x^n)^p,x, algorithm="maxima")
Output:
integrate((b*x^n + a)^p*x^(-n*(p - 3) - 1), x)
\[ \int x^{-1-n (-3+p)} \left (a+b x^n\right )^p \, dx=\int { {\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 3\right )} - 1} \,d x } \] Input:
integrate(x^(-1-n*(-3+p))*(a+b*x^n)^p,x, algorithm="giac")
Output:
integrate((b*x^n + a)^p*x^(-n*(p - 3) - 1), x)
Timed out. \[ \int x^{-1-n (-3+p)} \left (a+b x^n\right )^p \, dx=\int \frac {{\left (a+b\,x^n\right )}^p}{x^{n\,\left (p-3\right )+1}} \,d x \] Input:
int((a + b*x^n)^p/x^(n*(p - 3) + 1),x)
Output:
int((a + b*x^n)^p/x^(n*(p - 3) + 1), x)
\[ \int x^{-1-n (-3+p)} \left (a+b x^n\right )^p \, dx=\frac {2 x^{3 n} \left (x^{n} b +a \right )^{p} b^{2}+x^{2 n} \left (x^{n} b +a \right )^{p} a b p +x^{n} \left (x^{n} b +a \right )^{p} a^{2} p^{2}-2 x^{n} \left (x^{n} b +a \right )^{p} a^{2} p +x^{n p} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{3} n \,p^{3}-3 x^{n p} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{3} n \,p^{2}+2 x^{n p} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{3} n p}{6 x^{n p} b^{2} n} \] Input:
int(x^(-1-n*(-3+p))*(a+b*x^n)^p,x)
Output:
(2*x**(3*n)*(x**n*b + a)**p*b**2 + x**(2*n)*(x**n*b + a)**p*a*b*p + x**n*( x**n*b + a)**p*a**2*p**2 - 2*x**n*(x**n*b + a)**p*a**2*p + x**(n*p)*int((x **n*(x**n*b + a)**p)/(x**(n*p + n)*b*x + x**(n*p)*a*x),x)*a**3*n*p**3 - 3* x**(n*p)*int((x**n*(x**n*b + a)**p)/(x**(n*p + n)*b*x + x**(n*p)*a*x),x)*a **3*n*p**2 + 2*x**(n*p)*int((x**n*(x**n*b + a)**p)/(x**(n*p + n)*b*x + x** (n*p)*a*x),x)*a**3*n*p)/(6*x**(n*p)*b**2*n)