Integrand size = 22, antiderivative size = 69 \[ \int x^{-1-p q} \left (b x^n+a x^q\right )^p \, dx=-\frac {x^{-p q} \left (a+b x^{n-q}\right ) \left (b x^n+a x^q\right )^p \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b x^{n-q}}{a}\right )}{a (1+p) (n-q)} \] Output:
-(a+b*x^(n-q))*(b*x^n+a*x^q)^p*hypergeom([1, p+1],[2+p],1+b*x^(n-q)/a)/a/( p+1)/(n-q)/(x^(p*q))
Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.06 \[ \int x^{-1-p q} \left (b x^n+a x^q\right )^p \, dx=\frac {x^{-p q} \left (b x^n+a x^q\right )^p \left (1+\frac {a x^{-n+q}}{b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {a x^{-n+q}}{b}\right )}{p (n-q)} \] Input:
Integrate[x^(-1 - p*q)*(b*x^n + a*x^q)^p,x]
Output:
((b*x^n + a*x^q)^p*Hypergeometric2F1[-p, -p, 1 - p, -((a*x^(-n + q))/b)])/ (p*(n - q)*x^(p*q)*(1 + (a*x^(-n + q))/b)^p)
Time = 0.36 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1938, 798, 75}
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.
\(\displaystyle \int x^{-p q-1} \left (a x^q+b x^n\right )^p \, dx\) |
\(\Big \downarrow \) 1938 |
\(\displaystyle x^{-p q} \left (a+b x^{n-q}\right )^{-p} \left (a x^q+b x^n\right )^p \int \frac {\left (b x^{n-q}+a\right )^p}{x}dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {x^{-p q} \left (a+b x^{n-q}\right )^{-p} \left (a x^q+b x^n\right )^p \int x^{q-n} \left (b x^{n-q}+a\right )^pdx^{n-q}}{n-q}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle -\frac {x^{-p q} \left (a+b x^{n-q}\right ) \left (a x^q+b x^n\right )^p \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b x^{n-q}}{a}+1\right )}{a (p+1) (n-q)}\) |
Input:
Int[x^(-1 - p*q)*(b*x^n + a*x^q)^p,x]
Output:
-(((a + b*x^(n - q))*(b*x^n + a*x^q)^p*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*x^(n - q))/a])/(a*(1 + p)*(n - q)*x^(p*q)))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])) Int[x^(m + j* p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !Inte gerQ[p] && NeQ[n, j] && PosQ[n - j]
\[\int x^{-p q -1} \left (b \,x^{n}+a \,x^{q}\right )^{p}d x\]
Input:
int(x^(-p*q-1)*(b*x^n+a*x^q)^p,x)
Output:
int(x^(-p*q-1)*(b*x^n+a*x^q)^p,x)
\[ \int x^{-1-p q} \left (b x^n+a x^q\right )^p \, dx=\int { {\left (b x^{n} + a x^{q}\right )}^{p} x^{-p q - 1} \,d x } \] Input:
integrate(x^(-p*q-1)*(b*x^n+a*x^q)^p,x, algorithm="fricas")
Output:
integral((b*x^n + a*x^q)^p*x^(-p*q - 1), x)
\[ \int x^{-1-p q} \left (b x^n+a x^q\right )^p \, dx=\int x^{- p q - 1} \left (a x^{q} + b x^{n}\right )^{p}\, dx \] Input:
integrate(x**(-p*q-1)*(b*x**n+a*x**q)**p,x)
Output:
Integral(x**(-p*q - 1)*(a*x**q + b*x**n)**p, x)
\[ \int x^{-1-p q} \left (b x^n+a x^q\right )^p \, dx=\int { {\left (b x^{n} + a x^{q}\right )}^{p} x^{-p q - 1} \,d x } \] Input:
integrate(x^(-p*q-1)*(b*x^n+a*x^q)^p,x, algorithm="maxima")
Output:
integrate((b*x^n + a*x^q)^p*x^(-p*q - 1), x)
\[ \int x^{-1-p q} \left (b x^n+a x^q\right )^p \, dx=\int { {\left (b x^{n} + a x^{q}\right )}^{p} x^{-p q - 1} \,d x } \] Input:
integrate(x^(-p*q-1)*(b*x^n+a*x^q)^p,x, algorithm="giac")
Output:
integrate((b*x^n + a*x^q)^p*x^(-p*q - 1), x)
Timed out. \[ \int x^{-1-p q} \left (b x^n+a x^q\right )^p \, dx=\int \frac {{\left (b\,x^n+a\,x^q\right )}^p}{x^{p\,q+1}} \,d x \] Input:
int((b*x^n + a*x^q)^p/x^(p*q + 1),x)
Output:
int((b*x^n + a*x^q)^p/x^(p*q + 1), x)
\[ \int x^{-1-p q} \left (b x^n+a x^q\right )^p \, dx=\frac {\left (x^{n} b +x^{q} a \right )^{p}+x^{p q} \left (\int \frac {x^{q} \left (x^{n} b +x^{q} a \right )^{p}}{x^{p q +n} b x +x^{p q +q} a x}d x \right ) a n p -x^{p q} \left (\int \frac {x^{q} \left (x^{n} b +x^{q} a \right )^{p}}{x^{p q +n} b x +x^{p q +q} a x}d x \right ) a p q}{x^{p q} p \left (n -q \right )} \] Input:
int(x^(-p*q-1)*(b*x^n+a*x^q)^p,x)
Output:
((x**n*b + x**q*a)**p + x**(p*q)*int((x**q*(x**n*b + x**q*a)**p)/(x**(n + p*q)*b*x + x**(p*q + q)*a*x),x)*a*n*p - x**(p*q)*int((x**q*(x**n*b + x**q* a)**p)/(x**(n + p*q)*b*x + x**(p*q + q)*a*x),x)*a*p*q)/(x**(p*q)*p*(n - q) )