Integrand size = 27, antiderivative size = 60 \[ \int \frac {x^{-1-n p} \left (a+b x^n\right )^p}{c+d x^n} \, dx=-\frac {x^{-n p} \left (a+b x^n\right )^p \operatorname {Hypergeometric2F1}\left (1,-p,1-p,\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )}{c n p} \] Output:
-(a+b*x^n)^p*hypergeom([1, -p],[1-p],(-a*d+b*c)*x^n/c/(a+b*x^n))/c/n/p/(x^ (n*p))
Time = 0.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.47 \[ \int \frac {x^{-1-n p} \left (a+b x^n\right )^p}{c+d x^n} \, dx=-\frac {x^{-n p} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (1+\frac {d x^n}{c}\right )^p \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,\frac {(-b c+a d) x^n}{a \left (c+d x^n\right )}\right )}{c n p} \] Input:
Integrate[(x^(-1 - n*p)*(a + b*x^n)^p)/(c + d*x^n),x]
Output:
-(((a + b*x^n)^p*(1 + (d*x^n)/c)^p*Hypergeometric2F1[-p, -p, 1 - p, ((-(b* c) + a*d)*x^n)/(a*(c + d*x^n))])/(c*n*p*x^(n*p)*(1 + (b*x^n)/a)^p))
Time = 0.39 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.60, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1013, 1012}
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.
\(\displaystyle \int \frac {x^{-n p-1} \left (a+b x^n\right )^p}{c+d x^n} \, dx\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \int \frac {x^{-n p-1} \left (\frac {b x^n}{a}+1\right )^p}{d x^n+c}dx\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle -\frac {x^{-n p} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (\frac {d x^n}{c}+1\right )^p \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {\frac {b x^n}{a}-\frac {d x^n}{c}}{\frac {d x^n}{c}+1}\right )}{c n p}\) |
Input:
Int[(x^(-1 - n*p)*(a + b*x^n)^p)/(c + d*x^n),x]
Output:
-(((a + b*x^n)^p*(1 + (d*x^n)/c)^p*Hypergeometric2F1[-p, -p, 1 - p, -(((b* x^n)/a - (d*x^n)/c)/(1 + (d*x^n)/c))])/(c*n*p*x^(n*p)*(1 + (b*x^n)/a)^p))
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ n/a))^FracPart[p]) Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 0])
\[\int \frac {x^{-n p -1} \left (a +b \,x^{n}\right )^{p}}{c +d \,x^{n}}d x\]
Input:
int(x^(-n*p-1)*(a+b*x^n)^p/(c+d*x^n),x)
Output:
int(x^(-n*p-1)*(a+b*x^n)^p/(c+d*x^n),x)
\[ \int \frac {x^{-1-n p} \left (a+b x^n\right )^p}{c+d x^n} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} x^{-n p - 1}}{d x^{n} + c} \,d x } \] Input:
integrate(x^(-n*p-1)*(a+b*x^n)^p/(c+d*x^n),x, algorithm="fricas")
Output:
integral((b*x^n + a)^p*x^(-n*p - 1)/(d*x^n + c), x)
Exception generated. \[ \int \frac {x^{-1-n p} \left (a+b x^n\right )^p}{c+d x^n} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate(x**(-n*p-1)*(a+b*x**n)**p/(c+d*x**n),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int \frac {x^{-1-n p} \left (a+b x^n\right )^p}{c+d x^n} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} x^{-n p - 1}}{d x^{n} + c} \,d x } \] Input:
integrate(x^(-n*p-1)*(a+b*x^n)^p/(c+d*x^n),x, algorithm="maxima")
Output:
integrate((b*x^n + a)^p*x^(-n*p - 1)/(d*x^n + c), x)
\[ \int \frac {x^{-1-n p} \left (a+b x^n\right )^p}{c+d x^n} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} x^{-n p - 1}}{d x^{n} + c} \,d x } \] Input:
integrate(x^(-n*p-1)*(a+b*x^n)^p/(c+d*x^n),x, algorithm="giac")
Output:
integrate((b*x^n + a)^p*x^(-n*p - 1)/(d*x^n + c), x)
Timed out. \[ \int \frac {x^{-1-n p} \left (a+b x^n\right )^p}{c+d x^n} \, dx=\int \frac {{\left (a+b\,x^n\right )}^p}{x^{n\,p+1}\,\left (c+d\,x^n\right )} \,d x \] Input:
int((a + b*x^n)^p/(x^(n*p + 1)*(c + d*x^n)),x)
Output:
int((a + b*x^n)^p/(x^(n*p + 1)*(c + d*x^n)), x)
\[ \int \frac {x^{-1-n p} \left (a+b x^n\right )^p}{c+d x^n} \, dx=\frac {-\left (x^{n} b +a \right )^{p} b +x^{n p} \left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{n p +2 n} b d x +x^{n p +n} a d x +x^{n p +n} b c x +x^{n p} a c x}d x \right ) a^{2} d n p -x^{n p} \left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{n p +2 n} b d x +x^{n p +n} a d x +x^{n p +n} b c x +x^{n p} a c x}d x \right ) a b c n p}{x^{n p} a d n p} \] Input:
int(x^(-n*p-1)*(a+b*x^n)^p/(c+d*x^n),x)
Output:
( - (x**n*b + a)**p*b + x**(n*p)*int((x**n*b + a)**p/(x**(n*p + 2*n)*b*d*x + x**(n*p + n)*a*d*x + x**(n*p + n)*b*c*x + x**(n*p)*a*c*x),x)*a**2*d*n*p - x**(n*p)*int((x**n*b + a)**p/(x**(n*p + 2*n)*b*d*x + x**(n*p + n)*a*d*x + x**(n*p + n)*b*c*x + x**(n*p)*a*c*x),x)*a*b*c*n*p)/(x**(n*p)*a*d*n*p)