\(\int \frac {x^{-1-n (-1+p)} (a+b x^n)^p}{c+d x^n} \, dx\) [542]

Optimal result

Integrand size = 29, antiderivative size = 121 \[ \int \frac {x^{-1-n (-1+p)} \left (a+b x^n\right )^p}{c+d x^n} \, dx=\frac {x^{-n p} \left (a+b x^n\right )^p \operatorname {Hypergeometric2F1}\left (1,-p,1-p,\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )}{d n p}-\frac {x^{-n p} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^n}{a}\right )}{d n p} \] Output:

(a+b*x^n)^p*hypergeom([1, -p],[1-p],(-a*d+b*c)*x^n/c/(a+b*x^n))/d/n/p/(x^( 
n*p))-(a+b*x^n)^p*hypergeom([-p, -p],[1-p],-b*x^n/a)/d/n/p/(x^(n*p))/((1+b 
*x^n/a)^p)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.64 \[ \int \frac {x^{-1-n (-1+p)} \left (a+b x^n\right )^p}{c+d x^n} \, dx=-\frac {x^{-n (-1+p)} \left (a+b x^n\right )^p \left (\frac {a+b x^n}{a}\right )^{-p} \operatorname {AppellF1}\left (1-p,-p,1,2-p,-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c n (-1+p)} \] Input:

Integrate[(x^(-1 - n*(-1 + p))*(a + b*x^n)^p)/(c + d*x^n),x]
 

Output:

-(((a + b*x^n)^p*AppellF1[1 - p, -p, 1, 2 - p, -((b*x^n)/a), -((d*x^n)/c)] 
)/(c*n*(-1 + p)*x^(n*(-1 + p))*((a + b*x^n)/a)^p))
 

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.37 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.64, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, 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.

\(\displaystyle \int \frac {x^{-n (p-1)-1} \left (a+b x^n\right )^p}{c+d x^n} \, dx\)

\(\Big \downarrow \) 1013

\(\displaystyle \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \int \frac {x^{-p n+n-1} \left (\frac {b x^n}{a}+1\right )^p}{d x^n+c}dx\)

\(\Big \downarrow \) 1012

\(\displaystyle \frac {x^{n-n p} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \operatorname {AppellF1}\left (1-p,-p,1,2-p,-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c n (1-p)}\)

Input:

Int[(x^(-1 - n*(-1 + p))*(a + b*x^n)^p)/(c + d*x^n),x]
 

Output:

(x^(n - n*p)*(a + b*x^n)^p*AppellF1[1 - p, -p, 1, 2 - p, -((b*x^n)/a), -(( 
d*x^n)/c)])/(c*n*(1 - p)*(1 + (b*x^n)/a)^p)
 

Defintions of rubi rules used

rule 1012

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])
 

rule 1013

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])
 

Maple [F]

\[\int \frac {x^{-1-n \left (p -1\right )} \left (a +b \,x^{n}\right )^{p}}{c +d \,x^{n}}d x\]

Input:

int(x^(-1-n*(p-1))*(a+b*x^n)^p/(c+d*x^n),x)
 

Output:

int(x^(-1-n*(p-1))*(a+b*x^n)^p/(c+d*x^n),x)
 

Fricas [F]

\[ \int \frac {x^{-1-n (-1+p)} \left (a+b x^n\right )^p}{c+d x^n} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 1\right )} - 1}}{d x^{n} + c} \,d x } \] Input:

integrate(x^(-1-n*(-1+p))*(a+b*x^n)^p/(c+d*x^n),x, algorithm="fricas")
 

Output:

integral((b*x^n + a)^p*x^(-n*p + n - 1)/(d*x^n + c), x)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {x^{-1-n (-1+p)} \left (a+b x^n\right )^p}{c+d x^n} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate(x**(-1-n*(-1+p))*(a+b*x**n)**p/(c+d*x**n),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int \frac {x^{-1-n (-1+p)} \left (a+b x^n\right )^p}{c+d x^n} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 1\right )} - 1}}{d x^{n} + c} \,d x } \] Input:

integrate(x^(-1-n*(-1+p))*(a+b*x^n)^p/(c+d*x^n),x, algorithm="maxima")
 

Output:

integrate((b*x^n + a)^p*x^(-n*(p - 1) - 1)/(d*x^n + c), x)
 

Giac [F]

\[ \int \frac {x^{-1-n (-1+p)} \left (a+b x^n\right )^p}{c+d x^n} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 1\right )} - 1}}{d x^{n} + c} \,d x } \] Input:

integrate(x^(-1-n*(-1+p))*(a+b*x^n)^p/(c+d*x^n),x, algorithm="giac")
 

Output:

integrate((b*x^n + a)^p*x^(-n*(p - 1) - 1)/(d*x^n + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1-n (-1+p)} \left (a+b x^n\right )^p}{c+d x^n} \, dx=\int \frac {{\left (a+b\,x^n\right )}^p}{x^{n\,\left (p-1\right )+1}\,\left (c+d\,x^n\right )} \,d x \] Input:

int((a + b*x^n)^p/(x^(n*(p - 1) + 1)*(c + d*x^n)),x)
 

Output:

int((a + b*x^n)^p/(x^(n*(p - 1) + 1)*(c + d*x^n)), x)
 

Reduce [F]

\[ \int \frac {x^{-1-n (-1+p)} \left (a+b x^n\right )^p}{c+d x^n} \, dx=\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} d x +x^{n p} c x}d x \] Input:

int(x^(-1-n*(-1+p))*(a+b*x^n)^p/(c+d*x^n),x)
 

Output:

int((x**n*(x**n*b + a)**p)/(x**(n*p + n)*d*x + x**(n*p)*c*x),x)