\(\int (a+b x^n)^p (c+d x^n)^{-1-\frac {1}{n}-p} \, dx\) [320]

Optimal result

Integrand size = 28, antiderivative size = 93 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-1-\frac {1}{n}-p} \, dx=\frac {x \left (a+b x^n\right )^p \left (\frac {c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \left (c+d x^n\right )^{-\frac {1}{n}-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {388} \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-1-\frac {1}{n}-p} \, dx=\frac {x \left (a+b x^n\right )^p \left (c+d x^n\right )^{-\frac {1}{n}-p} \left (\frac {c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{c} \]

[In]

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

[Out]

(x*(a + b*x^n)^p*(c + d*x^n)^(-n^(-1) - p)*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -(((b*c - a*d)*x^n)/(a*(c
+ d*x^n)))])/(c*((c*(a + b*x^n))/(a*(c + d*x^n)))^p)

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(c*(c*((a
+ b*x^n)/(a*(c + d*x^n))))^p*(c + d*x^n)^(1/n + p)))*Hypergeometric2F1[1/n, -p, 1 + 1/n, (-(b*c - a*d))*(x^n/(
a*(c + d*x^n)))], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a+b x^n\right )^p \left (\frac {c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \left (c+d x^n\right )^{-\frac {1}{n}-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{c} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.16 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.01 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-1-\frac {1}{n}-p} \, dx=\frac {x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^{-\frac {1+n p}{n}} \left (1+\frac {d x^n}{c}\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},\frac {(-b c+a d) x^n}{a \left (c+d x^n\right )}\right )}{c} \]

[In]

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

[Out]

(x*(a + b*x^n)^p*(1 + (d*x^n)/c)^p*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), ((-(b*c) + a*d)*x^n)/(a*(c + d*x^
n))])/(c*(1 + (b*x^n)/a)^p*(c + d*x^n)^((1 + n*p)/n))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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