\(\int (a+b x^n)^p (c+d x^n)^{-2-\frac {1}{n}-p} \, dx\) [328]

Optimal result

Integrand size = 28, antiderivative size = 193 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \, dx=-\frac {b x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{-1-\frac {1}{n}-p}}{a (b c-a d) n (1+p)}+\frac {(b c+(b c-a d) n (1+p)) x \left (a+b x^n\right )^{1+p} \left (\frac {c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-1-p} \left (c+d x^n\right )^{-1-\frac {1}{n}-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-1-p,1+\frac {1}{n},-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a c (b c-a d) n (1+p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {390, 388} \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \, dx=\frac {x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{-\frac {1}{n}-p-1} \left (\frac {b}{n (p+1) (b c-a d)}+\frac {1}{c}\right ) \left (\frac {c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p-1,1+\frac {1}{n},-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a}-\frac {b x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{-\frac {1}{n}-p-1}}{a n (p+1) (b c-a d)} \]

[In]

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

[Out]

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

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]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {b x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{-1-\frac {1}{n}-p}}{a (b c-a d) n (1+p)}+\frac {\left (1+\frac {b c}{(b c-a d) n (1+p)}\right ) \int \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \, dx}{a} \\ & = -\frac {b x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{-1-\frac {1}{n}-p}}{a (b c-a d) n (1+p)}+\frac {\left (1+\frac {b c}{(b c-a d) n (1+p)}\right ) x \left (a+b x^n\right )^{1+p} \left (\frac {c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-1-p} \left (c+d x^n\right )^{-1-\frac {1}{n}-p} \, _2F_1\left (\frac {1}{n},-1-p;1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a c} \\ \end{align*}

Mathematica [F]

\[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \, dx=\int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \, dx \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )^{-2-\frac {1}{n}-p}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{-p - \frac {1}{n} - 2} \,d x } \]

[In]

integrate((a+b*x^n)^p*(c+d*x^n)^(-2-1/n-p),x, algorithm="fricas")

[Out]

integral((b*x^n + a)^p/(d*x^n + c)^((n*p + 2*n + 1)/n), x)

Sympy [F(-2)]

Exception generated. \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate((a+b*x**n)**p*(c+d*x**n)**(-2-1/n-p),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

\[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{-p - \frac {1}{n} - 2} \,d x } \]

[In]

integrate((a+b*x^n)^p*(c+d*x^n)^(-2-1/n-p),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{-p - \frac {1}{n} - 2} \,d x } \]

[In]

integrate((a+b*x^n)^p*(c+d*x^n)^(-2-1/n-p),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \, dx=\int \frac {{\left (a+b\,x^n\right )}^p}{{\left (c+d\,x^n\right )}^{p+\frac {1}{n}+2}} \,d x \]

[In]

int((a + b*x^n)^p/(c + d*x^n)^(p + 1/n + 2),x)

[Out]

int((a + b*x^n)^p/(c + d*x^n)^(p + 1/n + 2), x)