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]
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]
[Out]
Rule 388
Rule 390
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*}
\[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \, dx=\int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \, dx \]
[In]
[Out]
\[\int \left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )^{-2-\frac {1}{n}-p}d x\]
[In]
[Out]
\[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{-p - \frac {1}{n} - 2} \,d x } \]
[In]
[Out]
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]
[Out]
\[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{-p - \frac {1}{n} - 2} \,d x } \]
[In]
[Out]
\[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{-p - \frac {1}{n} - 2} \,d x } \]
[In]
[Out]
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]
[Out]