Integrand size = 27, antiderivative size = 241 \[ \int (e x)^{-1-n} \left (a+\frac {b}{\sqrt {c+d x^n}}\right )^p \, dx=-\frac {d x^n (e x)^{-n} \left (a+\frac {b}{\sqrt {c+d x^n}}\right )^p}{c e n \left (1-\frac {c}{c+d x^n}\right )}+\frac {b d x^n (e x)^{-n} \left (a+\frac {b}{\sqrt {c+d x^n}}\right )^p \operatorname {Hypergeometric2F1}\left (1,p,1+p,-\frac {\sqrt {c} \left (a+\frac {b}{\sqrt {c+d x^n}}\right )}{b-a \sqrt {c}}\right )}{2 \left (b-a \sqrt {c}\right ) c e n}+\frac {b d x^n (e x)^{-n} \left (a+\frac {b}{\sqrt {c+d x^n}}\right )^p \operatorname {Hypergeometric2F1}\left (1,p,1+p,\frac {\sqrt {c} \left (a+\frac {b}{\sqrt {c+d x^n}}\right )}{b+a \sqrt {c}}\right )}{2 \left (b+a \sqrt {c}\right ) c e n} \] Output:
-d*x^n*(a+b/(c+d*x^n)^(1/2))^p/c/e/n/((e*x)^n)/(1-c/(c+d*x^n))+1/2*b*d*x^n *(a+b/(c+d*x^n)^(1/2))^p*hypergeom([1, p],[p+1],-c^(1/2)*(a+b/(c+d*x^n)^(1 /2))/(b-a*c^(1/2)))/(b-a*c^(1/2))/c/e/n/((e*x)^n)+1/2*b*d*x^n*(a+b/(c+d*x^ n)^(1/2))^p*hypergeom([1, p],[p+1],c^(1/2)*(a+b/(c+d*x^n)^(1/2))/(b+a*c^(1 /2)))/(b+a*c^(1/2))/c/e/n/((e*x)^n)
\[ \int (e x)^{-1-n} \left (a+\frac {b}{\sqrt {c+d x^n}}\right )^p \, dx=\int (e x)^{-1-n} \left (a+\frac {b}{\sqrt {c+d x^n}}\right )^p \, dx \] Input:
Integrate[(e*x)^(-1 - n)*(a + b/Sqrt[c + d*x^n])^p,x]
Output:
Integrate[(e*x)^(-1 - n)*(a + b/Sqrt[c + d*x^n])^p, x]
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.
\(\displaystyle \int (e x)^{-n-1} \left (a+\frac {b}{\sqrt {c+d x^n}}\right )^p \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int (e x)^{-n-1} \left (a+\frac {b}{\sqrt {c+d x^n}}\right )^pdx\) |
Input:
Int[(e*x)^(-1 - n)*(a + b/Sqrt[c + d*x^n])^p,x]
Output:
$Aborted
\[\int \left (e x \right )^{-1-n} {\left (a +\frac {b}{\sqrt {c +d \,x^{n}}}\right )}^{p}d x\]
Input:
int((e*x)^(-1-n)*(a+b/(c+d*x^n)^(1/2))^p,x)
Output:
int((e*x)^(-1-n)*(a+b/(c+d*x^n)^(1/2))^p,x)
Exception generated. \[ \int (e x)^{-1-n} \left (a+\frac {b}{\sqrt {c+d x^n}}\right )^p \, dx=\text {Exception raised: TypeError} \] Input:
integrate((e*x)^(-1-n)*(a+b/(c+d*x^n)^(1/2))^p,x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: do_a lg_rde: unimplemented kernel
Timed out. \[ \int (e x)^{-1-n} \left (a+\frac {b}{\sqrt {c+d x^n}}\right )^p \, dx=\text {Timed out} \] Input:
integrate((e*x)**(-1-n)*(a+b/(c+d*x**n)**(1/2))**p,x)
Output:
Timed out
\[ \int (e x)^{-1-n} \left (a+\frac {b}{\sqrt {c+d x^n}}\right )^p \, dx=\int { \left (e x\right )^{-n - 1} {\left (a + \frac {b}{\sqrt {d x^{n} + c}}\right )}^{p} \,d x } \] Input:
integrate((e*x)^(-1-n)*(a+b/(c+d*x^n)^(1/2))^p,x, algorithm="maxima")
Output:
integrate((e*x)^(-n - 1)*(a + b/sqrt(d*x^n + c))^p, x)
\[ \int (e x)^{-1-n} \left (a+\frac {b}{\sqrt {c+d x^n}}\right )^p \, dx=\int { \left (e x\right )^{-n - 1} {\left (a + \frac {b}{\sqrt {d x^{n} + c}}\right )}^{p} \,d x } \] Input:
integrate((e*x)^(-1-n)*(a+b/(c+d*x^n)^(1/2))^p,x, algorithm="giac")
Output:
integrate((e*x)^(-n - 1)*(a + b/sqrt(d*x^n + c))^p, x)
Timed out. \[ \int (e x)^{-1-n} \left (a+\frac {b}{\sqrt {c+d x^n}}\right )^p \, dx=\int \frac {{\left (a+\frac {b}{\sqrt {c+d\,x^n}}\right )}^p}{{\left (e\,x\right )}^{n+1}} \,d x \] Input:
int((a + b/(c + d*x^n)^(1/2))^p/(e*x)^(n + 1),x)
Output:
int((a + b/(c + d*x^n)^(1/2))^p/(e*x)^(n + 1), x)
\[ \int (e x)^{-1-n} \left (a+\frac {b}{\sqrt {c+d x^n}}\right )^p \, dx=\frac {\int \frac {\left (\sqrt {x^{n} d +c}\, a +b \right )^{p}}{x^{n} \left (x^{n} d +c \right )^{\frac {p}{2}} x}d x}{e^{n} e} \] Input:
int((e*x)^(-1-n)*(a+b/(c+d*x^n)^(1/2))^p,x)
Output:
int((sqrt(x**n*d + c)*a + b)**p/(x**n*(x**n*d + c)**(p/2)*x),x)/(e**n*e)