Integrand size = 27, antiderivative size = 323 \[ \int (e x)^{-1-2 n} \left (a+b \sqrt {c+d x^n}\right )^p \, dx=-\frac {(e x)^{-2 n} \left (a+b \sqrt {c+d x^n}\right )^p}{2 e n}+\frac {b d p x^n (e x)^{-2 n} \left (b c-a \sqrt {c+d x^n}\right ) \left (a+b \sqrt {c+d x^n}\right )^p}{4 c \left (a^2-b^2 c\right ) e n}-\frac {b d^2 \left (a-b \sqrt {c} (2-p)\right ) x^{2 n} (e x)^{-2 n} \left (a+b \sqrt {c+d x^n}\right )^p \operatorname {Hypergeometric2F1}\left (1,p,1+p,\frac {a+b \sqrt {c+d x^n}}{a-b \sqrt {c}}\right )}{8 \left (a-b \sqrt {c}\right )^2 c^{3/2} e n}+\frac {b d^2 \left (a+b \sqrt {c} (2-p)\right ) x^{2 n} (e x)^{-2 n} \left (a+b \sqrt {c+d x^n}\right )^p \operatorname {Hypergeometric2F1}\left (1,p,1+p,\frac {a+b \sqrt {c+d x^n}}{a+b \sqrt {c}}\right )}{8 \left (a+b \sqrt {c}\right )^2 c^{3/2} e n} \] Output:
-1/2*(a+b*(c+d*x^n)^(1/2))^p/e/n/((e*x)^(2*n))+1/4*b*d*p*x^n*(b*c-a*(c+d*x ^n)^(1/2))*(a+b*(c+d*x^n)^(1/2))^p/c/(-b^2*c+a^2)/e/n/((e*x)^(2*n))-1/8*b* d^2*(a-b*c^(1/2)*(2-p))*x^(2*n)*(a+b*(c+d*x^n)^(1/2))^p*hypergeom([1, p],[ p+1],(a+b*(c+d*x^n)^(1/2))/(a-b*c^(1/2)))/(a-b*c^(1/2))^2/c^(3/2)/e/n/((e* x)^(2*n))+1/8*b*d^2*(a+b*c^(1/2)*(2-p))*x^(2*n)*(a+b*(c+d*x^n)^(1/2))^p*hy pergeom([1, p],[p+1],(a+b*(c+d*x^n)^(1/2))/(a+b*c^(1/2)))/(a+b*c^(1/2))^2/ c^(3/2)/e/n/((e*x)^(2*n))
\[ \int (e x)^{-1-2 n} \left (a+b \sqrt {c+d x^n}\right )^p \, dx=\int (e x)^{-1-2 n} \left (a+b \sqrt {c+d x^n}\right )^p \, dx \] Input:
Integrate[(e*x)^(-1 - 2*n)*(a + b*Sqrt[c + d*x^n])^p,x]
Output:
Integrate[(e*x)^(-1 - 2*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.
\(\displaystyle \int (e x)^{-2 n-1} \left (a+b \sqrt {c+d x^n}\right )^p \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int (e x)^{-2 n-1} \left (a+b \sqrt {c+d x^n}\right )^pdx\) |
Input:
Int[(e*x)^(-1 - 2*n)*(a + b*Sqrt[c + d*x^n])^p,x]
Output:
$Aborted
\[\int \left (e x \right )^{-1-2 n} \left (a +b \sqrt {c +d \,x^{n}}\right )^{p}d x\]
Input:
int((e*x)^(-1-2*n)*(a+b*(c+d*x^n)^(1/2))^p,x)
Output:
int((e*x)^(-1-2*n)*(a+b*(c+d*x^n)^(1/2))^p,x)
Exception generated. \[ \int (e x)^{-1-2 n} \left (a+b \sqrt {c+d x^n}\right )^p \, dx=\text {Exception raised: TypeError} \] Input:
integrate((e*x)^(-1-2*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. \[ \int (e x)^{-1-2 n} \left (a+b \sqrt {c+d x^n}\right )^p \, dx=\text {Timed out} \] Input:
integrate((e*x)**(-1-2*n)*(a+b*(c+d*x**n)**(1/2))**p,x)
Output:
Timed out
\[ \int (e x)^{-1-2 n} \left (a+b \sqrt {c+d x^n}\right )^p \, dx=\int { \left (e x\right )^{-2 \, n - 1} {\left (\sqrt {d x^{n} + c} b + a\right )}^{p} \,d x } \] Input:
integrate((e*x)^(-1-2*n)*(a+b*(c+d*x^n)^(1/2))^p,x, algorithm="maxima")
Output:
integrate((e*x)^(-2*n - 1)*(sqrt(d*x^n + c)*b + a)^p, x)
\[ \int (e x)^{-1-2 n} \left (a+b \sqrt {c+d x^n}\right )^p \, dx=\int { \left (e x\right )^{-2 \, n - 1} {\left (\sqrt {d x^{n} + c} b + a\right )}^{p} \,d x } \] Input:
integrate((e*x)^(-1-2*n)*(a+b*(c+d*x^n)^(1/2))^p,x, algorithm="giac")
Output:
integrate((e*x)^(-2*n - 1)*(sqrt(d*x^n + c)*b + a)^p, x)
Timed out. \[ \int (e x)^{-1-2 n} \left (a+b \sqrt {c+d x^n}\right )^p \, dx=\int \frac {{\left (a+b\,\sqrt {c+d\,x^n}\right )}^p}{{\left (e\,x\right )}^{2\,n+1}} \,d x \] Input:
int((a + b*(c + d*x^n)^(1/2))^p/(e*x)^(2*n + 1),x)
Output:
int((a + b*(c + d*x^n)^(1/2))^p/(e*x)^(2*n + 1), x)
\[ \int (e x)^{-1-2 n} \left (a+b \sqrt {c+d x^n}\right )^p \, dx=\frac {\int \frac {\left (\sqrt {x^{n} d +c}\, b +a \right )^{p}}{x^{2 n} x}d x}{e^{2 n} e} \] Input:
int((e*x)^(-1-2*n)*(a+b*(c+d*x^n)^(1/2))^p,x)
Output:
int((sqrt(x**n*d + c)*b + a)**p/(x**(2*n)*x),x)/(e**(2*n)*e)