Integrand size = 25, antiderivative size = 93 \[ \int (e x)^{-1+n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx=\frac {x^{-n} (e x)^n \left (c+d x^n\right ) \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \left (1+\frac {b \left (c+d x^n\right )^{3/2}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},-p,\frac {5}{3},-\frac {b \left (c+d x^n\right )^{3/2}}{a}\right )}{d e n} \] Output:
(e*x)^n*(c+d*x^n)*(a+b*(c+d*x^n)^(3/2))^p*hypergeom([2/3, -p],[5/3],-b*(c+ d*x^n)^(3/2)/a)/d/e/n/(x^n)/((1+b*(c+d*x^n)^(3/2)/a)^p)
Time = 0.33 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.01 \[ \int (e x)^{-1+n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx=\frac {x^{1-n} (e x)^{-1+n} \left (c+d x^n\right ) \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \left (1+\frac {b \left (c+d x^n\right )^{3/2}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},-p,\frac {5}{3},-\frac {b \left (c+d x^n\right )^{3/2}}{a}\right )}{d n} \] Input:
Integrate[(e*x)^(-1 + n)*(a + b*(c + d*x^n)^(3/2))^p,x]
Output:
(x^(1 - n)*(e*x)^(-1 + n)*(c + d*x^n)*(a + b*(c + d*x^n)^(3/2))^p*Hypergeo metric2F1[2/3, -p, 5/3, -((b*(c + d*x^n)^(3/2))/a)])/(d*n*(1 + (b*(c + d*x ^n)^(3/2))/a)^p)
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.
\(\displaystyle \int (e x)^{n-1} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int (e x)^{n-1} \left (a+b \left (c+d x^n\right )^{3/2}\right )^pdx\) |
Input:
Int[(e*x)^(-1 + n)*(a + b*(c + d*x^n)^(3/2))^p,x]
Output:
$Aborted
\[\int \left (e x \right )^{-1+n} {\left (a +b \left (c +d \,x^{n}\right )^{\frac {3}{2}}\right )}^{p}d x\]
Input:
int((e*x)^(-1+n)*(a+b*(c+d*x^n)^(3/2))^p,x)
Output:
int((e*x)^(-1+n)*(a+b*(c+d*x^n)^(3/2))^p,x)
Exception generated. \[ \int (e x)^{-1+n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx=\text {Exception raised: TypeError} \] Input:
integrate((e*x)^(-1+n)*(a+b*(c+d*x^n)^(3/2))^p,x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: do_a lg_rde: unimplemented kernel
Timed out. \[ \int (e x)^{-1+n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx=\text {Timed out} \] Input:
integrate((e*x)**(-1+n)*(a+b*(c+d*x**n)**(3/2))**p,x)
Output:
Timed out
\[ \int (e x)^{-1+n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx=\int { {\left ({\left (d x^{n} + c\right )}^{\frac {3}{2}} b + a\right )}^{p} \left (e x\right )^{n - 1} \,d x } \] Input:
integrate((e*x)^(-1+n)*(a+b*(c+d*x^n)^(3/2))^p,x, algorithm="maxima")
Output:
integrate(((d*x^n + c)^(3/2)*b + a)^p*(e*x)^(n - 1), x)
\[ \int (e x)^{-1+n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx=\int { {\left ({\left (d x^{n} + c\right )}^{\frac {3}{2}} b + a\right )}^{p} \left (e x\right )^{n - 1} \,d x } \] Input:
integrate((e*x)^(-1+n)*(a+b*(c+d*x^n)^(3/2))^p,x, algorithm="giac")
Output:
integrate(((d*x^n + c)^(3/2)*b + a)^p*(e*x)^(n - 1), x)
Timed out. \[ \int (e x)^{-1+n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx=\int {\left (e\,x\right )}^{n-1}\,{\left (a+b\,{\left (c+d\,x^n\right )}^{3/2}\right )}^p \,d x \] Input:
int((e*x)^(n - 1)*(a + b*(c + d*x^n)^(3/2))^p,x)
Output:
int((e*x)^(n - 1)*(a + b*(c + d*x^n)^(3/2))^p, x)
\[ \int (e x)^{-1+n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx=\frac {e^{n} \left (\int \frac {x^{n} \left (x^{n} \sqrt {x^{n} d +c}\, b d +\sqrt {x^{n} d +c}\, b c +a \right )^{p}}{x}d x \right )}{e} \] Input:
int((e*x)^(-1+n)*(a+b*(c+d*x^n)^(3/2))^p,x)
Output:
(e**n*int((x**n*(x**n*sqrt(x**n*d + c)*b*d + sqrt(x**n*d + c)*b*c + a)**p) /x,x))/e