\(\int (e x)^{-1+3 n} (a+b (c+d x^n)^{3/2})^p \, dx\) [363]

Optimal result

Integrand size = 27, antiderivative size = 319 \[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx=-\frac {2 a x^{-3 n} (e x)^{3 n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^{1+p}}{3 b^2 d^3 e n (1+p) (2+p)}+\frac {2 x^{-3 n} (e x)^{3 n} \left (c+d x^n\right )^{3/2} \left (a+b \left (c+d x^n\right )^{3/2}\right )^{1+p}}{3 b d^3 e n (2+p)}+\frac {c^2 x^{-3 n} (e x)^{3 n} \left (c+d x^n\right ) \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \left (1+\frac {b \left (c+d x^n\right )^{3/2}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},-p,\frac {5}{3},-\frac {b \left (c+d x^n\right )^{3/2}}{a}\right )}{d^3 e n}-\frac {c x^{-3 n} (e x)^{3 n} \left (c+d x^n\right )^2 \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \left (1+\frac {b \left (c+d x^n\right )^{3/2}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {4}{3},-p,\frac {7}{3},-\frac {b \left (c+d x^n\right )^{3/2}}{a}\right )}{d^3 e n} \] Output:

-2/3*a*(e*x)^(3*n)*(a+b*(c+d*x^n)^(3/2))^(p+1)/b^2/d^3/e/n/(p+1)/(2+p)/(x^ 
(3*n))+2/3*(e*x)^(3*n)*(c+d*x^n)^(3/2)*(a+b*(c+d*x^n)^(3/2))^(p+1)/b/d^3/e 
/n/(2+p)/(x^(3*n))+c^2*(e*x)^(3*n)*(c+d*x^n)*(a+b*(c+d*x^n)^(3/2))^p*hyper 
geom([2/3, -p],[5/3],-b*(c+d*x^n)^(3/2)/a)/d^3/e/n/(x^(3*n))/((1+b*(c+d*x^ 
n)^(3/2)/a)^p)-c*(e*x)^(3*n)*(c+d*x^n)^2*(a+b*(c+d*x^n)^(3/2))^p*hypergeom 
([4/3, -p],[7/3],-b*(c+d*x^n)^(3/2)/a)/d^3/e/n/(x^(3*n))/((1+b*(c+d*x^n)^( 
3/2)/a)^p)
 

Mathematica [F]

\[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx=\int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx \] Input:

Integrate[(e*x)^(-1 + 3*n)*(a + b*(c + d*x^n)^(3/2))^p,x]
 

Output:

Integrate[(e*x)^(-1 + 3*n)*(a + b*(c + d*x^n)^(3/2))^p, x]
 

Rubi [F]

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.

\(\displaystyle \int (e x)^{3 n-1} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int (e x)^{3 n-1} \left (a+b \left (c+d x^n\right )^{3/2}\right )^pdx\)

Input:

Int[(e*x)^(-1 + 3*n)*(a + b*(c + d*x^n)^(3/2))^p,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

\[\int \left (e x \right )^{-1+3 n} {\left (a +b \left (c +d \,x^{n}\right )^{\frac {3}{2}}\right )}^{p}d x\]

Input:

int((e*x)^(-1+3*n)*(a+b*(c+d*x^n)^(3/2))^p,x)
 

Output:

int((e*x)^(-1+3*n)*(a+b*(c+d*x^n)^(3/2))^p,x)
 

Fricas [F(-2)]

Exception generated. \[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x)^(-1+3*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
 

Sympy [F(-1)]

Timed out. \[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx=\text {Timed out} \] Input:

integrate((e*x)**(-1+3*n)*(a+b*(c+d*x**n)**(3/2))**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx=\int { {\left ({\left (d x^{n} + c\right )}^{\frac {3}{2}} b + a\right )}^{p} \left (e x\right )^{3 \, n - 1} \,d x } \] Input:

integrate((e*x)^(-1+3*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)^(3*n - 1), x)
 

Giac [F]

\[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx=\int { {\left ({\left (d x^{n} + c\right )}^{\frac {3}{2}} b + a\right )}^{p} \left (e x\right )^{3 \, n - 1} \,d x } \] Input:

integrate((e*x)^(-1+3*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)^(3*n - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx=\int {\left (e\,x\right )}^{3\,n-1}\,{\left (a+b\,{\left (c+d\,x^n\right )}^{3/2}\right )}^p \,d x \] Input:

int((e*x)^(3*n - 1)*(a + b*(c + d*x^n)^(3/2))^p,x)
 

Output:

int((e*x)^(3*n - 1)*(a + b*(c + d*x^n)^(3/2))^p, x)
 

Reduce [F]

\[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^{3/2}\right )^p \, dx=\frac {e^{3 n} \left (\int \frac {x^{3 n} \left (x^{n} \sqrt {x^{n} d +c}\, b d +\sqrt {x^{n} d +c}\, b c +a \right )^{p}}{x}d x \right )}{e} \] Input:

int((e*x)^(-1+3*n)*(a+b*(c+d*x^n)^(3/2))^p,x)
 

Output:

(e**(3*n)*int((x**(3*n)*(x**n*sqrt(x**n*d + c)*b*d + sqrt(x**n*d + c)*b*c 
+ a)**p)/x,x))/e