\(\int (e x)^{-1+3 n} (a+\frac {b}{(c+d x^n)^2})^p \, dx\) [358]

Optimal result

Integrand size = 25, antiderivative size = 238 \[ \int (e x)^{-1+3 n} \left (a+\frac {b}{\left (c+d x^n\right )^2}\right )^p \, dx=\frac {x^{-3 n} (e x)^{3 n} \left (c+d x^n\right )^3 \left (a+\frac {b}{\left (c+d x^n\right )^2}\right )^{1+p}}{3 a d^3 e n}+\frac {\left (3 a c^2-b (1-2 p)\right ) x^{-3 n} (e x)^{3 n} \left (c+d x^n\right ) \left (a+\frac {b}{\left (c+d x^n\right )^2}\right )^p \left (1+\frac {b}{a \left (c+d x^n\right )^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b}{a \left (c+d x^n\right )^2}\right )}{3 a d^3 e n}+\frac {b c x^{-3 n} (e x)^{3 n} \left (a+\frac {b}{\left (c+d x^n\right )^2}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (2,1+p,2+p,1+\frac {b}{a \left (c+d x^n\right )^2}\right )}{a^2 d^3 e n (1+p)} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 1.73 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.26, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {7273, 2089, 1804, 1802, 1291, 25, 27, 1269, 1118, 279, 278}

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.

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

\(\Big \downarrow \) 7273

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

\(\Big \downarrow \) 2089

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

\(\Big \downarrow \) 1804

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

\(\Big \downarrow \) 1802

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

\(\Big \downarrow \) 1291

\(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \left (c+d x^n\right )^{2 p} \left (a+\frac {b}{\left (c+d x^n\right )^2}\right )^p \left (a \left (c+d x^n\right )^2+b\right )^{-p} \left (\frac {\int -d^2 \left (d x^n+c\right )^{-2 p} \left (6 a c d x^n+3 a c^2+b-2 b p\right ) \left (2 a c d x^n+a d^2 x^{2 n}+a c^2+b\right )^pdx^n}{3 a d^4}+\frac {\left (c+d x^n\right )^{1-2 p} \left (a c^2+2 a c d x^n+a d^2 x^{2 n}+b\right )^{p+1}}{3 a d^3}\right )}{e n}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \left (c+d x^n\right )^{2 p} \left (a+\frac {b}{\left (c+d x^n\right )^2}\right )^p \left (a \left (c+d x^n\right )^2+b\right )^{-p} \left (\frac {\left (c+d x^n\right )^{1-2 p} \left (a c^2+2 a c d x^n+a d^2 x^{2 n}+b\right )^{p+1}}{3 a d^3}-\frac {\int d^2 \left (d x^n+c\right )^{-2 p} \left (6 a c d x^n+3 a c^2+b-2 b p\right ) \left (2 a c d x^n+a d^2 x^{2 n}+a c^2+b\right )^pdx^n}{3 a d^4}\right )}{e n}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \left (c+d x^n\right )^{2 p} \left (a+\frac {b}{\left (c+d x^n\right )^2}\right )^p \left (a \left (c+d x^n\right )^2+b\right )^{-p} \left (\frac {\left (c+d x^n\right )^{1-2 p} \left (a c^2+2 a c d x^n+a d^2 x^{2 n}+b\right )^{p+1}}{3 a d^3}-\frac {\int \left (d x^n+c\right )^{-2 p} \left (6 a c d x^n+3 a c^2+b-2 b p\right ) \left (2 a c d x^n+a d^2 x^{2 n}+a c^2+b\right )^pdx^n}{3 a d^2}\right )}{e n}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \left (c+d x^n\right )^{2 p} \left (a+\frac {b}{\left (c+d x^n\right )^2}\right )^p \left (a \left (c+d x^n\right )^2+b\right )^{-p} \left (\frac {\left (c+d x^n\right )^{1-2 p} \left (a c^2+2 a c d x^n+a d^2 x^{2 n}+b\right )^{p+1}}{3 a d^3}-\frac {6 a c \int \left (d x^n+c\right )^{1-2 p} \left (2 a c d x^n+a d^2 x^{2 n}+a c^2+b\right )^pdx^n+\left (-3 a c^2-2 b p+b\right ) \int \left (d x^n+c\right )^{-2 p} \left (2 a c d x^n+a d^2 x^{2 n}+a c^2+b\right )^pdx^n}{3 a d^2}\right )}{e n}\)

\(\Big \downarrow \) 1118

\(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \left (c+d x^n\right )^{2 p} \left (a+\frac {b}{\left (c+d x^n\right )^2}\right )^p \left (a \left (c+d x^n\right )^2+b\right )^{-p} \left (\frac {\left (c+d x^n\right )^{1-2 p} \left (a c^2+2 a c d x^n+a d^2 x^{2 n}+b\right )^{p+1}}{3 a d^3}-\frac {\frac {\left (-3 a c^2-2 b p+b\right ) \int \left (d x^n+c\right )^{-2 p} \left (a x^{2 n}+b\right )^pd\left (d x^n+c\right )}{d}+\frac {6 a c \int \left (d x^n+c\right )^{1-2 p} \left (a x^{2 n}+b\right )^pd\left (d x^n+c\right )}{d}}{3 a d^2}\right )}{e n}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \left (c+d x^n\right )^{2 p} \left (a+\frac {b}{\left (c+d x^n\right )^2}\right )^p \left (a \left (c+d x^n\right )^2+b\right )^{-p} \left (\frac {\left (c+d x^n\right )^{1-2 p} \left (a c^2+2 a c d x^n+a d^2 x^{2 n}+b\right )^{p+1}}{3 a d^3}-\frac {\frac {\left (-3 a c^2-2 b p+b\right ) \left (a x^{2 n}+b\right )^p \left (\frac {a x^{2 n}}{b}+1\right )^{-p} \int \left (d x^n+c\right )^{-2 p} \left (\frac {a x^{2 n}}{b}+1\right )^pd\left (d x^n+c\right )}{d}+\frac {6 a c \left (a x^{2 n}+b\right )^p \left (\frac {a x^{2 n}}{b}+1\right )^{-p} \int \left (d x^n+c\right )^{1-2 p} \left (\frac {a x^{2 n}}{b}+1\right )^pd\left (d x^n+c\right )}{d}}{3 a d^2}\right )}{e n}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \left (c+d x^n\right )^{2 p} \left (a+\frac {b}{\left (c+d x^n\right )^2}\right )^p \left (a \left (c+d x^n\right )^2+b\right )^{-p} \left (\frac {\left (c+d x^n\right )^{1-2 p} \left (a c^2+2 a c d x^n+a d^2 x^{2 n}+b\right )^{p+1}}{3 a d^3}-\frac {\frac {\left (-3 a c^2-2 b p+b\right ) \left (a x^{2 n}+b\right )^p \left (\frac {a x^{2 n}}{b}+1\right )^{-p} \left (c+d x^n\right )^{1-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),-p,\frac {1}{2} (3-2 p),-\frac {a x^{2 n}}{b}\right )}{d (1-2 p)}+\frac {3 a c \left (a x^{2 n}+b\right )^p \left (\frac {a x^{2 n}}{b}+1\right )^{-p} \left (c+d x^n\right )^{2-2 p} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {a x^{2 n}}{b}\right )}{d (1-p)}}{3 a d^2}\right )}{e n}\)

Input:

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

Output:

((e*x)^(3*n)*(c + d*x^n)^(2*p)*(a + b/(c + d*x^n)^2)^p*(((c + d*x^n)^(1 - 
2*p)*(b + a*c^2 + 2*a*c*d*x^n + a*d^2*x^(2*n))^(1 + p))/(3*a*d^3) - (((b - 
 3*a*c^2 - 2*b*p)*(c + d*x^n)^(1 - 2*p)*(b + a*x^(2*n))^p*Hypergeometric2F 
1[(1 - 2*p)/2, -p, (3 - 2*p)/2, -((a*x^(2*n))/b)])/(d*(1 - 2*p)*(1 + (a*x^ 
(2*n))/b)^p) + (3*a*c*(c + d*x^n)^(2 - 2*p)*(b + a*x^(2*n))^p*Hypergeometr 
ic2F1[1 - p, -p, 2 - p, -((a*x^(2*n))/b)])/(d*(1 - p)*(1 + (a*x^(2*n))/b)^ 
p))/(3*a*d^2)))/(e*n*x^(3*n)*(b + a*(c + d*x^n)^2)^p)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 1118

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[1/e   Subst[Int[x^m*(a - b^2/(4*c) + (c*x^2)/e^2)^p, x], x, 
d + e*x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1291

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b 
*x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m 
 + n + 2*p + 1))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m 
 + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d 
+ e*x)^(n - 2)*(b*d*e*(p + 1) + a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) 
 - e*(2*c*d - b*e)*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
, m, p}, x] && IGtQ[n, 1] && NeQ[m + n + 2*p + 1, 0]
 

rule 1802

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + ( 
e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1 
)/n] - 1)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, 
c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 1804

Int[((f_)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*(( 
d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[f^IntPart[m]*((f*x)^FracPar 
t[m]/x^FracPart[m])   Int[x^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], 
x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[ 
Simplify[(m + 1)/n]]
 

rule 2089

Int[(u_)^(p_.)*((f_.)*(x_))^(m_.)*(z_)^(q_.), x_Symbol] :> Int[(f*x)^m*Expa 
ndToSum[z, x]^q*ExpandToSum[u, x]^p, x] /; FreeQ[{f, m, p, q}, x] && Binomi 
alQ[z, x] && TrinomialQ[u, x] &&  !(BinomialMatchQ[z, x] && TrinomialMatchQ 
[u, x])
 

rule 7273

Int[(u_.)*((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*v^n)^Fra 
cPart[p]/(v^(n*FracPart[p])*(b + a/v^n)^FracPart[p])   Int[u*v^(n*p)*(b + a 
/v^n)^p, x], x] /; FreeQ[{a, b, p}, x] &&  !IntegerQ[p] && ILtQ[n, 0] && Bi 
nomialQ[v, x] &&  !LinearQ[v, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((e*x)^(-1+3*n)*(a+b/(c+d*x^n)^2)^p,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((e*x)^(-1+3*n)*(a+b/(c+d*x^n)^2)^p,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x)^(-1+3*n)*(a+b/(c+d*x^n)^2)^p,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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