\(\int (e x)^{-1+3 n} (a+b (c+d x^n)^2)^p \, dx\) [340]

Optimal result

Integrand size = 25, antiderivative size = 311 \[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^2\right )^p \, dx=-\frac {c (2+p) x^{-3 n} (e x)^{3 n} \left (a+b c^2+2 b c d x^n+b d^2 x^{2 n}\right )^{1+p}}{b d^3 e n (1+p) (3+2 p)}+\frac {x^{-2 n} (e x)^{3 n} \left (a+b c^2+2 b c d x^n+b d^2 x^{2 n}\right )^{1+p}}{b d^2 e n (3+2 p)}+\frac {2^p \left (a-b c^2 (3+2 p)\right ) x^{-3 n} (e x)^{3 n} \left (\frac {\sqrt {-a}-\sqrt {b} c-\sqrt {b} d x^n}{\sqrt {-a}}\right )^{-1-p} \left (a+b c^2+2 b c d x^n+b d^2 x^{2 n}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {\sqrt {-a}+\sqrt {b} c+\sqrt {b} d x^n}{2 \sqrt {-a}}\right )}{\sqrt {-a} b^{3/2} d^3 e n (1+p) (3+2 p)} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 1.45 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.60 \[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^2\right )^p \, dx=\frac {(e x)^{3 n} \left (\frac {\sqrt {-a b d^2}-b d \left (c+d x^n\right )}{-b c d+\sqrt {-a b d^2}}\right )^{-p} \left (\frac {\sqrt {-a b d^2}+b d \left (c+d x^n\right )}{b c d+\sqrt {-a b d^2}}\right )^{-p} \left (a+b \left (c+d x^n\right )^2\right )^p \operatorname {AppellF1}\left (3,-p,-p,4,-\frac {b d^2 x^n}{b c d+\sqrt {-a b d^2}},\frac {b d^2 x^n}{-b c d+\sqrt {-a b d^2}}\right )}{3 e n} \] Input:

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

Output:

((e*x)^(3*n)*(a + b*(c + d*x^n)^2)^p*AppellF1[3, -p, -p, 4, -((b*d^2*x^n)/ 
(b*c*d + Sqrt[-(a*b*d^2)])), (b*d^2*x^n)/(-(b*c*d) + Sqrt[-(a*b*d^2)])])/( 
3*e*n*((Sqrt[-(a*b*d^2)] - b*d*(c + d*x^n))/(-(b*c*d) + Sqrt[-(a*b*d^2)])) 
^p*((Sqrt[-(a*b*d^2)] + b*d*(c + d*x^n))/(b*c*d + Sqrt[-(a*b*d^2)]))^p)
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.89, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2086, 1694, 1693, 1166, 25, 1160, 1096}

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.

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

\(\Big \downarrow \) 2086

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

\(\Big \downarrow \) 1694

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

\(\Big \downarrow \) 1693

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

\(\Big \downarrow \) 1166

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1096

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

Input:

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

Output:

((e*x)^(3*n)*((x^n*(a + b*c^2 + 2*b*c*d*x^n + b*d^2*x^(2*n))^(1 + p))/(b*d 
^2*(3 + 2*p)) - ((c*(2 + p)*(a + b*c^2 + 2*b*c*d*x^n + b*d^2*x^(2*n))^(1 + 
 p))/(d*(1 + p)) - (2^p*(a - b*c^2*(3 + 2*p))*((Sqrt[-a] - Sqrt[b]*c - Sqr 
t[b]*d*x^n)/Sqrt[-a])^(-1 - p)*(a + b*c^2 + 2*b*c*d*x^n + b*d^2*x^(2*n))^( 
1 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, (Sqrt[-a] + Sqrt[b]*c + Sqrt[b] 
*d*x^n)/(2*Sqrt[-a])])/(Sqrt[-a]*Sqrt[b]*d*(1 + p)))/(b*d^2*(3 + 2*p))))/( 
e*n*x^(3*n))
 

Defintions of rubi rules used

rule 25

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

rule 1096

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 
 - 4*a*c, 2]}, Simp[(-(a + b*x + c*x^2)^(p + 1)/(q*(p + 1)*((q - b - 2*c*x) 
/(2*q))^(p + 1)))*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q) 
], x]] /; FreeQ[{a, b, c, p}, x] &&  !IntegerQ[4*p] &&  !IntegerQ[3*p]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1166

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

rule 1693

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

rule 1694

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

rule 2086

Int[(u_)^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[(d*x)^m*ExpandToSum[u, 
x]^p, x] /; FreeQ[{d, m, p}, x] && TrinomialQ[u, x] &&  !TrinomialMatchQ[u, 
 x]
 

Maple [F]

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

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]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

(e**(3*n)*(2*x**(3*n)*(x**(2*n)*b*d**2 + 2*x**n*b*c*d + a + b*c**2)**p*b** 
2*c*d**3*p**2 + 3*x**(3*n)*(x**(2*n)*b*d**2 + 2*x**n*b*c*d + a + b*c**2)** 
p*b**2*c*d**3*p + x**(3*n)*(x**(2*n)*b*d**2 + 2*x**n*b*c*d + a + b*c**2)** 
p*b**2*c*d**3 + 2*x**(2*n)*(x**(2*n)*b*d**2 + 2*x**n*b*c*d + a + b*c**2)** 
p*b**2*c**2*d**2*p**2 + x**(2*n)*(x**(2*n)*b*d**2 + 2*x**n*b*c*d + a + b*c 
**2)**p*b**2*c**2*d**2*p + 2*x**n*(x**(2*n)*b*d**2 + 2*x**n*b*c*d + a + b* 
c**2)**p*a*b*c*d*p**2 + 2*x**n*(x**(2*n)*b*d**2 + 2*x**n*b*c*d + a + b*c** 
2)**p*a*b*c*d*p - 2*x**n*(x**(2*n)*b*d**2 + 2*x**n*b*c*d + a + b*c**2)**p* 
b**2*c**3*d*p - (x**(2*n)*b*d**2 + 2*x**n*b*c*d + a + b*c**2)**p*a**2*p - 
(x**(2*n)*b*d**2 + 2*x**n*b*c*d + a + b*c**2)**p*a**2 - (x**(2*n)*b*d**2 + 
 2*x**n*b*c*d + a + b*c**2)**p*a*b*c**2*p + (x**(2*n)*b*d**2 + 2*x**n*b*c* 
d + a + b*c**2)**p*b**2*c**4 + 8*int((x**(2*n)*(x**(2*n)*b*d**2 + 2*x**n*b 
*c*d + a + b*c**2)**p)/(4*x**(2*n)*b*d**2*p**2*x + 8*x**(2*n)*b*d**2*p*x + 
 3*x**(2*n)*b*d**2*x + 8*x**n*b*c*d*p**2*x + 16*x**n*b*c*d*p*x + 6*x**n*b* 
c*d*x + 4*a*p**2*x + 8*a*p*x + 3*a*x + 4*b*c**2*p**2*x + 8*b*c**2*p*x + 3* 
b*c**2*x),x)*a**2*b*d**2*n*p**4 + 24*int((x**(2*n)*(x**(2*n)*b*d**2 + 2*x* 
*n*b*c*d + a + b*c**2)**p)/(4*x**(2*n)*b*d**2*p**2*x + 8*x**(2*n)*b*d**2*p 
*x + 3*x**(2*n)*b*d**2*x + 8*x**n*b*c*d*p**2*x + 16*x**n*b*c*d*p*x + 6*x** 
n*b*c*d*x + 4*a*p**2*x + 8*a*p*x + 3*a*x + 4*b*c**2*p**2*x + 8*b*c**2*p*x 
+ 3*b*c**2*x),x)*a**2*b*d**2*n*p**3 + 22*int((x**(2*n)*(x**(2*n)*b*d**2...