\(\int x^{-1+2 n} (d+e x^n)^q (a+c x^{2 n})^p \, dx\) [73]

Optimal result

Integrand size = 28, antiderivative size = 214 \[ \int x^{-1+2 n} \left (d+e x^n\right )^q \left (a+c x^{2 n}\right )^p \, dx=\frac {\left (d+e x^n\right )^q \left (a+c x^{2 n}\right )^{1+p}}{2 c n (1+p)}-\frac {\left (d+e x^n\right )^q \left (a+c x^{2 n}\right )^{1+p} \left (1-\frac {d+e x^n}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-1-p} \left (1-\frac {d+e x^n}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-1-p} \operatorname {AppellF1}\left (q,-1-p,-1-p,1+q,\frac {d+e x^n}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x^n}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{2 c n (1+p)} \] Output:

1/2*(d+e*x^n)^q*(a+c*x^(2*n))^(p+1)/c/n/(p+1)-1/2*(d+e*x^n)^q*(a+c*x^(2*n) 
)^(p+1)*(1-(d+e*x^n)/(d-(-a)^(1/2)*e/c^(1/2)))^(-1-p)*(1-(d+e*x^n)/(d+(-a) 
^(1/2)*e/c^(1/2)))^(-1-p)*AppellF1(q,-1-p,-1-p,1+q,(d+e*x^n)/(d-(-a)^(1/2) 
*e/c^(1/2)),(d+e*x^n)/(d+(-a)^(1/2)*e/c^(1/2)))/c/n/(p+1)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.50 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.57, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1803, 624, 514, 150}

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.

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

\(\Big \downarrow \) 1803

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

\(\Big \downarrow \) 624

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

\(\Big \downarrow \) 514

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

\(\Big \downarrow \) 150

\(\displaystyle \frac {\frac {\left (a+c x^{2 n}\right )^p \left (d+e x^n\right )^{q+2} \left (1-\frac {d+e x^n}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \left (1-\frac {d+e x^n}{\frac {\sqrt {-a} e}{\sqrt {c}}+d}\right )^{-p} \operatorname {AppellF1}\left (q+2,-p,-p,q+3,\frac {e x^n+d}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {e x^n+d}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e^2 (q+2)}-\frac {d \left (a+c x^{2 n}\right )^p \left (d+e x^n\right )^{q+1} \left (1-\frac {d+e x^n}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \left (1-\frac {d+e x^n}{\frac {\sqrt {-a} e}{\sqrt {c}}+d}\right )^{-p} \operatorname {AppellF1}\left (q+1,-p,-p,q+2,\frac {e x^n+d}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {e x^n+d}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e^2 (q+1)}}{n}\)

Input:

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

Output:

(-((d*(d + e*x^n)^(1 + q)*(a + c*x^(2*n))^p*AppellF1[1 + q, -p, -p, 2 + q, 
 (d + e*x^n)/(d - (Sqrt[-a]*e)/Sqrt[c]), (d + e*x^n)/(d + (Sqrt[-a]*e)/Sqr 
t[c])])/(e^2*(1 + q)*(1 - (d + e*x^n)/(d - (Sqrt[-a]*e)/Sqrt[c]))^p*(1 - ( 
d + e*x^n)/(d + (Sqrt[-a]*e)/Sqrt[c]))^p)) + ((d + e*x^n)^(2 + q)*(a + c*x 
^(2*n))^p*AppellF1[2 + q, -p, -p, 3 + q, (d + e*x^n)/(d - (Sqrt[-a]*e)/Sqr 
t[c]), (d + e*x^n)/(d + (Sqrt[-a]*e)/Sqrt[c])])/(e^2*(2 + q)*(1 - (d + e*x 
^n)/(d - (Sqrt[-a]*e)/Sqrt[c]))^p*(1 - (d + e*x^n)/(d + (Sqrt[-a]*e)/Sqrt[ 
c]))^p))/n
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 514

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Rt[-a/b, 2]}, Simp[(a + b*x^2)^p/(d*(1 - (c + d*x)/(c - d*q))^p*(1 - ( 
c + d*x)/(c + d*q))^p)   Subst[Int[x^n*Simp[1 - x/(c + d*q), x]^p*Simp[1 - 
x/(c - d*q), x]^p, x], x, c + d*x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && 
NeQ[b*c^2 + a*d^2, 0]
 

rule 624

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[1/d   Int[x^(m - 1)*(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] - Si 
mp[c/d   Int[x^(m - 1)*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, 
 d, n, p}, x] && IGtQ[m, 0]
 

rule 1803

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*x**(2*n)*(x**n*e + d)**q*(x**(2*n)*c + a)**p*c*e*p + x**(2*n)*(x**n*e + 
 d)**q*(x**(2*n)*c + a)**p*c*e*q + x**(2*n)*(x**n*e + d)**q*(x**(2*n)*c + 
a)**p*c*e + x**n*(x**n*e + d)**q*(x**(2*n)*c + a)**p*c*d*q + 2*(x**n*e + d 
)**q*(x**(2*n)*c + a)**p*a*e*p + 2*(x**n*e + d)**q*(x**(2*n)*c + a)**p*a*e 
*q + (x**n*e + d)**q*(x**(2*n)*c + a)**p*a*e - 16*int((x**(3*n)*(x**n*e + 
d)**q*(x**(2*n)*c + a)**p)/(4*x**(3*n)*c*e*p**2*x + 4*x**(3*n)*c*e*p*q*x + 
 6*x**(3*n)*c*e*p*x + x**(3*n)*c*e*q**2*x + 3*x**(3*n)*c*e*q*x + 2*x**(3*n 
)*c*e*x + 4*x**(2*n)*c*d*p**2*x + 4*x**(2*n)*c*d*p*q*x + 6*x**(2*n)*c*d*p* 
x + x**(2*n)*c*d*q**2*x + 3*x**(2*n)*c*d*q*x + 2*x**(2*n)*c*d*x + 4*x**n*a 
*e*p**2*x + 4*x**n*a*e*p*q*x + 6*x**n*a*e*p*x + x**n*a*e*q**2*x + 3*x**n*a 
*e*q*x + 2*x**n*a*e*x + 4*a*d*p**2*x + 4*a*d*p*q*x + 6*a*d*p*x + a*d*q**2* 
x + 3*a*d*q*x + 2*a*d*x),x)*a*c*e**2*n*p**3*q - 24*int((x**(3*n)*(x**n*e + 
 d)**q*(x**(2*n)*c + a)**p)/(4*x**(3*n)*c*e*p**2*x + 4*x**(3*n)*c*e*p*q*x 
+ 6*x**(3*n)*c*e*p*x + x**(3*n)*c*e*q**2*x + 3*x**(3*n)*c*e*q*x + 2*x**(3* 
n)*c*e*x + 4*x**(2*n)*c*d*p**2*x + 4*x**(2*n)*c*d*p*q*x + 6*x**(2*n)*c*d*p 
*x + x**(2*n)*c*d*q**2*x + 3*x**(2*n)*c*d*q*x + 2*x**(2*n)*c*d*x + 4*x**n* 
a*e*p**2*x + 4*x**n*a*e*p*q*x + 6*x**n*a*e*p*x + x**n*a*e*q**2*x + 3*x**n* 
a*e*q*x + 2*x**n*a*e*x + 4*a*d*p**2*x + 4*a*d*p*q*x + 6*a*d*p*x + a*d*q**2 
*x + 3*a*d*q*x + 2*a*d*x),x)*a*c*e**2*n*p**2*q**2 - 28*int((x**(3*n)*(x**n 
*e + d)**q*(x**(2*n)*c + a)**p)/(4*x**(3*n)*c*e*p**2*x + 4*x**(3*n)*c*e...