\(\int x^{-1+2 n} (d+e x^n)^q (a+b x^n+c x^{2 n})^p \, dx\) [195]

Optimal result

Integrand size = 33, antiderivative size = 411 \[ \int x^{-1+2 n} \left (d+e x^n\right )^q \left (a+b x^n+c x^{2 n}\right )^p \, dx=-\frac {d \left (d+e x^n\right )^{1+q} \left (a+b x^n+c x^{2 n}\right )^p \left (1-\frac {2 c \left (d+e x^n\right )}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac {2 c \left (d+e x^n\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \operatorname {AppellF1}\left (1+q,-p,-p,2+q,\frac {2 c \left (d+e x^n\right )}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c \left (d+e x^n\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e^2 n (1+q)}+\frac {\left (d+e x^n\right )^{2+q} \left (a+b x^n+c x^{2 n}\right )^p \left (1-\frac {2 c \left (d+e x^n\right )}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac {2 c \left (d+e x^n\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \operatorname {AppellF1}\left (2+q,-p,-p,3+q,\frac {2 c \left (d+e x^n\right )}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c \left (d+e x^n\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e^2 n (2+q)} \] Output:

-d*(d+e*x^n)^(1+q)*(a+b*x^n+c*x^(2*n))^p*AppellF1(1+q,-p,-p,2+q,2*c*(d+e*x 
^n)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e),2*c*(d+e*x^n)/(2*c*d-(b+(-4*a*c+b^2)^ 
(1/2))*e))/e^2/n/(1+q)/((1-2*c*(d+e*x^n)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)) 
^p)/((1-2*c*(d+e*x^n)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^p)+(d+e*x^n)^(2+q) 
*(a+b*x^n+c*x^(2*n))^p*AppellF1(2+q,-p,-p,3+q,2*c*(d+e*x^n)/(2*c*d-(b-(-4* 
a*c+b^2)^(1/2))*e),2*c*(d+e*x^n)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))/e^2/n/( 
2+q)/((1-2*c*(d+e*x^n)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e))^p)/((1-2*c*(d+e*x 
^n)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^p)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1802, 1269, 1179, 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.

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

\(\Big \downarrow \) 1802

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1179

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

\(\Big \downarrow \) 150

\(\displaystyle \frac {\frac {\left (d+e x^n\right )^{q+2} \left (a+b x^n+c x^{2 n}\right )^p \left (1-\frac {2 c \left (d+e x^n\right )}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c \left (d+e x^n\right )}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} \operatorname {AppellF1}\left (q+2,-p,-p,q+3,\frac {2 c \left (e x^n+d\right )}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c \left (e x^n+d\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e^2 (q+2)}-\frac {d \left (d+e x^n\right )^{q+1} \left (a+b x^n+c x^{2 n}\right )^p \left (1-\frac {2 c \left (d+e x^n\right )}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c \left (d+e x^n\right )}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} \operatorname {AppellF1}\left (q+1,-p,-p,q+2,\frac {2 c \left (e x^n+d\right )}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c \left (e x^n+d\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e^2 (q+1)}}{n}\)

Input:

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

Output:

(-((d*(d + e*x^n)^(1 + q)*(a + b*x^n + c*x^(2*n))^p*AppellF1[1 + q, -p, -p 
, 2 + q, (2*c*(d + e*x^n))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e), (2*c*(d + 
e*x^n))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(e^2*(1 + q)*(1 - (2*c*(d + 
e*x^n))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e))^p*(1 - (2*c*(d + e*x^n))/(2*c 
*d - (b + Sqrt[b^2 - 4*a*c])*e))^p)) + ((d + e*x^n)^(2 + q)*(a + b*x^n + c 
*x^(2*n))^p*AppellF1[2 + q, -p, -p, 3 + q, (2*c*(d + e*x^n))/(2*c*d - (b - 
 Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x^n))/(2*c*d - (b + Sqrt[b^2 - 4*a*c]) 
*e)])/(e^2*(2 + q)*(1 - (2*c*(d + e*x^n))/(2*c*d - (b - Sqrt[b^2 - 4*a*c]) 
*e))^p*(1 - (2*c*(d + e*x^n))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^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 1179

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

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 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]]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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