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