Integrand size = 21, antiderivative size = 299 \[ \int \left (d+e x^n\right )^3 \left (a+c x^{2 n}\right )^p \, dx=\frac {3 d e^2 x^{1+2 n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (2+\frac {1}{n}\right ),-p,\frac {1}{2} \left (4+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{1+2 n}+\frac {e^3 x^{1+3 n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (3+\frac {1}{n}\right ),-p,\frac {1}{2} \left (5+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{1+3 n}+d^3 x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2 n},-p,\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+\frac {3 d^2 e x^{1+n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2 n},-p,\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{1+n} \]
3*d*e^2*x^(1+2*n)*(a+c*x^(2*n))^p*hypergeom([-p, 1+1/2/n],[2+1/2/n],-c*x^( 2*n)/a)/(1+2*n)/((1+c*x^(2*n)/a)^p)+e^3*x^(1+3*n)*(a+c*x^(2*n))^p*hypergeo m([-p, 3/2+1/2/n],[5/2+1/2/n],-c*x^(2*n)/a)/(1+3*n)/((1+c*x^(2*n)/a)^p)+d^ 3*x*(a+c*x^(2*n))^p*hypergeom([-p, 1/2/n],[1+1/2/n],-c*x^(2*n)/a)/((1+c*x^ (2*n)/a)^p)+3*d^2*e*x^(1+n)*(a+c*x^(2*n))^p*hypergeom([-p, 1/2*(1+n)/n],[3 /2+1/2/n],-c*x^(2*n)/a)/(1+n)/((1+c*x^(2*n)/a)^p)
Time = 0.20 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.71 \[ \int \left (d+e x^n\right )^3 \left (a+c x^{2 n}\right )^p \, dx=x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \left (\frac {3 d e^2 x^{2 n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (2+\frac {1}{n}\right ),-p,\frac {1}{2} \left (4+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{1+2 n}+\frac {e^3 x^{3 n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (3+\frac {1}{n}\right ),-p,\frac {1}{2} \left (5+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{1+3 n}+d^2 \left (d \operatorname {Hypergeometric2F1}\left (\frac {1}{2 n},-p,\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+\frac {3 e x^n \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2 n},-p,\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{1+n}\right )\right ) \]
(x*(a + c*x^(2*n))^p*((3*d*e^2*x^(2*n)*Hypergeometric2F1[(2 + n^(-1))/2, - p, (4 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 + 2*n) + (e^3*x^(3*n)*Hypergeomet ric2F1[(3 + n^(-1))/2, -p, (5 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 + 3*n) + d^2*(d*Hypergeometric2F1[1/(2*n), -p, (2 + n^(-1))/2, -((c*x^(2*n))/a)] + (3*e*x^n*Hypergeometric2F1[(1 + n)/(2*n), -p, (3 + n^(-1))/2, -((c*x^(2*n) )/a)])/(1 + n))))/(1 + (c*x^(2*n))/a)^p
Time = 0.42 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1767, 2009}
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.
\(\displaystyle \int \left (d+e x^n\right )^3 \left (a+c x^{2 n}\right )^p \, dx\) |
\(\Big \downarrow \) 1767 |
\(\displaystyle \int \left (d^3 \left (a+c x^{2 n}\right )^p+3 d^2 e x^n \left (a+c x^{2 n}\right )^p+3 d e^2 x^{2 n} \left (a+c x^{2 n}\right )^p+e^3 x^{3 n} \left (a+c x^{2 n}\right )^p\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle d^3 x \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2 n},-p,\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+\frac {3 d^2 e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {n+1}{2 n},-p,\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{n+1}+\frac {3 d e^2 x^{2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (2+\frac {1}{n}\right ),-p,\frac {1}{2} \left (4+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{2 n+1}+\frac {e^3 x^{3 n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (3+\frac {1}{n}\right ),-p,\frac {1}{2} \left (5+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{3 n+1}\) |
(3*d*e^2*x^(1 + 2*n)*(a + c*x^(2*n))^p*Hypergeometric2F1[(2 + n^(-1))/2, - p, (4 + n^(-1))/2, -((c*x^(2*n))/a)])/((1 + 2*n)*(1 + (c*x^(2*n))/a)^p) + (e^3*x^(1 + 3*n)*(a + c*x^(2*n))^p*Hypergeometric2F1[(3 + n^(-1))/2, -p, ( 5 + n^(-1))/2, -((c*x^(2*n))/a)])/((1 + 3*n)*(1 + (c*x^(2*n))/a)^p) + (d^3 *x*(a + c*x^(2*n))^p*Hypergeometric2F1[1/(2*n), -p, (2 + n^(-1))/2, -((c*x ^(2*n))/a)])/(1 + (c*x^(2*n))/a)^p + (3*d^2*e*x^(1 + n)*(a + c*x^(2*n))^p* Hypergeometric2F1[(1 + n)/(2*n), -p, (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(( 1 + n)*(1 + (c*x^(2*n))/a)^p)
3.1.60.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a , c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && ((Integ ersQ[p, q] && !IntegerQ[n]) || IGtQ[p, 0] || (IGtQ[q, 0] && !IntegerQ[n]) )
\[\int \left (d +e \,x^{n}\right )^{3} \left (a +c \,x^{2 n}\right )^{p}d x\]
\[ \int \left (d+e x^n\right )^3 \left (a+c x^{2 n}\right )^p \, dx=\int { {\left (e x^{n} + d\right )}^{3} {\left (c x^{2 \, n} + a\right )}^{p} \,d x } \]
Timed out. \[ \int \left (d+e x^n\right )^3 \left (a+c x^{2 n}\right )^p \, dx=\text {Timed out} \]
\[ \int \left (d+e x^n\right )^3 \left (a+c x^{2 n}\right )^p \, dx=\int { {\left (e x^{n} + d\right )}^{3} {\left (c x^{2 \, n} + a\right )}^{p} \,d x } \]
Exception generated. \[ \int \left (d+e x^n\right )^3 \left (a+c x^{2 n}\right )^p \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{96,[1,0,6,4,3,5,4,1,2]%%%}+%%%{480,[1,0,6,4,3,4,4,1,2]%%%} +%%%{960,
Timed out. \[ \int \left (d+e x^n\right )^3 \left (a+c x^{2 n}\right )^p \, dx=\int {\left (a+c\,x^{2\,n}\right )}^p\,{\left (d+e\,x^n\right )}^3 \,d x \]