\(\int (a+b x^n)^p (c+d x^n)^2 \, dx\) [120]

Optimal result

Integrand size = 19, antiderivative size = 197 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^2 \, dx=-\frac {d (a d (1+n)-b (c+c n (3+p))) x \left (a+b x^n\right )^{1+p}}{b^2 (1+n+n p) (1+n (2+p))}+\frac {d x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )}{b (1+n (2+p))}-\frac {\left (c (a d-b (c+c n (2+p)))-\frac {a d (a d (1+n)-b (c+c n (3+p)))}{b (1+n+n p)}\right ) x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},-\frac {b x^n}{a}\right )}{b (1+n (2+p))} \] Output:

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

Mathematica [A] (verified)

Time = 5.22 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.71 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^2 \, dx=\frac {x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (2 c d (1+2 n) x^n \operatorname {Hypergeometric2F1}\left (1+\frac {1}{n},-p,2+\frac {1}{n},-\frac {b x^n}{a}\right )+(1+n) \left (d^2 x^{2 n} \operatorname {Hypergeometric2F1}\left (2+\frac {1}{n},-p,3+\frac {1}{n},-\frac {b x^n}{a}\right )+c^2 (1+2 n) \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},-\frac {b x^n}{a}\right )\right )\right )}{(1+n) (1+2 n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {933, 25, 913, 779, 778}

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.

\(\displaystyle \int \left (c+d x^n\right )^2 \left (a+b x^n\right )^p \, dx\)

\(\Big \downarrow \) 933

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 913

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

\(\Big \downarrow \) 779

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

\(\Big \downarrow \) 778

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 779

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x 
^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(1 + b*(x^n/a))^p, x], x 
] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Si 
mplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 913

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

rule 933

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

Maple [F]

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

Input:

int((a+b*x^n)^p*(c+d*x^n)^2,x)
 

Output:

int((a+b*x^n)^p*(c+d*x^n)^2,x)
 

Fricas [F]

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

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

Output:

integral((d^2*x^(2*n) + 2*c*d*x^n + c^2)*(b*x^n + a)^p, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 20.71 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.89 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^2 \, dx=\frac {a^{\frac {1}{n}} a^{p - \frac {1}{n}} c^{2} x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{n}, - p \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {1}{n}\right )} + \frac {2 a^{1 + \frac {1}{n}} a^{p - 1 - \frac {1}{n}} c d x^{n + 1} \Gamma \left (1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 + \frac {1}{n} \\ 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {1}{n}\right )} + \frac {a^{2 + \frac {1}{n}} a^{p - 2 - \frac {1}{n}} d^{2} x^{2 n + 1} \Gamma \left (2 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 2 + \frac {1}{n} \\ 3 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (3 + \frac {1}{n}\right )} \] Input:

integrate((a+b*x**n)**p*(c+d*x**n)**2,x)
 

Output:

a**(1/n)*a**(p - 1/n)*c**2*x*gamma(1/n)*hyper((1/n, -p), (1 + 1/n,), b*x** 
n*exp_polar(I*pi)/a)/(n*gamma(1 + 1/n)) + 2*a**(1 + 1/n)*a**(p - 1 - 1/n)* 
c*d*x**(n + 1)*gamma(1 + 1/n)*hyper((-p, 1 + 1/n), (2 + 1/n,), b*x**n*exp_ 
polar(I*pi)/a)/(n*gamma(2 + 1/n)) + a**(2 + 1/n)*a**(p - 2 - 1/n)*d**2*x** 
(2*n + 1)*gamma(2 + 1/n)*hyper((-p, 2 + 1/n), (3 + 1/n,), b*x**n*exp_polar 
(I*pi)/a)/(n*gamma(3 + 1/n))
 

Maxima [F]

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

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

Output:

integrate((d*x^n + c)^2*(b*x^n + a)^p, x)
 

Giac [F(-2)]

Exception generated. \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^2 \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

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%%%{-1,[1,0,4,3,1,3,3,2,0]%%%}+%%%{-3,[1,0,4,3,1,2,3,2,0]%%%}+ 
%%%{-3,[1
 

Mupad [F(-1)]

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

int((a + b*x^n)^p*(c + d*x^n)^2,x)
 

Output:

int((a + b*x^n)^p*(c + d*x^n)^2, x)
 

Reduce [F]

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

int((a+b*x^n)^p*(c+d*x^n)^2,x)
 

Output:

(x**(2*n)*(x**n*b + a)**p*b**2*d**2*n**2*p**2*x + x**(2*n)*(x**n*b + a)**p 
*b**2*d**2*n**2*p*x + 2*x**(2*n)*(x**n*b + a)**p*b**2*d**2*n*p*x + x**(2*n 
)*(x**n*b + a)**p*b**2*d**2*n*x + x**(2*n)*(x**n*b + a)**p*b**2*d**2*x + x 
**n*(x**n*b + a)**p*a*b*d**2*n**2*p**2*x + x**n*(x**n*b + a)**p*a*b*d**2*n 
*p*x + 2*x**n*(x**n*b + a)**p*b**2*c*d*n**2*p**2*x + 4*x**n*(x**n*b + a)** 
p*b**2*c*d*n**2*p*x + 4*x**n*(x**n*b + a)**p*b**2*c*d*n*p*x + 4*x**n*(x**n 
*b + a)**p*b**2*c*d*n*x + 2*x**n*(x**n*b + a)**p*b**2*c*d*x - (x**n*b + a) 
**p*a**2*d**2*n**2*p*x - (x**n*b + a)**p*a**2*d**2*n*p*x + 2*(x**n*b + a)* 
*p*a*b*c*d*n**2*p**2*x + 4*(x**n*b + a)**p*a*b*c*d*n**2*p*x + 2*(x**n*b + 
a)**p*a*b*c*d*n*p*x + (x**n*b + a)**p*b**2*c**2*n**2*p**2*x + 3*(x**n*b + 
a)**p*b**2*c**2*n**2*p*x + 2*(x**n*b + a)**p*b**2*c**2*n**2*x + 2*(x**n*b 
+ a)**p*b**2*c**2*n*p*x + 3*(x**n*b + a)**p*b**2*c**2*n*x + (x**n*b + a)** 
p*b**2*c**2*x + int((x**n*b + a)**p/(x**n*b*n**3*p**3 + 3*x**n*b*n**3*p**2 
 + 2*x**n*b*n**3*p + 3*x**n*b*n**2*p**2 + 6*x**n*b*n**2*p + 2*x**n*b*n**2 
+ 3*x**n*b*n*p + 3*x**n*b*n + x**n*b + a*n**3*p**3 + 3*a*n**3*p**2 + 2*a*n 
**3*p + 3*a*n**2*p**2 + 6*a*n**2*p + 2*a*n**2 + 3*a*n*p + 3*a*n + a),x)*a* 
*3*d**2*n**5*p**4 + 3*int((x**n*b + a)**p/(x**n*b*n**3*p**3 + 3*x**n*b*n** 
3*p**2 + 2*x**n*b*n**3*p + 3*x**n*b*n**2*p**2 + 6*x**n*b*n**2*p + 2*x**n*b 
*n**2 + 3*x**n*b*n*p + 3*x**n*b*n + x**n*b + a*n**3*p**3 + 3*a*n**3*p**2 + 
 2*a*n**3*p + 3*a*n**2*p**2 + 6*a*n**2*p + 2*a*n**2 + 3*a*n*p + 3*a*n +...