Integrand size = 17, antiderivative size = 89 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {d x \left (a+b x^n\right )^{1+p}}{b (1+n+n p)}+\left (c-\frac {a d}{b+b n+b n p}\right ) x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},-\frac {b x^n}{a}\right ) \] Output:
d*x*(a+b*x^n)^(p+1)/b/(n*p+n+1)+(c-a*d/(b*n*p+b*n+b))*x*(a+b*x^n)^p*hyperg eom([-p, 1/n],[1+1/n],-b*x^n/a)/((1+b*x^n/a)^p)
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.06 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (d \left (a+b x^n\right ) \left (1+\frac {b x^n}{a}\right )^p+(-a d+b c (1+n+n p)) \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},-\frac {b x^n}{a}\right )\right )}{b (1+n+n p)} \] Input:
Integrate[(a + b*x^n)^p*(c + d*x^n),x]
Output:
(x*(a + b*x^n)^p*(d*(a + b*x^n)*(1 + (b*x^n)/a)^p + (-(a*d) + b*c*(1 + n + n*p))*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*x^n)/a)]))/(b*(1 + n + n*p)*(1 + (b*x^n)/a)^p)
Time = 0.39 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {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.
\(\displaystyle \int \left (c+d x^n\right ) \left (a+b x^n\right )^p \, dx\) |
\(\Big \downarrow \) 913 |
\(\displaystyle \left (c-\frac {a d}{b n p+b n+b}\right ) \int \left (b x^n+a\right )^pdx+\frac {d x \left (a+b x^n\right )^{p+1}}{b (n p+n+1)}\) |
\(\Big \downarrow \) 779 |
\(\displaystyle \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c-\frac {a d}{b n p+b n+b}\right ) \int \left (\frac {b x^n}{a}+1\right )^pdx+\frac {d x \left (a+b x^n\right )^{p+1}}{b (n p+n+1)}\) |
\(\Big \downarrow \) 778 |
\(\displaystyle x \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c-\frac {a d}{b n p+b n+b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},-\frac {b x^n}{a}\right )+\frac {d x \left (a+b x^n\right )^{p+1}}{b (n p+n+1)}\) |
Input:
Int[(a + b*x^n)^p*(c + d*x^n),x]
Output:
(d*x*(a + b*x^n)^(1 + p))/(b*(1 + n + n*p)) + ((c - (a*d)/(b + b*n + b*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
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])
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])
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]
\[\int \left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )d x\]
Input:
int((a+b*x^n)^p*(c+d*x^n),x)
Output:
int((a+b*x^n)^p*(c+d*x^n),x)
\[ \int \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\int { {\left (d x^{n} + c\right )} {\left (b x^{n} + a\right )}^{p} \,d x } \] Input:
integrate((a+b*x^n)^p*(c+d*x^n),x, algorithm="fricas")
Output:
integral((d*x^n + c)*(b*x^n + a)^p, x)
Result contains complex when optimal does not.
Time = 3.00 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.20 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {a^{\frac {1}{n}} a^{p - \frac {1}{n}} c x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{n}, - p \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {1}{n}\right )} + \frac {a^{1 + \frac {1}{n}} a^{p - 1 - \frac {1}{n}} d x^{n + 1} \Gamma \left (1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 + \frac {1}{n} \\ 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {1}{n}\right )} \] Input:
integrate((a+b*x**n)**p*(c+d*x**n),x)
Output:
a**(1/n)*a**(p - 1/n)*c*x*gamma(1/n)*hyper((1/n, -p), (1 + 1/n,), b*x**n*e xp_polar(I*pi)/a)/(n*gamma(1 + 1/n)) + a**(1 + 1/n)*a**(p - 1 - 1/n)*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))
\[ \int \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\int { {\left (d x^{n} + c\right )} {\left (b x^{n} + a\right )}^{p} \,d x } \] Input:
integrate((a+b*x^n)^p*(c+d*x^n),x, algorithm="maxima")
Output:
integrate((d*x^n + c)*(b*x^n + a)^p, x)
Exception generated. \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*x^n)^p*(c+d*x^n),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,[0,0,2,2,1,2,1,0,1]%%%}+%%%{2,[0,0,2,2,1,1,1,0,1]%%%}+%% %{1,[0,0,
Timed out. \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\int {\left (a+b\,x^n\right )}^p\,\left (c+d\,x^n\right ) \,d x \] Input:
int((a + b*x^n)^p*(c + d*x^n),x)
Output:
int((a + b*x^n)^p*(c + d*x^n), x)
\[ \int \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx =\text {Too large to display} \] Input:
int((a+b*x^n)^p*(c+d*x^n),x)
Output:
(x**n*(x**n*b + a)**p*b*d*n*p*x + x**n*(x**n*b + a)**p*b*d*x + (x**n*b + a )**p*a*d*n*p*x + (x**n*b + a)**p*b*c*n*p*x + (x**n*b + a)**p*b*c*n*x + (x* *n*b + a)**p*b*c*x - int((x**n*b + a)**p/(x**n*b*n**2*p**2 + x**n*b*n**2*p + 2*x**n*b*n*p + x**n*b*n + x**n*b + a*n**2*p**2 + a*n**2*p + 2*a*n*p + a *n + a),x)*a**2*d*n**3*p**3 - int((x**n*b + a)**p/(x**n*b*n**2*p**2 + x**n *b*n**2*p + 2*x**n*b*n*p + x**n*b*n + x**n*b + a*n**2*p**2 + a*n**2*p + 2* a*n*p + a*n + a),x)*a**2*d*n**3*p**2 - 2*int((x**n*b + a)**p/(x**n*b*n**2* p**2 + x**n*b*n**2*p + 2*x**n*b*n*p + x**n*b*n + x**n*b + a*n**2*p**2 + a* n**2*p + 2*a*n*p + a*n + a),x)*a**2*d*n**2*p**2 - int((x**n*b + a)**p/(x** n*b*n**2*p**2 + x**n*b*n**2*p + 2*x**n*b*n*p + x**n*b*n + x**n*b + a*n**2* p**2 + a*n**2*p + 2*a*n*p + a*n + a),x)*a**2*d*n**2*p - int((x**n*b + a)** p/(x**n*b*n**2*p**2 + x**n*b*n**2*p + 2*x**n*b*n*p + x**n*b*n + x**n*b + a *n**2*p**2 + a*n**2*p + 2*a*n*p + a*n + a),x)*a**2*d*n*p + int((x**n*b + a )**p/(x**n*b*n**2*p**2 + x**n*b*n**2*p + 2*x**n*b*n*p + x**n*b*n + x**n*b + a*n**2*p**2 + a*n**2*p + 2*a*n*p + a*n + a),x)*a*b*c*n**4*p**4 + 2*int(( x**n*b + a)**p/(x**n*b*n**2*p**2 + x**n*b*n**2*p + 2*x**n*b*n*p + x**n*b*n + x**n*b + a*n**2*p**2 + a*n**2*p + 2*a*n*p + a*n + a),x)*a*b*c*n**4*p**3 + int((x**n*b + a)**p/(x**n*b*n**2*p**2 + x**n*b*n**2*p + 2*x**n*b*n*p + x**n*b*n + x**n*b + a*n**2*p**2 + a*n**2*p + 2*a*n*p + a*n + a),x)*a*b*c*n **4*p**2 + 3*int((x**n*b + a)**p/(x**n*b*n**2*p**2 + x**n*b*n**2*p + 2*...