Integrand size = 20, antiderivative size = 112 \[ \int x^m \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {d x^{1+m} \left (a+b x^n\right )^{1+p}}{b (1+m+n+n p)}+\left (\frac {c}{1+m}-\frac {a d}{b (1+m+n+n p)}\right ) x^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,\frac {1+m+n}{n},-\frac {b x^n}{a}\right ) \] Output:
d*x^(1+m)*(a+b*x^n)^(p+1)/b/(n*p+m+n+1)+(c/(1+m)-a*d/b/(n*p+m+n+1))*x^(1+m )*(a+b*x^n)^p*hypergeom([-p, (1+m)/n],[(1+m+n)/n],-b*x^n/a)/((1+b*x^n/a)^p )
Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02 \[ \int x^m \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {x^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c (1+m+n) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,\frac {1+m+n}{n},-\frac {b x^n}{a}\right )+d (1+m) x^n \operatorname {Hypergeometric2F1}\left (\frac {1+m+n}{n},-p,\frac {1+m+2 n}{n},-\frac {b x^n}{a}\right )\right )}{(1+m) (1+m+n)} \] Input:
Integrate[x^m*(a + b*x^n)^p*(c + d*x^n),x]
Output:
(x^(1 + m)*(a + b*x^n)^p*(c*(1 + m + n)*Hypergeometric2F1[(1 + m)/n, -p, ( 1 + m + n)/n, -((b*x^n)/a)] + d*(1 + m)*x^n*Hypergeometric2F1[(1 + m + n)/ n, -p, (1 + m + 2*n)/n, -((b*x^n)/a)]))/((1 + m)*(1 + m + n)*(1 + (b*x^n)/ a)^p)
Time = 0.42 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {959, 889, 888}
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.
| \(\displaystyle \int x^m \left (c+d x^n\right ) \left (a+b x^n\right )^p \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \left (c-\frac {a d (m+1)}{b (m+n p+n+1)}\right ) \int x^m \left (b x^n+a\right )^pdx+\frac {d x^{m+1} \left (a+b x^n\right )^{p+1}}{b (m+n p+n+1)}\) |
| \(\Big \downarrow \) 889 |
| \(\displaystyle \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c-\frac {a d (m+1)}{b (m+n p+n+1)}\right ) \int x^m \left (\frac {b x^n}{a}+1\right )^pdx+\frac {d x^{m+1} \left (a+b x^n\right )^{p+1}}{b (m+n p+n+1)}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle \frac {x^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c-\frac {a d (m+1)}{b (m+n p+n+1)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {m+1}{n},-p,\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{m+1}+\frac {d x^{m+1} \left (a+b x^n\right )^{p+1}}{b (m+n p+n+1)}\) |
Input:
Int[x^m*(a + b*x^n)^p*(c + d*x^n),x]
Output:
(d*x^(1 + m)*(a + b*x^n)^(1 + p))/(b*(1 + m + n + n*p)) + ((c - (a*d*(1 + m))/(b*(1 + m + n + n*p)))*x^(1 + m)*(a + b*x^n)^p*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)])/((1 + m)*(1 + (b*x^n)/a)^p)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(c*x) ^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0 ] && !(ILtQ[p, 0] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
\[\int x^{m} \left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )d x\]
Input:
int(x^m*(a+b*x^n)^p*(c+d*x^n),x)
Output:
int(x^m*(a+b*x^n)^p*(c+d*x^n),x)
\[ \int x^m \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\int { {\left (d x^{n} + c\right )} {\left (b x^{n} + a\right )}^{p} x^{m} \,d x } \] Input:
integrate(x^m*(a+b*x^n)^p*(c+d*x^n),x, algorithm="fricas")
Output:
integral((d*x^m*x^n + c*x^m)*(b*x^n + a)^p, x)
Result contains complex when optimal does not.
Time = 38.17 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.37 \[ \int x^m \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {a^{\frac {m}{n} + \frac {1}{n}} a^{- \frac {m}{n} + p - \frac {1}{n}} c x^{m + 1} \Gamma \left (\frac {m}{n} + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} + \frac {1}{n} \\ \frac {m}{n} + 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {a^{\frac {m}{n} + 1 + \frac {1}{n}} a^{- \frac {m}{n} + p - 1 - \frac {1}{n}} d x^{m + n + 1} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} + 1 + \frac {1}{n} \\ \frac {m}{n} + 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} \] Input:
integrate(x**m*(a+b*x**n)**p*(c+d*x**n),x)
Output:
a**(m/n + 1/n)*a**(-m/n + p - 1/n)*c*x**(m + 1)*gamma(m/n + 1/n)*hyper((-p , m/n + 1/n), (m/n + 1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(m/n + 1 + 1/n)) + a**(m/n + 1 + 1/n)*a**(-m/n + p - 1 - 1/n)*d*x**(m + n + 1)*gam ma(m/n + 1 + 1/n)*hyper((-p, m/n + 1 + 1/n), (m/n + 2 + 1/n,), b*x**n*exp_ polar(I*pi)/a)/(n*gamma(m/n + 2 + 1/n))
\[ \int x^m \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\int { {\left (d x^{n} + c\right )} {\left (b x^{n} + a\right )}^{p} x^{m} \,d x } \] Input:
integrate(x^m*(a+b*x^n)^p*(c+d*x^n),x, algorithm="maxima")
Output:
integrate((d*x^n + c)*(b*x^n + a)^p*x^m, x)
\[ \int x^m \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\int { {\left (d x^{n} + c\right )} {\left (b x^{n} + a\right )}^{p} x^{m} \,d x } \] Input:
integrate(x^m*(a+b*x^n)^p*(c+d*x^n),x, algorithm="giac")
Output:
integrate((d*x^n + c)*(b*x^n + a)^p*x^m, x)
Timed out. \[ \int x^m \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\int x^m\,{\left (a+b\,x^n\right )}^p\,\left (c+d\,x^n\right ) \,d x \] Input:
int(x^m*(a + b*x^n)^p*(c + d*x^n),x)
Output:
int(x^m*(a + b*x^n)^p*(c + d*x^n), x)
\[ \int x^m \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\text {too large to display} \] Input:
int(x^m*(a+b*x^n)^p*(c+d*x^n),x)
Output:
(x**(m + n)*(x**n*b + a)**p*b*d*m*x + x**(m + n)*(x**n*b + a)**p*b*d*n*p*x + x**(m + n)*(x**n*b + a)**p*b*d*x + x**m*(x**n*b + a)**p*a*d*n*p*x + x** m*(x**n*b + a)**p*b*c*m*x + x**m*(x**n*b + a)**p*b*c*n*p*x + x**m*(x**n*b + a)**p*b*c*n*x + x**m*(x**n*b + a)**p*b*c*x - int((x**m*(x**n*b + a)**p)/ (x**n*b*m**2 + 2*x**n*b*m*n*p + x**n*b*m*n + 2*x**n*b*m + 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*m**2 + 2*a*m*n*p + a*m*n + 2*a*m + a*n**2*p**2 + a*n**2*p + 2*a*n*p + a*n + a),x)*a**2*d*m** 3*n*p - 2*int((x**m*(x**n*b + a)**p)/(x**n*b*m**2 + 2*x**n*b*m*n*p + x**n* b*m*n + 2*x**n*b*m + 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*m**2 + 2*a*m*n*p + a*m*n + 2*a*m + a*n**2*p**2 + a*n**2 *p + 2*a*n*p + a*n + a),x)*a**2*d*m**2*n**2*p**2 - int((x**m*(x**n*b + a)* *p)/(x**n*b*m**2 + 2*x**n*b*m*n*p + x**n*b*m*n + 2*x**n*b*m + 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*m**2 + 2*a*m*n *p + a*m*n + 2*a*m + a*n**2*p**2 + a*n**2*p + 2*a*n*p + a*n + a),x)*a**2*d *m**2*n**2*p - 3*int((x**m*(x**n*b + a)**p)/(x**n*b*m**2 + 2*x**n*b*m*n*p + x**n*b*m*n + 2*x**n*b*m + 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*m**2 + 2*a*m*n*p + a*m*n + 2*a*m + a*n**2*p**2 + a*n**2*p + 2*a*n*p + a*n + a),x)*a**2*d*m**2*n*p - int((x**m*(x**n*b + a) **p)/(x**n*b*m**2 + 2*x**n*b*m*n*p + x**n*b*m*n + 2*x**n*b*m + 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*m**2 + 2*a...