Integrand size = 24, antiderivative size = 142 \[ \int x^{m-3 n} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {c x^{1+m-3 n} \left (a+b x^n\right )^{1+p}}{a (1+m-3 n)}+\frac {(a d (1+m-3 n)-b c (1+m-n (2-p))) x^{1+m-2 n} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m-2 n}{n},-p,\frac {1+m-n}{n},-\frac {b x^n}{a}\right )}{a (1+m-3 n) (1+m-2 n)} \] Output:
c*x^(1+m-3*n)*(a+b*x^n)^(p+1)/a/(1+m-3*n)+(a*d*(1+m-3*n)-b*c*(1+m-n*(-p+2) ))*x^(1+m-2*n)*(a+b*x^n)^p*hypergeom([-p, (1+m-2*n)/n],[(1+m-n)/n],-b*x^n/ a)/a/(1+m-3*n)/(1+m-2*n)/((1+b*x^n/a)^p)
Time = 0.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.94 \[ \int x^{m-3 n} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {x^{1+m-3 n} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c (1+m-2 n) \operatorname {Hypergeometric2F1}\left (\frac {1+m-3 n}{n},-p,\frac {1+m-2 n}{n},-\frac {b x^n}{a}\right )+d (1+m-3 n) x^n \operatorname {Hypergeometric2F1}\left (\frac {1+m-2 n}{n},-p,\frac {1+m-n}{n},-\frac {b x^n}{a}\right )\right )}{(1+m-3 n) (1+m-2 n)} \] Input:
Integrate[x^(m - 3*n)*(a + b*x^n)^p*(c + d*x^n),x]
Output:
(x^(1 + m - 3*n)*(a + b*x^n)^p*(c*(1 + m - 2*n)*Hypergeometric2F1[(1 + m - 3*n)/n, -p, (1 + m - 2*n)/n, -((b*x^n)/a)] + d*(1 + m - 3*n)*x^n*Hypergeo metric2F1[(1 + m - 2*n)/n, -p, (1 + m - n)/n, -((b*x^n)/a)]))/((1 + m - 3* n)*(1 + m - 2*n)*(1 + (b*x^n)/a)^p)
Time = 0.49 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.
| \(\displaystyle \int x^{m-3 n} \left (c+d x^n\right ) \left (a+b x^n\right )^p \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \left (c-\frac {a d (m-3 n+1)}{b (m-n (2-p)+1)}\right ) \int x^{m-3 n} \left (b x^n+a\right )^pdx+\frac {d x^{m-3 n+1} \left (a+b x^n\right )^{p+1}}{b (m-n (2-p)+1)}\) |
| \(\Big \downarrow \) 889 |
| \(\displaystyle \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c-\frac {a d (m-3 n+1)}{b (m-n (2-p)+1)}\right ) \int x^{m-3 n} \left (\frac {b x^n}{a}+1\right )^pdx+\frac {d x^{m-3 n+1} \left (a+b x^n\right )^{p+1}}{b (m-n (2-p)+1)}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle \frac {x^{m-3 n+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c-\frac {a d (m-3 n+1)}{b (m-n (2-p)+1)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {m-3 n+1}{n},-p,\frac {m-2 n+1}{n},-\frac {b x^n}{a}\right )}{m-3 n+1}+\frac {d x^{m-3 n+1} \left (a+b x^n\right )^{p+1}}{b (m-n (2-p)+1)}\) |
Input:
Int[x^(m - 3*n)*(a + b*x^n)^p*(c + d*x^n),x]
Output:
(d*x^(1 + m - 3*n)*(a + b*x^n)^(1 + p))/(b*(1 + m - n*(2 - p))) + ((c - (a *d*(1 + m - 3*n))/(b*(1 + m - n*(2 - p))))*x^(1 + m - 3*n)*(a + b*x^n)^p*H ypergeometric2F1[(1 + m - 3*n)/n, -p, (1 + m - 2*n)/n, -((b*x^n)/a)])/((1 + m - 3*n)*(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]
\[\int x^{m -3 n} \left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )d x\]
Input:
int(x^(m-3*n)*(a+b*x^n)^p*(c+d*x^n),x)
Output:
int(x^(m-3*n)*(a+b*x^n)^p*(c+d*x^n),x)
\[ \int x^{m-3 n} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\int { {\left (d x^{n} + c\right )} {\left (b x^{n} + a\right )}^{p} x^{m - 3 \, n} \,d x } \] Input:
integrate(x^(m-3*n)*(a+b*x^n)^p*(c+d*x^n),x, algorithm="fricas")
Output:
integral((d*x^(m - 3*n)*x^n + c*x^(m - 3*n))*(b*x^n + a)^p, x)
Result contains complex when optimal does not.
Time = 26.73 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.16 \[ \int x^{m-3 n} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {a^{\frac {m}{n} - 3 + \frac {1}{n}} a^{- \frac {m}{n} + p + 3 - \frac {1}{n}} c x^{m - 3 n + 1} \Gamma \left (\frac {m}{n} - 3 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} - 3 + \frac {1}{n} \\ \frac {m}{n} - 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} - 2 + \frac {1}{n}\right )} + \frac {a^{\frac {m}{n} - 2 + \frac {1}{n}} a^{- \frac {m}{n} + p + 2 - \frac {1}{n}} d x^{m - 2 n + 1} \Gamma \left (\frac {m}{n} - 2 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} - 2 + \frac {1}{n} \\ \frac {m}{n} - 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} - 1 + \frac {1}{n}\right )} \] Input:
integrate(x**(m-3*n)*(a+b*x**n)**p*(c+d*x**n),x)
Output:
a**(m/n - 3 + 1/n)*a**(-m/n + p + 3 - 1/n)*c*x**(m - 3*n + 1)*gamma(m/n - 3 + 1/n)*hyper((-p, m/n - 3 + 1/n), (m/n - 2 + 1/n,), b*x**n*exp_polar(I*p i)/a)/(n*gamma(m/n - 2 + 1/n)) + a**(m/n - 2 + 1/n)*a**(-m/n + p + 2 - 1/n )*d*x**(m - 2*n + 1)*gamma(m/n - 2 + 1/n)*hyper((-p, m/n - 2 + 1/n), (m/n - 1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(m/n - 1 + 1/n))
\[ \int x^{m-3 n} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\int { {\left (d x^{n} + c\right )} {\left (b x^{n} + a\right )}^{p} x^{m - 3 \, n} \,d x } \] Input:
integrate(x^(m-3*n)*(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 - 3*n), x)
Exception generated. \[ \int x^{m-3 n} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^(m-3*n)*(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,[0,0,2,2,1,2,0,1,0,1]%%%}+%%%{-1,[0,0,2,2,1,1,0,1,0,1]%% %}+%%%{-2
Timed out. \[ \int x^{m-3 n} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\int x^{m-3\,n}\,{\left (a+b\,x^n\right )}^p\,\left (c+d\,x^n\right ) \,d x \] Input:
int(x^(m - 3*n)*(a + b*x^n)^p*(c + d*x^n),x)
Output:
int(x^(m - 3*n)*(a + b*x^n)^p*(c + d*x^n), x)
\[ \int x^{m-3 n} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\text {too large to display} \] Input:
int(x^(m-3*n)*(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 - 3*x**(m + n)*(x**n*b + a)**p*b*d*n*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 - 2*x**m*(x**n*b + a)**p*b*c*n*x + x**m*(x**n*b + a)**p*b*c*x - x**(3*n)*int((x**m*(x**n*b + a)**p)/(x**(4*n)*b*m**2 + 2*x** (4*n)*b*m*n*p - 5*x**(4*n)*b*m*n + 2*x**(4*n)*b*m + x**(4*n)*b*n**2*p**2 - 5*x**(4*n)*b*n**2*p + 6*x**(4*n)*b*n**2 + 2*x**(4*n)*b*n*p - 5*x**(4*n)*b *n + x**(4*n)*b + x**(3*n)*a*m**2 + 2*x**(3*n)*a*m*n*p - 5*x**(3*n)*a*m*n + 2*x**(3*n)*a*m + x**(3*n)*a*n**2*p**2 - 5*x**(3*n)*a*n**2*p + 6*x**(3*n) *a*n**2 + 2*x**(3*n)*a*n*p - 5*x**(3*n)*a*n + x**(3*n)*a),x)*a**2*d*m**3*n *p - 2*x**(3*n)*int((x**m*(x**n*b + a)**p)/(x**(4*n)*b*m**2 + 2*x**(4*n)*b *m*n*p - 5*x**(4*n)*b*m*n + 2*x**(4*n)*b*m + x**(4*n)*b*n**2*p**2 - 5*x**( 4*n)*b*n**2*p + 6*x**(4*n)*b*n**2 + 2*x**(4*n)*b*n*p - 5*x**(4*n)*b*n + x* *(4*n)*b + x**(3*n)*a*m**2 + 2*x**(3*n)*a*m*n*p - 5*x**(3*n)*a*m*n + 2*x** (3*n)*a*m + x**(3*n)*a*n**2*p**2 - 5*x**(3*n)*a*n**2*p + 6*x**(3*n)*a*n**2 + 2*x**(3*n)*a*n*p - 5*x**(3*n)*a*n + x**(3*n)*a),x)*a**2*d*m**2*n**2*p** 2 + 8*x**(3*n)*int((x**m*(x**n*b + a)**p)/(x**(4*n)*b*m**2 + 2*x**(4*n)*b* m*n*p - 5*x**(4*n)*b*m*n + 2*x**(4*n)*b*m + x**(4*n)*b*n**2*p**2 - 5*x**(4 *n)*b*n**2*p + 6*x**(4*n)*b*n**2 + 2*x**(4*n)*b*n*p - 5*x**(4*n)*b*n + x** (4*n)*b + x**(3*n)*a*m**2 + 2*x**(3*n)*a*m*n*p - 5*x**(3*n)*a*m*n + 2*x...