Integrand size = 22, antiderivative size = 55 \[ \int x^m \left (2+b x^n\right )^p \left (3+d x^n\right )^q \, dx=\frac {2^p 3^q x^{1+m} \operatorname {AppellF1}\left (\frac {1+m}{n},-p,-q,\frac {1+m+n}{n},-\frac {b x^n}{2},-\frac {d x^n}{3}\right )}{1+m} \] Output:
2^p*3^q*x^(1+m)*AppellF1((1+m)/n,-p,-q,(1+m+n)/n,-1/2*b*x^n,-1/3*d*x^n)/(1 +m)
Time = 0.16 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02 \[ \int x^m \left (2+b x^n\right )^p \left (3+d x^n\right )^q \, dx=\frac {2^p 3^q x^{1+m} \operatorname {AppellF1}\left (\frac {1+m}{n},-p,-q,1+\frac {1+m}{n},-\frac {b x^n}{2},-\frac {d x^n}{3}\right )}{1+m} \] Input:
Integrate[x^m*(2 + b*x^n)^p*(3 + d*x^n)^q,x]
Output:
(2^p*3^q*x^(1 + m)*AppellF1[(1 + m)/n, -p, -q, 1 + (1 + m)/n, -1/2*(b*x^n) , -1/3*(d*x^n)])/(1 + m)
Time = 0.31 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1012}
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.
\(\displaystyle \int x^m \left (b x^n+2\right )^p \left (d x^n+3\right )^q \, dx\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {2^p 3^q x^{m+1} \operatorname {AppellF1}\left (\frac {m+1}{n},-p,-q,\frac {m+n+1}{n},-\frac {b x^n}{2},-\frac {d x^n}{3}\right )}{m+1}\) |
Input:
Int[x^m*(2 + b*x^n)^p*(3 + d*x^n)^q,x]
Output:
(2^p*3^q*x^(1 + m)*AppellF1[(1 + m)/n, -p, -q, (1 + m + n)/n, -1/2*(b*x^n) , -1/3*(d*x^n)])/(1 + m)
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
\[\int x^{m} \left (2+b \,x^{n}\right )^{p} \left (3+d \,x^{n}\right )^{q}d x\]
Input:
int(x^m*(2+b*x^n)^p*(3+d*x^n)^q,x)
Output:
int(x^m*(2+b*x^n)^p*(3+d*x^n)^q,x)
\[ \int x^m \left (2+b x^n\right )^p \left (3+d x^n\right )^q \, dx=\int { {\left (b x^{n} + 2\right )}^{p} {\left (d x^{n} + 3\right )}^{q} x^{m} \,d x } \] Input:
integrate(x^m*(2+b*x^n)^p*(3+d*x^n)^q,x, algorithm="fricas")
Output:
integral((b*x^n + 2)^p*(d*x^n + 3)^q*x^m, x)
Timed out. \[ \int x^m \left (2+b x^n\right )^p \left (3+d x^n\right )^q \, dx=\text {Timed out} \] Input:
integrate(x**m*(2+b*x**n)**p*(3+d*x**n)**q,x)
Output:
Timed out
\[ \int x^m \left (2+b x^n\right )^p \left (3+d x^n\right )^q \, dx=\int { {\left (b x^{n} + 2\right )}^{p} {\left (d x^{n} + 3\right )}^{q} x^{m} \,d x } \] Input:
integrate(x^m*(2+b*x^n)^p*(3+d*x^n)^q,x, algorithm="maxima")
Output:
integrate((b*x^n + 2)^p*(d*x^n + 3)^q*x^m, x)
\[ \int x^m \left (2+b x^n\right )^p \left (3+d x^n\right )^q \, dx=\int { {\left (b x^{n} + 2\right )}^{p} {\left (d x^{n} + 3\right )}^{q} x^{m} \,d x } \] Input:
integrate(x^m*(2+b*x^n)^p*(3+d*x^n)^q,x, algorithm="giac")
Output:
integrate((b*x^n + 2)^p*(d*x^n + 3)^q*x^m, x)
Timed out. \[ \int x^m \left (2+b x^n\right )^p \left (3+d x^n\right )^q \, dx=\int x^m\,{\left (b\,x^n+2\right )}^p\,{\left (d\,x^n+3\right )}^q \,d x \] Input:
int(x^m*(b*x^n + 2)^p*(d*x^n + 3)^q,x)
Output:
int(x^m*(b*x^n + 2)^p*(d*x^n + 3)^q, x)
\[ \int x^m \left (2+b x^n\right )^p \left (3+d x^n\right )^q \, dx=\text {too large to display} \] Input:
int(x^m*(2+b*x^n)^p*(3+d*x^n)^q,x)
Output:
(3*x**m*(x**n*d + 3)**q*(x**n*b + 2)**p*b*x + 2*x**m*(x**n*d + 3)**q*(x**n *b + 2)**p*d*x - 9*int((x**(m + 2*n)*(x**n*d + 3)**q*(x**n*b + 2)**p)/(3*x **(2*n)*b**2*d*m + 3*x**(2*n)*b**2*d*n*p + 3*x**(2*n)*b**2*d + 2*x**(2*n)* b*d**2*m + 2*x**(2*n)*b*d**2*n*q + 2*x**(2*n)*b*d**2 + 9*x**n*b**2*m + 9*x **n*b**2*n*p + 9*x**n*b**2 + 12*x**n*b*d*m + 6*x**n*b*d*n*p + 6*x**n*b*d*n *q + 12*x**n*b*d + 4*x**n*d**2*m + 4*x**n*d**2*n*q + 4*x**n*d**2 + 18*b*m + 18*b*n*p + 18*b + 12*d*m + 12*d*n*q + 12*d),x)*b**3*d*m*n*q - 9*int((x** (m + 2*n)*(x**n*d + 3)**q*(x**n*b + 2)**p)/(3*x**(2*n)*b**2*d*m + 3*x**(2* n)*b**2*d*n*p + 3*x**(2*n)*b**2*d + 2*x**(2*n)*b*d**2*m + 2*x**(2*n)*b*d** 2*n*q + 2*x**(2*n)*b*d**2 + 9*x**n*b**2*m + 9*x**n*b**2*n*p + 9*x**n*b**2 + 12*x**n*b*d*m + 6*x**n*b*d*n*p + 6*x**n*b*d*n*q + 12*x**n*b*d + 4*x**n*d **2*m + 4*x**n*d**2*n*q + 4*x**n*d**2 + 18*b*m + 18*b*n*p + 18*b + 12*d*m + 12*d*n*q + 12*d),x)*b**3*d*n**2*p*q - 9*int((x**(m + 2*n)*(x**n*d + 3)** q*(x**n*b + 2)**p)/(3*x**(2*n)*b**2*d*m + 3*x**(2*n)*b**2*d*n*p + 3*x**(2* n)*b**2*d + 2*x**(2*n)*b*d**2*m + 2*x**(2*n)*b*d**2*n*q + 2*x**(2*n)*b*d** 2 + 9*x**n*b**2*m + 9*x**n*b**2*n*p + 9*x**n*b**2 + 12*x**n*b*d*m + 6*x**n *b*d*n*p + 6*x**n*b*d*n*q + 12*x**n*b*d + 4*x**n*d**2*m + 4*x**n*d**2*n*q + 4*x**n*d**2 + 18*b*m + 18*b*n*p + 18*b + 12*d*m + 12*d*n*q + 12*d),x)*b* *3*d*n*q - 6*int((x**(m + 2*n)*(x**n*d + 3)**q*(x**n*b + 2)**p)/(3*x**(2*n )*b**2*d*m + 3*x**(2*n)*b**2*d*n*p + 3*x**(2*n)*b**2*d + 2*x**(2*n)*b*d...