Integrand size = 22, antiderivative size = 54 \[ \int x^m \left (2+b x^4\right )^p \left (3+d x^4\right )^q \, dx=\frac {2^p 3^q x^{1+m} \operatorname {AppellF1}\left (\frac {1+m}{4},-p,-q,\frac {5+m}{4},-\frac {b x^4}{2},-\frac {d x^4}{3}\right )}{1+m} \] Output:
2^p*3^q*x^(1+m)*AppellF1(1/4+1/4*m,-p,-q,5/4+1/4*m,-1/2*b*x^4,-1/3*d*x^4)/ (1+m)
Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04 \[ \int x^m \left (2+b x^4\right )^p \left (3+d x^4\right )^q \, dx=\frac {2^p 3^q x^{1+m} \operatorname {AppellF1}\left (\frac {1+m}{4},-p,-q,1+\frac {1+m}{4},-\frac {b x^4}{2},-\frac {d x^4}{3}\right )}{1+m} \] Input:
Integrate[x^m*(2 + b*x^4)^p*(3 + d*x^4)^q,x]
Output:
(2^p*3^q*x^(1 + m)*AppellF1[(1 + m)/4, -p, -q, 1 + (1 + m)/4, -1/2*(b*x^4) , -1/3*(d*x^4)])/(1 + m)
Time = 0.29 (sec) , antiderivative size = 54, 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.
\(\displaystyle \int x^m \left (b x^4+2\right )^p \left (d x^4+3\right )^q \, dx\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {2^p 3^q x^{m+1} \operatorname {AppellF1}\left (\frac {m+1}{4},-p,-q,\frac {m+5}{4},-\frac {b x^4}{2},-\frac {d x^4}{3}\right )}{m+1}\) |
Input:
Int[x^m*(2 + b*x^4)^p*(3 + d*x^4)^q,x]
Output:
(2^p*3^q*x^(1 + m)*AppellF1[(1 + m)/4, -p, -q, (5 + m)/4, -1/2*(b*x^4), -1 /3*(d*x^4)])/(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])
\[\int x^{m} \left (b \,x^{4}+2\right )^{p} \left (d \,x^{4}+3\right )^{q}d x\]
Input:
int(x^m*(b*x^4+2)^p*(d*x^4+3)^q,x)
Output:
int(x^m*(b*x^4+2)^p*(d*x^4+3)^q,x)
\[ \int x^m \left (2+b x^4\right )^p \left (3+d x^4\right )^q \, dx=\int { {\left (b x^{4} + 2\right )}^{p} {\left (d x^{4} + 3\right )}^{q} x^{m} \,d x } \] Input:
integrate(x^m*(b*x^4+2)^p*(d*x^4+3)^q,x, algorithm="fricas")
Output:
integral((b*x^4 + 2)^p*(d*x^4 + 3)^q*x^m, x)
Timed out. \[ \int x^m \left (2+b x^4\right )^p \left (3+d x^4\right )^q \, dx=\text {Timed out} \] Input:
integrate(x**m*(b*x**4+2)**p*(d*x**4+3)**q,x)
Output:
Timed out
\[ \int x^m \left (2+b x^4\right )^p \left (3+d x^4\right )^q \, dx=\int { {\left (b x^{4} + 2\right )}^{p} {\left (d x^{4} + 3\right )}^{q} x^{m} \,d x } \] Input:
integrate(x^m*(b*x^4+2)^p*(d*x^4+3)^q,x, algorithm="maxima")
Output:
integrate((b*x^4 + 2)^p*(d*x^4 + 3)^q*x^m, x)
\[ \int x^m \left (2+b x^4\right )^p \left (3+d x^4\right )^q \, dx=\int { {\left (b x^{4} + 2\right )}^{p} {\left (d x^{4} + 3\right )}^{q} x^{m} \,d x } \] Input:
integrate(x^m*(b*x^4+2)^p*(d*x^4+3)^q,x, algorithm="giac")
Output:
integrate((b*x^4 + 2)^p*(d*x^4 + 3)^q*x^m, x)
Timed out. \[ \int x^m \left (2+b x^4\right )^p \left (3+d x^4\right )^q \, dx=\int x^m\,{\left (b\,x^4+2\right )}^p\,{\left (d\,x^4+3\right )}^q \,d x \] Input:
int(x^m*(b*x^4 + 2)^p*(d*x^4 + 3)^q,x)
Output:
int(x^m*(b*x^4 + 2)^p*(d*x^4 + 3)^q, x)
\[ \int x^m \left (2+b x^4\right )^p \left (3+d x^4\right )^q \, dx=\text {too large to display} \] Input:
int(x^m*(b*x^4+2)^p*(d*x^4+3)^q,x)
Output:
(x**m*(d*x**4 + 3)**q*(b*x**4 + 2)**p*x + 12*int((x**m*(d*x**4 + 3)**q*(b* x**4 + 2)**p*x**4)/(b*d*m*x**8 + 4*b*d*p*x**8 + 4*b*d*q*x**8 + b*d*x**8 + 3*b*m*x**4 + 12*b*p*x**4 + 12*b*q*x**4 + 3*b*x**4 + 2*d*m*x**4 + 8*d*p*x** 4 + 8*d*q*x**4 + 2*d*x**4 + 6*m + 24*p + 24*q + 6),x)*b*m*q + 48*int((x**m *(d*x**4 + 3)**q*(b*x**4 + 2)**p*x**4)/(b*d*m*x**8 + 4*b*d*p*x**8 + 4*b*d* q*x**8 + b*d*x**8 + 3*b*m*x**4 + 12*b*p*x**4 + 12*b*q*x**4 + 3*b*x**4 + 2* d*m*x**4 + 8*d*p*x**4 + 8*d*q*x**4 + 2*d*x**4 + 6*m + 24*p + 24*q + 6),x)* b*p*q + 48*int((x**m*(d*x**4 + 3)**q*(b*x**4 + 2)**p*x**4)/(b*d*m*x**8 + 4 *b*d*p*x**8 + 4*b*d*q*x**8 + b*d*x**8 + 3*b*m*x**4 + 12*b*p*x**4 + 12*b*q* x**4 + 3*b*x**4 + 2*d*m*x**4 + 8*d*p*x**4 + 8*d*q*x**4 + 2*d*x**4 + 6*m + 24*p + 24*q + 6),x)*b*q**2 + 12*int((x**m*(d*x**4 + 3)**q*(b*x**4 + 2)**p* x**4)/(b*d*m*x**8 + 4*b*d*p*x**8 + 4*b*d*q*x**8 + b*d*x**8 + 3*b*m*x**4 + 12*b*p*x**4 + 12*b*q*x**4 + 3*b*x**4 + 2*d*m*x**4 + 8*d*p*x**4 + 8*d*q*x** 4 + 2*d*x**4 + 6*m + 24*p + 24*q + 6),x)*b*q + 8*int((x**m*(d*x**4 + 3)**q *(b*x**4 + 2)**p*x**4)/(b*d*m*x**8 + 4*b*d*p*x**8 + 4*b*d*q*x**8 + b*d*x** 8 + 3*b*m*x**4 + 12*b*p*x**4 + 12*b*q*x**4 + 3*b*x**4 + 2*d*m*x**4 + 8*d*p *x**4 + 8*d*q*x**4 + 2*d*x**4 + 6*m + 24*p + 24*q + 6),x)*d*m*p + 32*int(( x**m*(d*x**4 + 3)**q*(b*x**4 + 2)**p*x**4)/(b*d*m*x**8 + 4*b*d*p*x**8 + 4* b*d*q*x**8 + b*d*x**8 + 3*b*m*x**4 + 12*b*p*x**4 + 12*b*q*x**4 + 3*b*x**4 + 2*d*m*x**4 + 8*d*p*x**4 + 8*d*q*x**4 + 2*d*x**4 + 6*m + 24*p + 24*q +...