Integrand size = 31, antiderivative size = 214 \[ \int (g x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^q \left (e+f x^n\right ) \, dx=\frac {e (g x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1+m}{n},-p,-q,\frac {1+m+n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{g (1+m)}+\frac {f x^n (g x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1+m+n}{n},-p,-q,\frac {1+m+2 n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{g (1+m+n)} \] Output:
e*(g*x)^(1+m)*(a+b*x^n)^p*(c+d*x^n)^q*AppellF1((1+m)/n,-p,-q,(1+m+n)/n,-b* x^n/a,-d*x^n/c)/g/(1+m)/((1+b*x^n/a)^p)/((1+d*x^n/c)^q)+f*x^n*(g*x)^(1+m)* (a+b*x^n)^p*(c+d*x^n)^q*AppellF1((1+m+n)/n,-p,-q,(1+m+2*n)/n,-b*x^n/a,-d*x ^n/c)/g/(1+m+n)/((1+b*x^n/a)^p)/((1+d*x^n/c)^q)
Time = 0.54 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.76 \[ \int (g x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^q \left (e+f x^n\right ) \, dx=\frac {x (g x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \left (e (1+m+n) \operatorname {AppellF1}\left (\frac {1+m}{n},-p,-q,\frac {1+m+n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+f (1+m) x^n \operatorname {AppellF1}\left (\frac {1+m+n}{n},-p,-q,\frac {1+m+2 n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}{(1+m) (1+m+n)} \] Input:
Integrate[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n),x]
Output:
(x*(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e*(1 + m + n)*AppellF1[(1 + m)/n, -p, -q, (1 + m + n)/n, -((b*x^n)/a), -((d*x^n)/c)] + f*(1 + m)*x^n*AppellF 1[(1 + m + n)/n, -p, -q, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)]))/(( 1 + m)*(1 + m + n)*(1 + (b*x^n)/a)^p*(1 + (d*x^n)/c)^q)
Time = 0.71 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1068, 1013, 1013, 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.
\(\displaystyle \int (g x)^m \left (e+f x^n\right ) \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx\) |
\(\Big \downarrow \) 1068 |
\(\displaystyle e \int (g x)^m \left (b x^n+a\right )^p \left (d x^n+c\right )^qdx+f x^{-m} (g x)^m \int x^{m+n} \left (b x^n+a\right )^p \left (d x^n+c\right )^qdx\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle e \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \int (g x)^m \left (\frac {b x^n}{a}+1\right )^p \left (d x^n+c\right )^qdx+f x^{-m} (g x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \int x^{m+n} \left (\frac {b x^n}{a}+1\right )^p \left (d x^n+c\right )^qdx\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle e \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac {d x^n}{c}+1\right )^{-q} \int (g x)^m \left (\frac {b x^n}{a}+1\right )^p \left (\frac {d x^n}{c}+1\right )^qdx+f x^{-m} (g x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac {d x^n}{c}+1\right )^{-q} \int x^{m+n} \left (\frac {b x^n}{a}+1\right )^p \left (\frac {d x^n}{c}+1\right )^qdx\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {e (g x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac {d x^n}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {m+1}{n},-p,-q,\frac {m+n+1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{g (m+1)}+\frac {f x^{n+1} (g x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac {d x^n}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {m+n+1}{n},-p,-q,\frac {m+2 n+1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{m+n+1}\) |
Input:
Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n),x]
Output:
(e*(g*x)^(1 + m)*(a + b*x^n)^p*(c + d*x^n)^q*AppellF1[(1 + m)/n, -p, -q, ( 1 + m + n)/n, -((b*x^n)/a), -((d*x^n)/c)])/(g*(1 + m)*(1 + (b*x^n)/a)^p*(1 + (d*x^n)/c)^q) + (f*x^(1 + n)*(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*Appell F1[(1 + m + n)/n, -p, -q, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)])/(( 1 + m + n)*(1 + (b*x^n)/a)^p*(1 + (d*x^n)/c)^q)
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ n/a))^FracPart[p]) Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], 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])
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e Int[(g*x)^m*(a + b* x^n)^p*(c + d*x^n)^q, x], x] + Simp[f*((g*x)^m/x^m) Int[x^(m + n)*(a + b* x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q}, x]
\[\int \left (g x \right )^{m} \left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )^{q} \left (e +f \,x^{n}\right )d x\]
Input:
int((g*x)^m*(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n),x)
Output:
int((g*x)^m*(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n),x)
\[ \int (g x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^q \left (e+f x^n\right ) \, dx=\int { {\left (f x^{n} + e\right )} {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q} \left (g x\right )^{m} \,d x } \] Input:
integrate((g*x)^m*(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n),x, algorithm="fricas")
Output:
integral((f*x^n + e)*(b*x^n + a)^p*(d*x^n + c)^q*(g*x)^m, x)
Exception generated. \[ \int (g x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^q \left (e+f x^n\right ) \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((g*x)**m*(a+b*x**n)**p*(c+d*x**n)**q*(e+f*x**n),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int (g x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^q \left (e+f x^n\right ) \, dx=\int { {\left (f x^{n} + e\right )} {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q} \left (g x\right )^{m} \,d x } \] Input:
integrate((g*x)^m*(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n),x, algorithm="maxima")
Output:
integrate((f*x^n + e)*(b*x^n + a)^p*(d*x^n + c)^q*(g*x)^m, x)
\[ \int (g x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^q \left (e+f x^n\right ) \, dx=\int { {\left (f x^{n} + e\right )} {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q} \left (g x\right )^{m} \,d x } \] Input:
integrate((g*x)^m*(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n),x, algorithm="giac")
Output:
integrate((f*x^n + e)*(b*x^n + a)^p*(d*x^n + c)^q*(g*x)^m, x)
Timed out. \[ \int (g x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^q \left (e+f x^n\right ) \, dx=\int {\left (g\,x\right )}^m\,{\left (a+b\,x^n\right )}^p\,{\left (c+d\,x^n\right )}^q\,\left (e+f\,x^n\right ) \,d x \] Input:
int((g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n),x)
Output:
int((g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n), x)
\[ \int (g x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^q \left (e+f x^n\right ) \, dx=\int \left (g x \right )^{m} \left (x^{n} b +a \right )^{p} \left (x^{n} d +c \right )^{q} \left (e +f \,x^{n}\right )d x \] Input:
int((g*x)^m*(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n),x)
Output:
int((g*x)^m*(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n),x)