Integrand size = 22, antiderivative size = 97 \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )^q}{x} \, dx=-\frac {\left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^q \left (\frac {b \left (c+d x^n\right )}{b c-a d}\right )^{-q} \operatorname {AppellF1}\left (1+p,1,-q,2+p,1+\frac {b x^n}{a},-\frac {d \left (a+b x^n\right )}{b c-a d}\right )}{a n (1+p)} \] Output:
-(a+b*x^n)^(p+1)*(c+d*x^n)^q*AppellF1(p+1,-q,1,2+p,-d*(a+b*x^n)/(-a*d+b*c) ,1+b*x^n/a)/a/n/(p+1)/((b*(c+d*x^n)/(-a*d+b*c))^q)
Time = 0.15 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )^q}{x} \, dx=\frac {\left (1+\frac {a x^{-n}}{b}\right )^{-p} \left (1+\frac {c x^{-n}}{d}\right )^{-q} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \operatorname {AppellF1}\left (-p-q,-p,-q,1-p-q,-\frac {a x^{-n}}{b},-\frac {c x^{-n}}{d}\right )}{n (p+q)} \] Input:
Integrate[((a + b*x^n)^p*(c + d*x^n)^q)/x,x]
Output:
((a + b*x^n)^p*(c + d*x^n)^q*AppellF1[-p - q, -p, -q, 1 - p - q, -(a/(b*x^ n)), -(c/(d*x^n))])/(n*(p + q)*(1 + a/(b*x^n))^p*(1 + c/(d*x^n))^q)
Time = 0.39 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {948, 154, 153}
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.
\(\displaystyle \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )^q}{x} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {\int x^{-n} \left (b x^n+a\right )^p \left (d x^n+c\right )^qdx^n}{n}\) |
\(\Big \downarrow \) 154 |
\(\displaystyle \frac {\left (c+d x^n\right )^q \left (\frac {b \left (c+d x^n\right )}{b c-a d}\right )^{-q} \int x^{-n} \left (b x^n+a\right )^p \left (\frac {b d x^n}{b c-a d}+\frac {b c}{b c-a d}\right )^qdx^n}{n}\) |
\(\Big \downarrow \) 153 |
\(\displaystyle -\frac {\left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^q \left (\frac {b \left (c+d x^n\right )}{b c-a d}\right )^{-q} \operatorname {AppellF1}\left (p+1,-q,1,p+2,-\frac {d \left (b x^n+a\right )}{b c-a d},\frac {b x^n+a}{a}\right )}{a n (p+1)}\) |
Input:
Int[((a + b*x^n)^p*(c + d*x^n)^q)/x,x]
Output:
-(((a + b*x^n)^(1 + p)*(c + d*x^n)^q*AppellF1[1 + p, -q, 1, 2 + p, -((d*(a + b*x^n))/(b*c - a*d)), (a + b*x^n)/a])/(a*n*(1 + p)*((b*(c + d*x^n))/(b* c - a*d))^q))
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(b*e - a*f)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*Simp lify[b/(b*c - a*d)]^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && IntegerQ[p] && GtQ[Simplify[b/( b*c - a*d)], 0] && !(GtQ[Simplify[d/(d*a - c*b)], 0] && SimplerQ[c + d*x, a + b*x])
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(c + d*x)^FracPart[n]/(Simplify[b/(b*c - a*d)]^IntPart[n ]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d)), x]^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && IntegerQ[p] && !G tQ[Simplify[b/(b*c - a*d)], 0] && !SimplerQ[c + d*x, a + b*x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
\[\int \frac {\left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )^{q}}{x}d x\]
Input:
int((a+b*x^n)^p*(c+d*x^n)^q/x,x)
Output:
int((a+b*x^n)^p*(c+d*x^n)^q/x,x)
\[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )^q}{x} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q}}{x} \,d x } \] Input:
integrate((a+b*x^n)^p*(c+d*x^n)^q/x,x, algorithm="fricas")
Output:
integral((b*x^n + a)^p*(d*x^n + c)^q/x, x)
\[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )^q}{x} \, dx=\int \frac {\left (a + b x^{n}\right )^{p} \left (c + d x^{n}\right )^{q}}{x}\, dx \] Input:
integrate((a+b*x**n)**p*(c+d*x**n)**q/x,x)
Output:
Integral((a + b*x**n)**p*(c + d*x**n)**q/x, x)
\[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )^q}{x} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q}}{x} \,d x } \] Input:
integrate((a+b*x^n)^p*(c+d*x^n)^q/x,x, algorithm="maxima")
Output:
integrate((b*x^n + a)^p*(d*x^n + c)^q/x, x)
\[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )^q}{x} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q}}{x} \,d x } \] Input:
integrate((a+b*x^n)^p*(c+d*x^n)^q/x,x, algorithm="giac")
Output:
integrate((b*x^n + a)^p*(d*x^n + c)^q/x, x)
Timed out. \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )^q}{x} \, dx=\int \frac {{\left (a+b\,x^n\right )}^p\,{\left (c+d\,x^n\right )}^q}{x} \,d x \] Input:
int(((a + b*x^n)^p*(c + d*x^n)^q)/x,x)
Output:
int(((a + b*x^n)^p*(c + d*x^n)^q)/x, x)
\[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )^q}{x} \, dx =\text {Too large to display} \] Input:
int((a+b*x^n)^p*(c+d*x^n)^q/x,x)
Output:
((x**n*d + c)**q*(x**n*b + a)**p*a*d + (x**n*d + c)**q*(x**n*b + a)**p*b*c + int(((x**n*d + c)**q*(x**n*b + a)**p)/(x**(2*n)*a*b*d**2*q*x + x**(2*n) *b**2*c*d*p*x + x**n*a**2*d**2*q*x + x**n*a*b*c*d*p*x + x**n*a*b*c*d*q*x + x**n*b**2*c**2*p*x + a**2*c*d*q*x + a*b*c**2*p*x),x)*a**3*c*d**2*n*q**2 + 2*int(((x**n*d + c)**q*(x**n*b + a)**p)/(x**(2*n)*a*b*d**2*q*x + x**(2*n) *b**2*c*d*p*x + x**n*a**2*d**2*q*x + x**n*a*b*c*d*p*x + x**n*a*b*c*d*q*x + x**n*b**2*c**2*p*x + a**2*c*d*q*x + a*b*c**2*p*x),x)*a**2*b*c**2*d*n*p*q + int(((x**n*d + c)**q*(x**n*b + a)**p)/(x**(2*n)*a*b*d**2*q*x + x**(2*n)* b**2*c*d*p*x + x**n*a**2*d**2*q*x + x**n*a*b*c*d*p*x + x**n*a*b*c*d*q*x + x**n*b**2*c**2*p*x + a**2*c*d*q*x + a*b*c**2*p*x),x)*a*b**2*c**3*n*p**2 - int((x**(2*n)*(x**n*d + c)**q*(x**n*b + a)**p)/(x**(2*n)*a*b*d**2*q*x + x* *(2*n)*b**2*c*d*p*x + x**n*a**2*d**2*q*x + x**n*a*b*c*d*p*x + x**n*a*b*c*d *q*x + x**n*b**2*c**2*p*x + a**2*c*d*q*x + a*b*c**2*p*x),x)*a**2*b*d**3*n* p*q - int((x**(2*n)*(x**n*d + c)**q*(x**n*b + a)**p)/(x**(2*n)*a*b*d**2*q* x + x**(2*n)*b**2*c*d*p*x + x**n*a**2*d**2*q*x + x**n*a*b*c*d*p*x + x**n*a *b*c*d*q*x + x**n*b**2*c**2*p*x + a**2*c*d*q*x + a*b*c**2*p*x),x)*a*b**2*c *d**2*n*p**2 - int((x**(2*n)*(x**n*d + c)**q*(x**n*b + a)**p)/(x**(2*n)*a* b*d**2*q*x + x**(2*n)*b**2*c*d*p*x + x**n*a**2*d**2*q*x + x**n*a*b*c*d*p*x + x**n*a*b*c*d*q*x + x**n*b**2*c**2*p*x + a**2*c*d*q*x + a*b*c**2*p*x),x) *a*b**2*c*d**2*n*q**2 - int((x**(2*n)*(x**n*d + c)**q*(x**n*b + a)**p)/...