Integrand size = 20, antiderivative size = 68 \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{x} \, dx=\frac {d \left (a+b x^n\right )^{1+p}}{b n (1+p)}-\frac {c \left (a+b x^n\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b x^n}{a}\right )}{a n (1+p)} \] Output:
d*(a+b*x^n)^(p+1)/b/n/(p+1)-c*(a+b*x^n)^(p+1)*hypergeom([1, p+1],[2+p],1+b *x^n/a)/a/n/(p+1)
Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{x} \, dx=\frac {\left (a+b x^n\right )^{1+p} \left (a d-b c \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b x^n}{a}\right )\right )}{a b n (1+p)} \] Input:
Integrate[((a + b*x^n)^p*(c + d*x^n))/x,x]
Output:
((a + b*x^n)^(1 + p)*(a*d - b*c*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b* x^n)/a]))/(a*b*n*(1 + p))
Time = 0.31 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {948, 90, 75}
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.
\(\displaystyle \int \frac {\left (c+d x^n\right ) \left (a+b x^n\right )^p}{x} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {\int x^{-n} \left (b x^n+a\right )^p \left (d x^n+c\right )dx^n}{n}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {c \int x^{-n} \left (b x^n+a\right )^pdx^n+\frac {d \left (a+b x^n\right )^{p+1}}{b (p+1)}}{n}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle \frac {\frac {d \left (a+b x^n\right )^{p+1}}{b (p+1)}-\frac {c \left (a+b x^n\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b x^n}{a}+1\right )}{a (p+1)}}{n}\) |
Input:
Int[((a + b*x^n)^p*(c + d*x^n))/x,x]
Output:
((d*(a + b*x^n)^(1 + p))/(b*(1 + p)) - (c*(a + b*x^n)^(1 + p)*Hypergeometr ic2F1[1, 1 + p, 2 + p, 1 + (b*x^n)/a])/(a*(1 + p)))/n
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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]]
\[\int \frac {\left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )}{x}d x\]
Input:
int((a+b*x^n)^p*(c+d*x^n)/x,x)
Output:
int((a+b*x^n)^p*(c+d*x^n)/x,x)
\[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{x} \, dx=\int { \frac {{\left (d x^{n} + c\right )} {\left (b x^{n} + a\right )}^{p}}{x} \,d x } \] Input:
integrate((a+b*x^n)^p*(c+d*x^n)/x,x, algorithm="fricas")
Output:
integral((d*x^n + c)*(b*x^n + a)^p/x, x)
Time = 12.34 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{x} \, dx=- \frac {b^{p} c x^{n p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {a x^{- n} e^{i \pi }}{b}} \right )}}{n \Gamma \left (1 - p\right )} - d \left (\begin {cases} - \left (a + b\right )^{p} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\begin {cases} a^{p} x^{n} & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x^{n}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b x^{n} \right )} & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases}}{n} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((a+b*x**n)**p*(c+d*x**n)/x,x)
Output:
-b**p*c*x**(n*p)*gamma(-p)*hyper((-p, -p), (1 - p,), a*exp_polar(I*pi)/(b* x**n))/(n*gamma(1 - p)) - d*Piecewise((-(a + b)**p*log(x), Eq(n, 0)), (-Pi ecewise((a**p*x**n, Eq(b, 0)), (Piecewise(((a + b*x**n)**(p + 1)/(p + 1), Ne(p, -1)), (log(a + b*x**n), True))/b, True))/n, True))
\[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{x} \, dx=\int { \frac {{\left (d x^{n} + c\right )} {\left (b x^{n} + a\right )}^{p}}{x} \,d x } \] Input:
integrate((a+b*x^n)^p*(c+d*x^n)/x,x, algorithm="maxima")
Output:
integrate((d*x^n + c)*(b*x^n + a)^p/x, x)
\[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{x} \, dx=\int { \frac {{\left (d x^{n} + c\right )} {\left (b x^{n} + a\right )}^{p}}{x} \,d x } \] Input:
integrate((a+b*x^n)^p*(c+d*x^n)/x,x, algorithm="giac")
Output:
integrate((d*x^n + c)*(b*x^n + a)^p/x, x)
Timed out. \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{x} \, dx=\int \frac {{\left (a+b\,x^n\right )}^p\,\left (c+d\,x^n\right )}{x} \,d x \] Input:
int(((a + b*x^n)^p*(c + d*x^n))/x,x)
Output:
int(((a + b*x^n)^p*(c + d*x^n))/x, x)
\[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{x} \, dx=\frac {x^{n} \left (x^{n} b +a \right )^{p} b d p +\left (x^{n} b +a \right )^{p} a d p +\left (x^{n} b +a \right )^{p} b c p +\left (x^{n} b +a \right )^{p} b c +\left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{n} b x +a x}d x \right ) a b c n \,p^{2}+\left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{n} b x +a x}d x \right ) a b c n p}{b n p \left (p +1\right )} \] Input:
int((a+b*x^n)^p*(c+d*x^n)/x,x)
Output:
(x**n*(x**n*b + a)**p*b*d*p + (x**n*b + a)**p*a*d*p + (x**n*b + a)**p*b*c* p + (x**n*b + a)**p*b*c + int((x**n*b + a)**p/(x**n*b*x + a*x),x)*a*b*c*n* p**2 + int((x**n*b + a)**p/(x**n*b*x + a*x),x)*a*b*c*n*p)/(b*n*p*(p + 1))