\(\int \frac {(a+b x)^n (c+d x)^p}{x} \, dx\) [961]

Optimal result

Integrand size = 18, antiderivative size = 85 \[ \int \frac {(a+b x)^n (c+d x)^p}{x} \, dx=-\frac {(a+b x)^{1+n} (c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \operatorname {AppellF1}\left (1+n,-p,1,2+n,-\frac {d (a+b x)}{b c-a d},\frac {a+b x}{a}\right )}{a (1+n)} \]

[Out]

-(b*x+a)^(1+n)*(d*x+c)^p*AppellF1(1+n,1,-p,2+n,(b*x+a)/a,-d*(b*x+a)/(-a*d+b*c))/a/(1+n)/((b*(d*x+c)/(-a*d+b*c)
)^p)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {142, 141} \[ \int \frac {(a+b x)^n (c+d x)^p}{x} \, dx=-\frac {(a+b x)^{n+1} (c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \operatorname {AppellF1}\left (n+1,-p,1,n+2,-\frac {d (a+b x)}{b c-a d},\frac {a+b x}{a}\right )}{a (n+1)} \]

[In]

Int[((a + b*x)^n*(c + d*x)^p)/x,x]

[Out]

-(((a + b*x)^(1 + n)*(c + d*x)^p*AppellF1[1 + n, -p, 1, 2 + n, -((d*(a + b*x))/(b*c - a*d)), (a + b*x)/a])/(a*
(1 + n)*((b*(c + d*x))/(b*c - a*d))^p))

Rule 141

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(b*e - a*f
)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*(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] &&  !Int
egerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])

Rule 142

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^
FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*(b*(c/(b*c -
a*d)) + b*d*(x/(b*c - a*d)))^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&
!IntegerQ[n] && IntegerQ[p] &&  !GtQ[b/(b*c - a*d), 0] &&  !SimplerQ[c + d*x, a + b*x]

Rubi steps \begin{align*} \text {integral}& = \left ((c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p}\right ) \int \frac {(a+b x)^n \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^p}{x} \, dx \\ & = -\frac {(a+b x)^{1+n} (c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} F_1\left (1+n;-p,1;2+n;-\frac {d (a+b x)}{b c-a d},\frac {a+b x}{a}\right )}{a (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b x)^n (c+d x)^p}{x} \, dx=\frac {\left (1+\frac {a}{b x}\right )^{-n} \left (1+\frac {c}{d x}\right )^{-p} (a+b x)^n (c+d x)^p \operatorname {AppellF1}\left (-n-p,-n,-p,1-n-p,-\frac {a}{b x},-\frac {c}{d x}\right )}{n+p} \]

[In]

Integrate[((a + b*x)^n*(c + d*x)^p)/x,x]

[Out]

((a + b*x)^n*(c + d*x)^p*AppellF1[-n - p, -n, -p, 1 - n - p, -(a/(b*x)), -(c/(d*x))])/((n + p)*(1 + a/(b*x))^n
*(1 + c/(d*x))^p)

Maple [F]

\[\int \frac {\left (b x +a \right )^{n} \left (d x +c \right )^{p}}{x}d x\]

[In]

int((b*x+a)^n*(d*x+c)^p/x,x)

[Out]

int((b*x+a)^n*(d*x+c)^p/x,x)

Fricas [F]

\[ \int \frac {(a+b x)^n (c+d x)^p}{x} \, dx=\int { \frac {{\left (b x + a\right )}^{n} {\left (d x + c\right )}^{p}}{x} \,d x } \]

[In]

integrate((b*x+a)^n*(d*x+c)^p/x,x, algorithm="fricas")

[Out]

integral((b*x + a)^n*(d*x + c)^p/x, x)

Sympy [F]

\[ \int \frac {(a+b x)^n (c+d x)^p}{x} \, dx=\int \frac {\left (a + b x\right )^{n} \left (c + d x\right )^{p}}{x}\, dx \]

[In]

integrate((b*x+a)**n*(d*x+c)**p/x,x)

[Out]

Integral((a + b*x)**n*(c + d*x)**p/x, x)

Maxima [F]

\[ \int \frac {(a+b x)^n (c+d x)^p}{x} \, dx=\int { \frac {{\left (b x + a\right )}^{n} {\left (d x + c\right )}^{p}}{x} \,d x } \]

[In]

integrate((b*x+a)^n*(d*x+c)^p/x,x, algorithm="maxima")

[Out]

integrate((b*x + a)^n*(d*x + c)^p/x, x)

Giac [F]

\[ \int \frac {(a+b x)^n (c+d x)^p}{x} \, dx=\int { \frac {{\left (b x + a\right )}^{n} {\left (d x + c\right )}^{p}}{x} \,d x } \]

[In]

integrate((b*x+a)^n*(d*x+c)^p/x,x, algorithm="giac")

[Out]

integrate((b*x + a)^n*(d*x + c)^p/x, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^n (c+d x)^p}{x} \, dx=\int \frac {{\left (a+b\,x\right )}^n\,{\left (c+d\,x\right )}^p}{x} \,d x \]

[In]

int(((a + b*x)^n*(c + d*x)^p)/x,x)

[Out]

int(((a + b*x)^n*(c + d*x)^p)/x, x)