\(\int \frac {(a+b x)^n}{x (c+d x)} \, dx\) [940]

Optimal result

Integrand size = 18, antiderivative size = 95 \[ \int \frac {(a+b x)^n}{x (c+d x)} \, dx=-\frac {d (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{c (b c-a d) (1+n)}-\frac {(a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a c (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {88, 67, 70} \[ \int \frac {(a+b x)^n}{x (c+d x)} \, dx=-\frac {d (a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,-\frac {d (a+b x)}{b c-a d}\right )}{c (n+1) (b c-a d)}-\frac {(a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b x}{a}+1\right )}{a c (n+1)} \]

[In]

Int[(a + b*x)^n/(x*(c + d*x)),x]

[Out]

-((d*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, -((d*(a + b*x))/(b*c - a*d))])/(c*(b*c - a*d)*(1 + n
))) - ((a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a*c*(1 + n))

Rule 67

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] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 88

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), In
t[(e + f*x)^p/(a + b*x), x], x] - Dist[d/(b*c - a*d), Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] &&  !IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(a+b x)^n}{x} \, dx}{c}-\frac {d \int \frac {(a+b x)^n}{c+d x} \, dx}{c} \\ & = -\frac {d (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{c (b c-a d) (1+n)}-\frac {(a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a c (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^n}{x (c+d x)} \, dx=\frac {(a+b x)^{1+n} \left (a d \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )+(b c-a d) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )\right )}{a c (-b c+a d) (1+n)} \]

[In]

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

[Out]

((a + b*x)^(1 + n)*(a*d*Hypergeometric2F1[1, 1 + n, 2 + n, (d*(a + b*x))/(-(b*c) + a*d)] + (b*c - a*d)*Hyperge
ometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a]))/(a*c*(-(b*c) + a*d)*(1 + n))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b*x + a)^n/(d*x^2 + c*x), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*x)^n/(x*(c + d*x)),x)

[Out]

int((a + b*x)^n/(x*(c + d*x)), x)