Integrand size = 18, antiderivative size = 95 \[ \int \frac {(a+b x)^n}{x (c+d x)} \, dx=-\frac {d (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{c (b c-a d) (1+n)}-\frac {(a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a c (1+n)} \] Output:
-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)
Time = 0.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^n}{x (c+d x)} \, dx=\frac {(a+b x)^{1+n} \left (a d \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )+(b c-a d) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )\right )}{a c (-b c+a d) (1+n)} \] Input:
Integrate[(a + b*x)^n/(x*(c + d*x)),x]
Output:
((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)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/ a]))/(a*c*(-(b*c) + a*d)*(1 + n))
Time = 0.20 (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 = {97, 75, 78}
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.
\(\displaystyle \int \frac {(a+b x)^n}{x (c+d x)} \, dx\) |
\(\Big \downarrow \) 97 |
\(\displaystyle \frac {\int \frac {(a+b x)^n}{x}dx}{c}-\frac {d \int \frac {(a+b x)^n}{c+d x}dx}{c}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle -\frac {d \int \frac {(a+b x)^n}{c+d x}dx}{c}-\frac {(a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b x}{a}+1\right )}{a c (n+1)}\) |
\(\Big \downarrow \) 78 |
\(\displaystyle -\frac {d (a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,-\frac {d (a+b x)}{b c-a d}\right )}{c (n+1) (b c-a d)}-\frac {(a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b x}{a}+1\right )}{a c (n+1)}\) |
Input:
Int[(a + b*x)^n/(x*(c + d*x)),x]
Output:
-((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)*Hypergeometri c2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a*c*(1 + 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_))^(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] && !IntegerQ[m] && IntegerQ[n]
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[b/(b*c - a*d) Int[(e + f*x)^p/(a + b*x), x], x] - Simp[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]
\[\int \frac {\left (b x +a \right )^{n}}{x \left (x d +c \right )}d x\]
Input:
int((b*x+a)^n/x/(d*x+c),x)
Output:
int((b*x+a)^n/x/(d*x+c),x)
\[ \int \frac {(a+b x)^n}{x (c+d x)} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )} x} \,d x } \] Input:
integrate((b*x+a)^n/x/(d*x+c),x, algorithm="fricas")
Output:
integral((b*x + a)^n/(d*x^2 + c*x), x)
\[ \int \frac {(a+b x)^n}{x (c+d x)} \, dx=\int \frac {\left (a + b x\right )^{n}}{x \left (c + d x\right )}\, dx \] Input:
integrate((b*x+a)**n/x/(d*x+c),x)
Output:
Integral((a + b*x)**n/(x*(c + d*x)), x)
\[ \int \frac {(a+b x)^n}{x (c+d x)} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )} x} \,d x } \] Input:
integrate((b*x+a)^n/x/(d*x+c),x, algorithm="maxima")
Output:
integrate((b*x + a)^n/((d*x + c)*x), x)
\[ \int \frac {(a+b x)^n}{x (c+d x)} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )} x} \,d x } \] Input:
integrate((b*x+a)^n/x/(d*x+c),x, algorithm="giac")
Output:
integrate((b*x + a)^n/((d*x + c)*x), x)
Timed out. \[ \int \frac {(a+b x)^n}{x (c+d x)} \, dx=\int \frac {{\left (a+b\,x\right )}^n}{x\,\left (c+d\,x\right )} \,d x \] Input:
int((a + b*x)^n/(x*(c + d*x)),x)
Output:
int((a + b*x)^n/(x*(c + d*x)), x)
\[ \int \frac {(a+b x)^n}{x (c+d x)} \, dx=\int \frac {\left (b x +a \right )^{n}}{d \,x^{2}+c x}d x \] Input:
int((b*x+a)^n/x/(d*x+c),x)
Output:
int((a + b*x)**n/(c*x + d*x**2),x)