\(\int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx\) [560]

Optimal result

Integrand size = 18, antiderivative size = 124 \[ \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx=-\frac {(a+b x)^{1+n}}{a c x}+\frac {d^2 (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{c^2 (b c-a d) (1+n)}+\frac {(a d-b c n) (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a^2 c^2 (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx=-\frac {(a+b x)^{1+n} \left (a^2 d^2 x \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )+(-b c+a d) \left (a c (1+n)+(-a d x+b c n x) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )\right )\right )}{a^2 c^2 (-b c+a d) (1+n) x} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {114, 174, 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.

\(\displaystyle \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx\)

\(\Big \downarrow \) 114

\(\displaystyle -\frac {\int \frac {(a+b x)^n (a d-b n x d-b c n)}{x (c+d x)}dx}{a c}-\frac {(a+b x)^{n+1}}{a c x}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {\frac {(a d-b c n) \int \frac {(a+b x)^n}{x}dx}{c}-\frac {a d^2 \int \frac {(a+b x)^n}{c+d x}dx}{c}}{a c}-\frac {(a+b x)^{n+1}}{a c x}\)

\(\Big \downarrow \) 75

\(\displaystyle -\frac {-\frac {a d^2 \int \frac {(a+b x)^n}{c+d x}dx}{c}-\frac {(a+b x)^{n+1} (a d-b c n) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b x}{a}+1\right )}{a c (n+1)}}{a c}-\frac {(a+b x)^{n+1}}{a c x}\)

\(\Big \downarrow \) 78

\(\displaystyle -\frac {-\frac {a d^2 (a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,-\frac {d (a+b x)}{b c-a d}\right )}{c (n+1) (b c-a d)}-\frac {(a+b x)^{n+1} (a d-b c n) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b x}{a}+1\right )}{a c (n+1)}}{a c}-\frac {(a+b x)^{n+1}}{a c x}\)

Input:

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

Output:

-((a + b*x)^(1 + n)/(a*c*x)) - (-((a*d^2*(a + b*x)^(1 + n)*Hypergeometric2 
F1[1, 1 + n, 2 + n, -((d*(a + b*x))/(b*c - a*d))])/(c*(b*c - a*d)*(1 + n)) 
) - ((a*d - b*c*n)*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 
+ (b*x)/a])/(a*c*(1 + n)))/(a*c)
 

Defintions of rubi rules used

rule 75

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])
 

rule 78

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]
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

Maple [F]

\[\int \frac {\left (b x +a \right )^{n}}{x^{2} \left (x d +c \right )}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )} x^{2}} \,d x } \] Input:

integrate((b*x+a)^n/x^2/(d*x+c),x, algorithm="fricas")
 

Output:

integral((b*x + a)^n/(d*x^3 + c*x^2), x)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )} x^{2}} \,d x } \] Input:

integrate((b*x+a)^n/x^2/(d*x+c),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )} x^{2}} \,d x } \] Input:

integrate((b*x+a)^n/x^2/(d*x+c),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx=\int \frac {{\left (a+b\,x\right )}^n}{x^2\,\left (c+d\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx=\int \frac {\left (b x +a \right )^{n}}{d \,x^{3}+c \,x^{2}}d x \] Input:

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

Output:

int((a + b*x)**n/(c*x**2 + d*x**3),x)