3.10.47 \(\int \frac {(a+b x)^n}{x^2 (c+d x)^2} \, dx\) [947]

3.10.47.1 Optimal result

Integrand size = 18, antiderivative size = 190 \[ \int \frac {(a+b x)^n}{x^2 (c+d x)^2} \, dx=-\frac {d (b c-2 a d) (a+b x)^{1+n}}{a c^2 (b c-a d) (c+d x)}-\frac {(a+b x)^{1+n}}{a c x (c+d x)}-\frac {d^2 (2 a d-b c (2-n)) (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{c^3 (b c-a d)^2 (1+n)}+\frac {(2 a d-b c n) (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a^2 c^3 (1+n)} \]

output

-d*(-2*a*d+b*c)*(b*x+a)^(1+n)/a/c^2/(-a*d+b*c)/(d*x+c)-(b*x+a)^(1+n)/a/c/x 
/(d*x+c)-d^2*(2*a*d-b*c*(2-n))*(b*x+a)^(1+n)*hypergeom([1, 1+n],[2+n],-d*( 
b*x+a)/(-a*d+b*c))/c^3/(-a*d+b*c)^2/(1+n)+(-b*c*n+2*a*d)*(b*x+a)^(1+n)*hyp 
ergeom([1, 1+n],[2+n],1+b*x/a)/a^2/c^3/(1+n)
 

3.10.47.2 Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^n}{x^2 (c+d x)^2} \, dx=-\frac {(a+b x)^{1+n} \left (a c^2 (b c-a d)^2 (1+n)+a c d (-b c+a d) (-b c+2 a d) (1+n) x-x (c+d x) \left (-a^2 d^2 (2 a d+b c (-2+n)) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )+(b c-a d)^2 (2 a d-b c n) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )\right )\right )}{a^2 c^3 (b c-a d)^2 (1+n) x (c+d x)} \]

input

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

output

-(((a + b*x)^(1 + n)*(a*c^2*(b*c - a*d)^2*(1 + n) + a*c*d*(-(b*c) + a*d)*( 
-(b*c) + 2*a*d)*(1 + n)*x - x*(c + d*x)*(-(a^2*d^2*(2*a*d + b*c*(-2 + n))* 
Hypergeometric2F1[1, 1 + n, 2 + n, (d*(a + b*x))/(-(b*c) + a*d)]) + (b*c - 
 a*d)^2*(2*a*d - b*c*n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a]))) 
/(a^2*c^3*(b*c - a*d)^2*(1 + n)*x*(c + d*x)))
 

3.10.47.3 Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {114, 168, 25, 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.

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 75

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

\(\Big \downarrow \) 78

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

input

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

output

-((a + b*x)^(1 + n)/(a*c*x*(c + d*x))) - ((d*(b*c - 2*a*d)*(a + b*x)^(1 + 
n))/(c*(b*c - a*d)*(c + d*x)) + ((a*d^2*(2*a*d - b*c*(2 - n))*(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)) - ((b*c - a*d)*(2*a*d - b*c*n)*(a + b*x)^(1 + n)*Hyp 
ergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a*c*(1 + n)))/(c*(b*c - a*d 
)))/(a*c)
 

3.10.47.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S 
imp[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*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

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]
 

3.10.47.4 Maple [F]

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

input

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

output

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

3.10.47.5 Fricas [F]

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

input

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

output

integral((b*x + a)^n/(d^2*x^4 + 2*c*d*x^3 + c^2*x^2), x)
 

3.10.47.6 Sympy [F]

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

input

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

output

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

3.10.47.7 Maxima [F]

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

input

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

output

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

3.10.47.8 Giac [F]

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

input

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

output

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

3.10.47.9 Mupad [F(-1)]

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

input

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

output

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