\(\int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^5} \, dx\) [343]

Optimal result

Integrand size = 23, antiderivative size = 139 \[ \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^5} \, dx=-\frac {(a-b x)^{1-n} (a+b x)^{2+n}}{4 a^2 x^4}-\frac {b (1+2 n) (a-b x)^{1-n} (a+b x)^{2+n}}{12 a^3 x^3}-\frac {4 b^4 \left (1+n+n^2\right ) (a-b x)^{1-n} (a+b x)^{-1+n} \operatorname {Hypergeometric2F1}\left (3,1-n,2-n,\frac {a-b x}{a+b x}\right )}{3 a^3 (1-n)} \] Output:

-1/4*(-b*x+a)^(1-n)*(b*x+a)^(2+n)/a^2/x^4-1/12*b*(1+2*n)*(-b*x+a)^(1-n)*(b 
*x+a)^(2+n)/a^3/x^3-4/3*b^4*(n^2+n+1)*(-b*x+a)^(1-n)*(b*x+a)^(-1+n)*hyperg 
eom([3, 1-n],[2-n],(-b*x+a)/(b*x+a))/a^3/(1-n)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.73 \[ \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^5} \, dx=\frac {(a-b x)^{1-n} (a+b x)^{-1+n} \left (-\left ((-1+n) (a+b x)^3 (3 a+b (1+2 n) x)\right )+16 b^4 \left (1+n+n^2\right ) x^4 \operatorname {Hypergeometric2F1}\left (3,1-n,2-n,\frac {a-b x}{a+b x}\right )\right )}{12 a^3 (-1+n) x^4} \] Input:

Integrate[(a + b*x)^(1 + n)/(x^5*(a - b*x)^n),x]
 

Output:

((a - b*x)^(1 - n)*(a + b*x)^(-1 + n)*(-((-1 + n)*(a + b*x)^3*(3*a + b*(1 
+ 2*n)*x)) + 16*b^4*(1 + n + n^2)*x^4*Hypergeometric2F1[3, 1 - n, 2 - n, ( 
a - b*x)/(a + b*x)]))/(12*a^3*(-1 + n)*x^4)
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {114, 25, 27, 168, 27, 141}

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.

\(\displaystyle \int \frac {(a-b x)^{-n} (a+b x)^{n+1}}{x^5} \, dx\)

\(\Big \downarrow \) 114

\(\displaystyle -\frac {\int -\frac {b (a-b x)^{-n} (a+b x)^{n+1} (a (2 n+1)+b x)}{x^4}dx}{4 a^2}-\frac {(a+b x)^{n+2} (a-b x)^{1-n}}{4 a^2 x^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {b (a-b x)^{-n} (a+b x)^{n+1} (a (2 n+1)+b x)}{x^4}dx}{4 a^2}-\frac {(a-b x)^{1-n} (a+b x)^{n+2}}{4 a^2 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \int \frac {(a-b x)^{-n} (a+b x)^{n+1} (a (2 n+1)+b x)}{x^4}dx}{4 a^2}-\frac {(a-b x)^{1-n} (a+b x)^{n+2}}{4 a^2 x^4}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {b \left (-\frac {\int -\frac {4 a^2 b \left (n^2+n+1\right ) (a-b x)^{-n} (a+b x)^{n+1}}{x^3}dx}{3 a^2}-\frac {(2 n+1) (a+b x)^{n+2} (a-b x)^{1-n}}{3 a x^3}\right )}{4 a^2}-\frac {(a-b x)^{1-n} (a+b x)^{n+2}}{4 a^2 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {4}{3} b \left (n^2+n+1\right ) \int \frac {(a-b x)^{-n} (a+b x)^{n+1}}{x^3}dx-\frac {(2 n+1) (a-b x)^{1-n} (a+b x)^{n+2}}{3 a x^3}\right )}{4 a^2}-\frac {(a-b x)^{1-n} (a+b x)^{n+2}}{4 a^2 x^4}\)

\(\Big \downarrow \) 141

\(\displaystyle \frac {b \left (-\frac {16 b^3 \left (n^2+n+1\right ) (a+b x)^{n-1} (a-b x)^{1-n} \operatorname {Hypergeometric2F1}\left (3,1-n,2-n,\frac {a-b x}{a+b x}\right )}{3 a (1-n)}-\frac {(2 n+1) (a+b x)^{n+2} (a-b x)^{1-n}}{3 a x^3}\right )}{4 a^2}-\frac {(a-b x)^{1-n} (a+b x)^{n+2}}{4 a^2 x^4}\)

Input:

Int[(a + b*x)^(1 + n)/(x^5*(a - b*x)^n),x]
 

Output:

-1/4*((a - b*x)^(1 - n)*(a + b*x)^(2 + n))/(a^2*x^4) + (b*(-1/3*((1 + 2*n) 
*(a - b*x)^(1 - n)*(a + b*x)^(2 + n))/(a*x^3) - (16*b^3*(1 + n + n^2)*(a - 
 b*x)^(1 - n)*(a + b*x)^(-1 + n)*Hypergeometric2F1[3, 1 - n, 2 - n, (a - b 
*x)/(a + b*x)])/(3*a*(1 - n))))/(4*a^2)
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 141

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( 
n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f 
))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, 
p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !Su 
mSimplerQ[p, 1]) &&  !ILtQ[m, 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]
 

Maple [F]

\[\int \frac {\left (b x +a \right )^{1+n} \left (-b x +a \right )^{-n}}{x^{5}}d x\]

Input:

int((b*x+a)^(1+n)/x^5/((-b*x+a)^n),x)
 

Output:

int((b*x+a)^(1+n)/x^5/((-b*x+a)^n),x)
 

Fricas [F]

\[ \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^5} \, dx=\int { \frac {{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{5}} \,d x } \] Input:

integrate((b*x+a)^(1+n)/x^5/((-b*x+a)^n),x, algorithm="fricas")
 

Output:

integral((b*x + a)^(n + 1)/((-b*x + a)^n*x^5), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^5} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**(1+n)/x**5/((-b*x+a)**n),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^5} \, dx=\int { \frac {{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{5}} \,d x } \] Input:

integrate((b*x+a)^(1+n)/x^5/((-b*x+a)^n),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(n + 1)/((-b*x + a)^n*x^5), x)
 

Giac [F]

\[ \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^5} \, dx=\int { \frac {{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{5}} \,d x } \] Input:

integrate((b*x+a)^(1+n)/x^5/((-b*x+a)^n),x, algorithm="giac")
 

Output:

integrate((b*x + a)^(n + 1)/((-b*x + a)^n*x^5), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^5} \, dx=\int \frac {{\left (a+b\,x\right )}^{n+1}}{x^5\,{\left (a-b\,x\right )}^n} \,d x \] Input:

int((a + b*x)^(n + 1)/(x^5*(a - b*x)^n),x)
 

Output:

int((a + b*x)^(n + 1)/(x^5*(a - b*x)^n), x)
 

Reduce [F]

\[ \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^5} \, dx=\left (\int \frac {\left (b x +a \right )^{n}}{\left (-b x +a \right )^{n} x^{5}}d x \right ) a +\left (\int \frac {\left (b x +a \right )^{n}}{\left (-b x +a \right )^{n} x^{4}}d x \right ) b \] Input:

int((b*x+a)^(1+n)/x^5/((-b*x+a)^n),x)
 

Output:

int((a + b*x)**n/((a - b*x)**n*x**5),x)*a + int((a + b*x)**n/((a - b*x)**n 
*x**4),x)*b