\(\int \frac {(a+\frac {b}{c+d x^2})^p}{x^4} \, dx\) [221]

Optimal result

Integrand size = 19, antiderivative size = 87 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^4} \, dx=-\frac {\left (1+\frac {d x^2}{c}\right )^p \left (1+\frac {a d x^2}{b+a c}\right )^{-p} \left (a+\frac {b}{c+d x^2}\right )^p \operatorname {AppellF1}\left (-\frac {3}{2},p,-p,-\frac {1}{2},-\frac {d x^2}{c},-\frac {a d x^2}{b+a c}\right )}{3 x^3} \] Output:

-1/3*(1+d*x^2/c)^p*(a+b/(d*x^2+c))^p*AppellF1(-3/2,p,-p,-1/2,-d*x^2/c,-a*d 
*x^2/(a*c+b))/x^3/((1+a*d*x^2/(a*c+b))^p)
 

Mathematica [F]

\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^4} \, dx=\int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^4} \, dx \] Input:

Integrate[(a + b/(c + d*x^2))^p/x^4,x]
 

Output:

Integrate[(a + b/(c + d*x^2))^p/x^4, x]
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2057, 2058, 395, 395, 394}

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.

\(\displaystyle \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^4} \, dx\)

\(\Big \downarrow \) 2057

\(\displaystyle \int \frac {\left (\frac {a c+a d x^2+b}{c+d x^2}\right )^p}{x^4}dx\)

\(\Big \downarrow \) 2058

\(\displaystyle \left (c+d x^2\right )^p \left (a c+a d x^2+b\right )^{-p} \left (\frac {a c+a d x^2+b}{c+d x^2}\right )^p \int \frac {\left (d x^2+c\right )^{-p} \left (a d x^2+b+a c\right )^p}{x^4}dx\)

\(\Big \downarrow \) 395

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

\(\Big \downarrow \) 395

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

\(\Big \downarrow \) 394

\(\displaystyle -\frac {\left (\frac {d x^2}{c}+1\right )^p \left (\frac {a c+a d x^2+b}{c+d x^2}\right )^p \left (\frac {a d x^2}{a c+b}+1\right )^{-p} \operatorname {AppellF1}\left (-\frac {3}{2},p,-p,-\frac {1}{2},-\frac {d x^2}{c},-\frac {a d x^2}{b+a c}\right )}{3 x^3}\)

Input:

Int[(a + b/(c + d*x^2))^p/x^4,x]
 

Output:

-1/3*(((b + a*c + a*d*x^2)/(c + d*x^2))^p*(1 + (d*x^2)/c)^p*AppellF1[-3/2, 
 p, -p, -1/2, -((d*x^2)/c), -((a*d*x^2)/(b + a*c))])/(x^3*(1 + (a*d*x^2)/( 
b + a*c))^p)
 

Defintions of rubi rules used

rule 394

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 
, -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, 
 d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int 
egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 395

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^ 
FracPart[p])   Int[(e*x)^m*(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ 
[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 
1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 2057

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* 
((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

Maple [F]

\[\int \frac {\left (a +\frac {b}{d \,x^{2}+c}\right )^{p}}{x^{4}}d x\]

Input:

int((a+b/(d*x^2+c))^p/x^4,x)
 

Output:

int((a+b/(d*x^2+c))^p/x^4,x)
 

Fricas [F]

\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^4} \, dx=\int { \frac {{\left (a + \frac {b}{d x^{2} + c}\right )}^{p}}{x^{4}} \,d x } \] Input:

integrate((a+b/(d*x^2+c))^p/x^4,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^4} \, dx=\text {Timed out} \] Input:

integrate((a+b/(d*x**2+c))**p/x**4,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^4} \, dx=\int { \frac {{\left (a + \frac {b}{d x^{2} + c}\right )}^{p}}{x^{4}} \,d x } \] Input:

integrate((a+b/(d*x^2+c))^p/x^4,x, algorithm="maxima")
 

Output:

integrate((a + b/(d*x^2 + c))^p/x^4, x)
 

Giac [F]

\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^4} \, dx=\int { \frac {{\left (a + \frac {b}{d x^{2} + c}\right )}^{p}}{x^{4}} \,d x } \] Input:

integrate((a+b/(d*x^2+c))^p/x^4,x, algorithm="giac")
 

Output:

integrate((a + b/(d*x^2 + c))^p/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^4} \, dx=\int \frac {{\left (a+\frac {b}{d\,x^2+c}\right )}^p}{x^4} \,d x \] Input:

int((a + b/(c + d*x^2))^p/x^4,x)
 

Output:

int((a + b/(c + d*x^2))^p/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^4} \, dx=\int \frac {\left (a d \,x^{2}+a c +b \right )^{p}}{\left (d \,x^{2}+c \right )^{p} x^{4}}d x \] Input:

int((a+b/(d*x^2+c))^p/x^4,x)
 

Output:

int((a*c + a*d*x**2 + b)**p/((c + d*x**2)**p*x**4),x)