\(\int \frac {(e x)^{-2-2 p} (a+b x^2)^p}{(c+d x^2)^3} \, dx\) [1634]

Optimal result

Integrand size = 28, antiderivative size = 88 \[ \int \frac {(e x)^{-2-2 p} \left (a+b x^2\right )^p}{\left (c+d x^2\right )^3} \, dx=-\frac {(e x)^{-1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2} (-1-2 p),-p,3,\frac {1}{2} (1-2 p),-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c^3 e (1+2 p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93 \[ \int \frac {(e x)^{-2-2 p} \left (a+b x^2\right )^p}{\left (c+d x^2\right )^3} \, dx=-\frac {(e x)^{-1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (-\frac {1}{2}-p,-p,3,\frac {1}{2}-p,-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c^3 (e+2 e p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {393, 152, 150}

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.

\(\displaystyle \int \frac {(e x)^{-2 p-2} \left (a+b x^2\right )^p}{\left (c+d x^2\right )^3} \, dx\)

\(\Big \downarrow \) 393

\(\displaystyle \frac {\left (x^2\right )^{p+\frac {3}{2}} (e x)^{-2 (p+1)} \int \frac {\left (x^2\right )^{-p-\frac {3}{2}} \left (b x^2+a\right )^p}{\left (d x^2+c\right )^3}dx^2}{2 x}\)

\(\Big \downarrow \) 152

\(\displaystyle \frac {\left (x^2\right )^{p+\frac {3}{2}} (e x)^{-2 (p+1)} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int \frac {\left (x^2\right )^{-p-\frac {3}{2}} \left (\frac {b x^2}{a}+1\right )^p}{\left (d x^2+c\right )^3}dx^2}{2 x}\)

\(\Big \downarrow \) 150

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 152

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) 
Int[(b*x)^m*(1 + d*(x/c))^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !GtQ[c, 0]
 

rule 393

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

Maple [F]

\[\int \frac {\left (e x \right )^{-2 p -2} \left (b \,x^{2}+a \right )^{p}}{\left (x^{2} d +c \right )^{3}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((e*x)^(-2-2*p)*(b*x^2+a)^p/(d*x^2+c)^3,x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)^p*(e*x)^(-2*p - 2)/(d^3*x^6 + 3*c*d^2*x^4 + 3*c^2*d*x 
^2 + c^3), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{-2-2 p} \left (a+b x^2\right )^p}{\left (c+d x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((e*x)**(-2-2*p)*(b*x**2+a)**p/(d*x**2+c)**3,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((e*x)^(-2-2*p)*(b*x^2+a)^p/(d*x^2+c)^3,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x)^(-2-2*p)*(b*x^2+a)^p/(d*x^2+c)^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{-2-2 p} \left (a+b x^2\right )^p}{\left (c+d x^2\right )^3} \, dx=\int \frac {{\left (b\,x^2+a\right )}^p}{{\left (e\,x\right )}^{2\,p+2}\,{\left (d\,x^2+c\right )}^3} \,d x \] Input:

int((a + b*x^2)^p/((e*x)^(2*p + 2)*(c + d*x^2)^3),x)
 

Output:

int((a + b*x^2)^p/((e*x)^(2*p + 2)*(c + d*x^2)^3), x)
 

Reduce [F]

\[ \int \frac {(e x)^{-2-2 p} \left (a+b x^2\right )^p}{\left (c+d x^2\right )^3} \, dx=\frac {\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} c^{3} x^{2}+3 x^{2 p} c^{2} d \,x^{4}+3 x^{2 p} c \,d^{2} x^{6}+x^{2 p} d^{3} x^{8}}d x}{e^{2 p} e^{2}} \] Input:

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

Output:

int((a + b*x**2)**p/(x**(2*p)*c**3*x**2 + 3*x**(2*p)*c**2*d*x**4 + 3*x**(2 
*p)*c*d**2*x**6 + x**(2*p)*d**3*x**8),x)/(e**(2*p)*e**2)