\(\int \frac {(e x)^{-2-2 p} (a+b x^2)^p}{(c+d x)^2} \, dx\) [1901]

Optimal result

Integrand size = 26, antiderivative size = 315 \[ \int \frac {(e x)^{-2-2 p} \left (a+b x^2\right )^p}{(c+d x)^2} \, dx=\frac {d (e x)^{-2 p} \left (a+b x^2\right )^{1+p}}{a c e^2 p \left (c^2-d^2 x^2\right )}-\frac {(e x)^{-1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2} (-1-2 p),-p,2,\frac {1}{2} (1-2 p),-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 e (1+2 p)}+\frac {d^2 (e x)^{1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2} (1-2 p),-p,2,\frac {1}{2} (3-2 p),-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^4 e^3 (1-2 p)}-\frac {d \left (b c^2+a d^2 (1+p)\right ) (e x)^{2-2 p} \left (a+b x^2\right )^{-1+p} \operatorname {Hypergeometric2F1}\left (2,1-p,2-p,\frac {\left (b+\frac {a d^2}{c^2}\right ) x^2}{a+b x^2}\right )}{c^5 e^4 (1-p) p} \] Output:

d*(b*x^2+a)^(p+1)/a/c/e^2/p/((e*x)^(2*p))/(-d^2*x^2+c^2)-(e*x)^(-1-2*p)*(b 
*x^2+a)^p*AppellF1(-1/2-p,2,-p,1/2-p,d^2*x^2/c^2,-b*x^2/a)/c^2/e/(1+2*p)/( 
(1+b*x^2/a)^p)+d^2*(e*x)^(1-2*p)*(b*x^2+a)^p*AppellF1(1/2-p,2,-p,3/2-p,d^2 
*x^2/c^2,-b*x^2/a)/c^4/e^3/(1-2*p)/((1+b*x^2/a)^p)-d*(b*c^2+a*d^2*(p+1))*( 
e*x)^(2-2*p)*(b*x^2+a)^(-1+p)*hypergeom([2, 1-p],[2-p],(b+a*d^2/c^2)*x^2/( 
b*x^2+a))/c^5/e^4/(1-p)/p
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {623, 622, 2009}

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.

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

\(\Big \downarrow \) 623

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

\(\Big \downarrow \) 622

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

\(\Big \downarrow \) 2009

\(\displaystyle x^{2 (p+1)} (e x)^{-2 (p+1)} \left (-\frac {x^{-2 p-1} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (-p-\frac {1}{2},-p,2,\frac {1}{2}-p,-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 (2 p+1)}+\frac {d^2 x^{1-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2}-p,-p,2,\frac {3}{2}-p,-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^4 (1-2 p)}+\frac {d x^{-2 p} \left (a+b x^2\right )^{p+1}}{a c p \left (c^2-d^2 x^2\right )}-\frac {d x^{2 (1-p)} \left (a+b x^2\right )^{p-1} \left (a d^2 (p+1)+b c^2\right ) \operatorname {Hypergeometric2F1}\left (2,1-p,2-p,\frac {\left (\frac {a d^2}{c^2}+b\right ) x^2}{b x^2+a}\right )}{c^5 (1-p) p}\right )\)

Input:

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

Output:

(x^(2*(1 + p))*((d*(a + b*x^2)^(1 + p))/(a*c*p*x^(2*p)*(c^2 - d^2*x^2)) - 
(x^(-1 - 2*p)*(a + b*x^2)^p*AppellF1[-1/2 - p, -p, 2, 1/2 - p, -((b*x^2)/a 
), (d^2*x^2)/c^2])/(c^2*(1 + 2*p)*(1 + (b*x^2)/a)^p) + (d^2*x^(1 - 2*p)*(a 
 + b*x^2)^p*AppellF1[1/2 - p, -p, 2, 3/2 - p, -((b*x^2)/a), (d^2*x^2)/c^2] 
)/(c^4*(1 - 2*p)*(1 + (b*x^2)/a)^p) - (d*(b*c^2 + a*d^2*(1 + p))*x^(2*(1 - 
 p))*(a + b*x^2)^(-1 + p)*Hypergeometric2F1[2, 1 - p, 2 - p, ((b + (a*d^2) 
/c^2)*x^2)/(a + b*x^2)])/(c^5*(1 - p)*p)))/(e*x)^(2*(1 + p))
 

Defintions of rubi rules used

rule 622

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[x^m*(a + b*x^2)^p, (c/(c^2 - d^2*x^2) - d*(x/(c^2 
 - d^2*x^2)))^(-n), x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[n, -1]
 

rule 623

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 6*(a + b*x**2)**p*a*c**2*d*p - 4*(a + b*x**2)**p*a*c*d**2*p**2*x - 6*( 
a + b*x**2)**p*a*c*d**2*p*x - 2*(a + b*x**2)**p*a*c*d**2*x - 4*(a + b*x**2 
)**p*a*d**3*p**2*x**2 - 6*(a + b*x**2)**p*a*d**3*p*x**2 - 2*(a + b*x**2)** 
p*a*d**3*x**2 + (a + b*x**2)**p*b*c**3*p*x + 2*(a + b*x**2)**p*b*c**3*x + 
(a + b*x**2)**p*b*c**2*d*p*x**2 + 2*(a + b*x**2)**p*b*c**2*d*x**2 - 16*x** 
(2*p)*int((a + b*x**2)**p/(x**(2*p)*a*c**2 + 2*x**(2*p)*a*c*d*x + x**(2*p) 
*a*d**2*x**2 + x**(2*p)*b*c**2*x**2 + 2*x**(2*p)*b*c*d*x**3 + x**(2*p)*b*d 
**2*x**4),x)*a**2*c**2*d**3*p**3*x - 24*x**(2*p)*int((a + b*x**2)**p/(x**( 
2*p)*a*c**2 + 2*x**(2*p)*a*c*d*x + x**(2*p)*a*d**2*x**2 + x**(2*p)*b*c**2* 
x**2 + 2*x**(2*p)*b*c*d*x**3 + x**(2*p)*b*d**2*x**4),x)*a**2*c**2*d**3*p** 
2*x - 8*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a*c**2 + 2*x**(2*p)*a*c*d*x 
 + x**(2*p)*a*d**2*x**2 + x**(2*p)*b*c**2*x**2 + 2*x**(2*p)*b*c*d*x**3 + x 
**(2*p)*b*d**2*x**4),x)*a**2*c**2*d**3*p*x - 16*x**(2*p)*int((a + b*x**2)* 
*p/(x**(2*p)*a*c**2 + 2*x**(2*p)*a*c*d*x + x**(2*p)*a*d**2*x**2 + x**(2*p) 
*b*c**2*x**2 + 2*x**(2*p)*b*c*d*x**3 + x**(2*p)*b*d**2*x**4),x)*a**2*c*d** 
4*p**3*x**2 - 24*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a*c**2 + 2*x**(2*p 
)*a*c*d*x + x**(2*p)*a*d**2*x**2 + x**(2*p)*b*c**2*x**2 + 2*x**(2*p)*b*c*d 
*x**3 + x**(2*p)*b*d**2*x**4),x)*a**2*c*d**4*p**2*x**2 - 8*x**(2*p)*int((a 
 + b*x**2)**p/(x**(2*p)*a*c**2 + 2*x**(2*p)*a*c*d*x + x**(2*p)*a*d**2*x**2 
 + x**(2*p)*b*c**2*x**2 + 2*x**(2*p)*b*c*d*x**3 + x**(2*p)*b*d**2*x**4)...