\(\int \frac {(e x)^{-2 p} (a+b x^2)^p}{(c+d x)^2} \, dx\) [307]

Optimal result

Integrand size = 24, antiderivative size = 252 \[ \int \frac {(e x)^{-2 p} \left (a+b x^2\right )^p}{(c+d x)^2} \, dx=\frac {(e x)^{1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2} (1-2 p),-p,2,\frac {1}{2} (3-2 p),-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 e (1-2 p)}+\frac {d^2 (e x)^{3-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2} (3-2 p),-p,2,\frac {1}{2} (5-2 p),-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^4 e^3 (3-2 p)}-\frac {a d (e x)^{2-2 p} \left (a+b x^2\right )^{-1+p} \operatorname {Hypergeometric2F1}\left (2,1-p,2-p,\frac {\left (b+\frac {a d^2}{c^2}\right ) x^2}{a+b x^2}\right )}{c^3 e^2 (1-p)} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.

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

\(\Big \downarrow \) 623

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

\(\Big \downarrow \) 622

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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