\(\int \frac {(e x)^{-2 p} (a+b x^2)^p}{c+d x^2} \, dx\) [1618]

Optimal result

Integrand size = 26, antiderivative size = 87 \[ \int \frac {(e x)^{-2 p} \left (a+b x^2\right )^p}{c+d x^2} \, dx=\frac {(e x)^{1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2} (1-2 p),-p,1,\frac {1}{2} (3-2 p),-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c e (1-2 p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.89 \[ \int \frac {(e x)^{-2 p} \left (a+b x^2\right )^p}{c+d x^2} \, dx=\frac {x (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2}-p,-p,1,\frac {3}{2}-p,-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c-2 c p} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, 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.

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

\(\Big \downarrow \) 393

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

\(\Big \downarrow \) 152

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

\(\Big \downarrow \) 150

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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