\(\int \frac {(e x)^{-2 p} (a+b x^2)^p}{(c+d x)^3} \, dx\) [308]

Optimal result

Integrand size = 24, antiderivative size = 407 \[ \int \frac {(e x)^{-2 p} \left (a+b x^2\right )^p}{(c+d x)^3} \, dx=-\frac {3 d (e x)^{2-2 p} \left (a+b x^2\right )^{1+p}}{2 a e^2 (1-p) \left (c^2-d^2 x^2\right )^2}+\frac {(e x)^{1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2} (1-2 p),-p,3,\frac {1}{2} (3-2 p),-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^3 e (1-2 p)}+\frac {3 d^2 (e x)^{3-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2} (3-2 p),-p,3,\frac {1}{2} (5-2 p),-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^5 e^3 (3-2 p)}-\frac {a^2 d^3 (e x)^{4-2 p} \left (a+b x^2\right )^{-2+p} \operatorname {Hypergeometric2F1}\left (3,2-p,3-p,\frac {\left (b+\frac {a d^2}{c^2}\right ) x^2}{a+b x^2}\right )}{2 c^6 e^4 (2-p)}+\frac {3 a d \left (2 b c^2+a d^2 (1+p)\right ) (e x)^{4-2 p} \left (a+b x^2\right )^{-2+p} \operatorname {Hypergeometric2F1}\left (3,2-p,3-p,\frac {\left (b+\frac {a d^2}{c^2}\right ) x^2}{a+b x^2}\right )}{2 c^6 e^4 (1-p) (2-p)} \] Output:

-3/2*d*(e*x)^(2-2*p)*(b*x^2+a)^(p+1)/a/e^2/(1-p)/(-d^2*x^2+c^2)^2+(e*x)^(1 
-2*p)*(b*x^2+a)^p*AppellF1(1/2-p,3,-p,3/2-p,d^2*x^2/c^2,-b*x^2/a)/c^3/e/(1 
-2*p)/((1+b*x^2/a)^p)+3*d^2*(e*x)^(3-2*p)*(b*x^2+a)^p*AppellF1(3/2-p,3,-p, 
5/2-p,d^2*x^2/c^2,-b*x^2/a)/c^5/e^3/(3-2*p)/((1+b*x^2/a)^p)-1/2*a^2*d^3*(e 
*x)^(4-2*p)*(b*x^2+a)^(-2+p)*hypergeom([3, 2-p],[3-p],(b+a*d^2/c^2)*x^2/(b 
*x^2+a))/c^6/e^4/(2-p)+3/2*a*d*(2*b*c^2+a*d^2*(p+1))*(e*x)^(4-2*p)*(b*x^2+ 
a)^(-2+p)*hypergeom([3, 2-p],[3-p],(b+a*d^2/c^2)*x^2/(b*x^2+a))/c^6/e^4/(1 
-p)/(2-p)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.70 (sec) , antiderivative size = 398, normalized size of antiderivative = 0.98, 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.

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

\(\Big \downarrow \) 623

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

\(\Big \downarrow \) 622

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

\(\Big \downarrow \) 2009

\(\displaystyle x^{2 p} (e x)^{-2 p} \left (-\frac {a^2 d^3 x^{4-2 p} \left (a+b x^2\right )^{p-2} \operatorname {Hypergeometric2F1}\left (3,2-p,3-p,\frac {\left (\frac {a d^2}{c^2}+b\right ) x^2}{b x^2+a}\right )}{2 c^6 (2-p)}+\frac {3 d^2 x^{3-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2}-p,-p,3,\frac {5}{2}-p,-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^5 (3-2 p)}+\frac {x^{1-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2}-p,-p,3,\frac {3}{2}-p,-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^3 (1-2 p)}-\frac {3 d^3 x^{2-2 p} \left (a+b x^2\right )^{p+1}}{4 \left (c^2-d^2 x^2\right )^2 \left (a d^2+b c^2\right )}-\frac {3 a d x^{2-2 p} \left (a+b x^2\right )^{p-1} \left (a d^2 (p+1)+2 b c^2\right ) \operatorname {Hypergeometric2F1}\left (2,1-p,2-p,\frac {\left (\frac {a d^2}{c^2}+b\right ) x^2}{b x^2+a}\right )}{4 c^4 (1-p) \left (a d^2+b c^2\right )}\right )\)

Input:

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

Output:

(x^(2*p)*((-3*d^3*x^(2 - 2*p)*(a + b*x^2)^(1 + p))/(4*(b*c^2 + a*d^2)*(c^2 
 - d^2*x^2)^2) + (x^(1 - 2*p)*(a + b*x^2)^p*AppellF1[1/2 - p, -p, 3, 3/2 - 
 p, -((b*x^2)/a), (d^2*x^2)/c^2])/(c^3*(1 - 2*p)*(1 + (b*x^2)/a)^p) + (3*d 
^2*x^(3 - 2*p)*(a + b*x^2)^p*AppellF1[3/2 - p, -p, 3, 5/2 - p, -((b*x^2)/a 
), (d^2*x^2)/c^2])/(c^5*(3 - 2*p)*(1 + (b*x^2)/a)^p) - (3*a*d*(2*b*c^2 + a 
*d^2*(1 + p))*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)])/(4*c^4*(b*c^2 + a*d^2)*(1 - 
p)) - (a^2*d^3*x^(4 - 2*p)*(a + b*x^2)^(-2 + p)*Hypergeometric2F1[3, 2 - p 
, 3 - p, ((b + (a*d^2)/c^2)*x^2)/(a + b*x^2)])/(2*c^6*(2 - 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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