\(\int \frac {(e x)^m}{(c+d x)^2 (a+b x^2)^2} \, dx\) [128]

Optimal result

Integrand size = 22, antiderivative size = 372 \[ \int \frac {(e x)^m}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\frac {b d^2 \left (3 b c^2-a d^2\right ) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a \left (b c^2+a d^2\right )^3 e (1+m)}+\frac {4 b d^4 (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d x}{c}\right )}{\left (b c^2+a d^2\right )^3 e (1+m)}-\frac {4 b^2 c d^3 (e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )}{a \left (b c^2+a d^2\right )^3 e^2 (2+m)}+\frac {b \left (b c^2-a d^2\right ) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a^2 \left (b c^2+a d^2\right )^2 e (1+m)}+\frac {d^4 (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,-\frac {d x}{c}\right )}{c^2 \left (b c^2+a d^2\right )^2 e (1+m)}-\frac {2 b^2 c d (e x)^{2+m} \operatorname {Hypergeometric2F1}\left (2,\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )}{a^2 \left (b c^2+a d^2\right )^2 e^2 (2+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.78 \[ \int \frac {(e x)^m}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\frac {x (e x)^m \left (\frac {b d^2 \left (3 b c^2-a d^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a (1+m)}+\frac {4 b d^4 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d x}{c}\right )}{1+m}-\frac {4 b^2 c d^3 x \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )}{a (2+m)}+\frac {b \left (b c^2-a d^2\right ) \left (b c^2+a d^2\right ) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a^2 (1+m)}+\frac {d^4 \left (b c^2+a d^2\right ) \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,-\frac {d x}{c}\right )}{c^2 (1+m)}-\frac {2 b^2 c d \left (b c^2+a d^2\right ) x \operatorname {Hypergeometric2F1}\left (2,\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )}{a^2 (2+m)}\right )}{\left (b c^2+a d^2\right )^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {615, 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.

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

\(\Big \downarrow \) 615

\(\displaystyle \int \left (-\frac {b d^2 (e x)^m \left (a d^2-3 b c^2+4 b c d x\right )}{\left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}+\frac {b (e x)^m \left (-a d^2+b c^2-2 b c d x\right )}{\left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^2}+\frac {4 b c d^4 (e x)^m}{(c+d x) \left (a d^2+b c^2\right )^3}+\frac {d^4 (e x)^m}{(c+d x)^2 \left (a d^2+b c^2\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 615

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

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((e*x)^m/(b^2*d^2*x^6 + 2*b^2*c*d*x^5 + 4*a*b*c*d*x^3 + 2*a^2*c*d* 
x + (b^2*c^2 + 2*a*b*d^2)*x^4 + a^2*c^2 + (2*a*b*c^2 + a^2*d^2)*x^2), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e x)^m}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=e^{m} \left (\int \frac {x^{m}}{b^{2} d^{2} x^{6}+2 b^{2} c d \,x^{5}+2 a b \,d^{2} x^{4}+b^{2} c^{2} x^{4}+4 a b c d \,x^{3}+a^{2} d^{2} x^{2}+2 a b \,c^{2} x^{2}+2 a^{2} c d x +a^{2} c^{2}}d x \right ) \] Input:

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

Output:

e**m*int(x**m/(a**2*c**2 + 2*a**2*c*d*x + a**2*d**2*x**2 + 2*a*b*c**2*x**2 
 + 4*a*b*c*d*x**3 + 2*a*b*d**2*x**4 + b**2*c**2*x**4 + 2*b**2*c*d*x**5 + b 
**2*d**2*x**6),x)