\(\int \frac {(e x)^m (c+d x)^n}{(a+b x^2)^2} \, dx\) [282]

Optimal result

Integrand size = 22, antiderivative size = 413 \[ \int \frac {(e x)^m (c+d x)^n}{\left (a+b x^2\right )^2} \, dx=\frac {b (e x)^{1+m} (c-d x) (c+d x)^{1+n}}{2 a \left (b c^2+a d^2\right ) e \left (a+b x^2\right )}+\frac {\left (\sqrt {-a} b c^2 (1-m)+\sqrt {-a} a d^2 (1-m-n)-a \sqrt {b} c d n\right ) (e x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (1+m,-n,1,2+m,-\frac {d x}{c},-\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{4 (-a)^{5/2} \left (b c^2+a d^2\right ) e (1+m)}+\frac {\left (\sqrt {-a} b c^2 (1-m)+\sqrt {-a} a d^2 (1-m-n)+a \sqrt {b} c d n\right ) (e x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (1+m,-n,1,2+m,-\frac {d x}{c},\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{4 (-a)^{5/2} \left (b c^2+a d^2\right ) e (1+m)}+\frac {d^2 (1+m+n) (e x)^{1+m} (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d x}{c}\right )}{2 a \left (b c^2+a d^2\right ) e (1+m)} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.71, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(e x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (m+1,-n,1,m+2,-\frac {d x}{c},-\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{4 a^2 e (m+1)}+\frac {(e x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (m+1,-n,1,m+2,-\frac {d x}{c},\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{4 a^2 e (m+1)}+\frac {(e x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (m+1,-n,2,m+2,-\frac {d x}{c},-\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{4 a^2 e (m+1)}+\frac {(e x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (m+1,-n,2,m+2,-\frac {d x}{c},\frac {\sqrt {b} x}{\sqrt {-a}}\right )}{4 a^2 e (m+1)}\)

Input:

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

Output:

((e*x)^(1 + m)*(c + d*x)^n*AppellF1[1 + m, -n, 1, 2 + m, -((d*x)/c), -((Sq 
rt[b]*x)/Sqrt[-a])])/(4*a^2*e*(1 + m)*(1 + (d*x)/c)^n) + ((e*x)^(1 + m)*(c 
 + d*x)^n*AppellF1[1 + m, -n, 1, 2 + m, -((d*x)/c), (Sqrt[b]*x)/Sqrt[-a]]) 
/(4*a^2*e*(1 + m)*(1 + (d*x)/c)^n) + ((e*x)^(1 + m)*(c + d*x)^n*AppellF1[1 
 + m, -n, 2, 2 + m, -((d*x)/c), -((Sqrt[b]*x)/Sqrt[-a])])/(4*a^2*e*(1 + m) 
*(1 + (d*x)/c)^n) + ((e*x)^(1 + m)*(c + d*x)^n*AppellF1[1 + m, -n, 2, 2 + 
m, -((d*x)/c), (Sqrt[b]*x)/Sqrt[-a]])/(4*a^2*e*(1 + m)*(1 + (d*x)/c)^n)
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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