\(\int \frac {(e x)^m (a+b x^2)^p (A+B x^2)}{c+d x^2} \, dx\) [47]

Optimal result

Integrand size = 31, antiderivative size = 162 \[ \int \frac {(e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right )}{c+d x^2} \, dx=-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m}{2},-p,1,\frac {3+m}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c d e (1+m)}+\frac {B (e x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {b x^2}{a}\right )}{d e (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.73 \[ \int \frac {(e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right )}{c+d x^2} \, dx=\frac {x (e x)^m \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left ((-B c+A d) \operatorname {AppellF1}\left (\frac {1+m}{2},-p,1,\frac {3+m}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+B c \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {b x^2}{a}\right )\right )}{c d (1+m)} \] Input:

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

Output:

(x*(e*x)^m*(a + b*x^2)^p*((-(B*c) + A*d)*AppellF1[(1 + m)/2, -p, 1, (3 + m 
)/2, -((b*x^2)/a), -((d*x^2)/c)] + B*c*Hypergeometric2F1[(1 + m)/2, -p, (3 
 + m)/2, -((b*x^2)/a)]))/(c*d*(1 + m)*(1 + (b*x^2)/a)^p)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {446, 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.

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

\(\Big \downarrow \) 446

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-(((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^2)^p*AppellF1[(1 + m)/2, -p, 1, (3 + 
 m)/2, -((b*x^2)/a), -((d*x^2)/c)])/(c*d*e*(1 + m)*(1 + (b*x^2)/a)^p)) + ( 
B*(e*x)^(1 + m)*(a + b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, 
-((b*x^2)/a)])/(d*e*(1 + m)*(1 + (b*x^2)/a)^p)
 

Defintions of rubi rules used

rule 446

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

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right )}{c+d x^2} \, dx=\text {too large to display} \] Input:

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

Output:

(e**m*(x**m*(a + b*x**2)**p*b*x + int((x**m*(a + b*x**2)**p*x**2)/(a*c*m + 
 2*a*c*p + a*c + a*d*m*x**2 + 2*a*d*p*x**2 + a*d*x**2 + b*c*m*x**2 + 2*b*c 
*p*x**2 + b*c*x**2 + b*d*m*x**4 + 2*b*d*p*x**4 + b*d*x**4),x)*a*b*d*m**2 + 
 6*int((x**m*(a + b*x**2)**p*x**2)/(a*c*m + 2*a*c*p + a*c + a*d*m*x**2 + 2 
*a*d*p*x**2 + a*d*x**2 + b*c*m*x**2 + 2*b*c*p*x**2 + b*c*x**2 + b*d*m*x**4 
 + 2*b*d*p*x**4 + b*d*x**4),x)*a*b*d*m*p + 2*int((x**m*(a + b*x**2)**p*x** 
2)/(a*c*m + 2*a*c*p + a*c + a*d*m*x**2 + 2*a*d*p*x**2 + a*d*x**2 + b*c*m*x 
**2 + 2*b*c*p*x**2 + b*c*x**2 + b*d*m*x**4 + 2*b*d*p*x**4 + b*d*x**4),x)*a 
*b*d*m + 8*int((x**m*(a + b*x**2)**p*x**2)/(a*c*m + 2*a*c*p + a*c + a*d*m* 
x**2 + 2*a*d*p*x**2 + a*d*x**2 + b*c*m*x**2 + 2*b*c*p*x**2 + b*c*x**2 + b* 
d*m*x**4 + 2*b*d*p*x**4 + b*d*x**4),x)*a*b*d*p**2 + 6*int((x**m*(a + b*x** 
2)**p*x**2)/(a*c*m + 2*a*c*p + a*c + a*d*m*x**2 + 2*a*d*p*x**2 + a*d*x**2 
+ b*c*m*x**2 + 2*b*c*p*x**2 + b*c*x**2 + b*d*m*x**4 + 2*b*d*p*x**4 + b*d*x 
**4),x)*a*b*d*p + int((x**m*(a + b*x**2)**p*x**2)/(a*c*m + 2*a*c*p + a*c + 
 a*d*m*x**2 + 2*a*d*p*x**2 + a*d*x**2 + b*c*m*x**2 + 2*b*c*p*x**2 + b*c*x* 
*2 + b*d*m*x**4 + 2*b*d*p*x**4 + b*d*x**4),x)*a*b*d - int((x**m*(a + b*x** 
2)**p*x**2)/(a*c*m + 2*a*c*p + a*c + a*d*m*x**2 + 2*a*d*p*x**2 + a*d*x**2 
+ b*c*m*x**2 + 2*b*c*p*x**2 + b*c*x**2 + b*d*m*x**4 + 2*b*d*p*x**4 + b*d*x 
**4),x)*b**2*c*m**2 - 4*int((x**m*(a + b*x**2)**p*x**2)/(a*c*m + 2*a*c*p + 
 a*c + a*d*m*x**2 + 2*a*d*p*x**2 + a*d*x**2 + b*c*m*x**2 + 2*b*c*p*x**2...