\(\int \frac {(e x)^m (a+b x^2)}{(c+d x)^2} \, dx\) [105]

Optimal result

Integrand size = 20, antiderivative size = 109 \[ \int \frac {(e x)^m \left (a+b x^2\right )}{(c+d x)^2} \, dx=\frac {b (e x)^{1+m}}{d^2 e (1+m)}+\frac {\left (b c^2+a d^2\right ) (e x)^{1+m}}{c d^2 e (c+d x)}-\frac {\left (a d^2 m+b c^2 (2+m)\right ) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d x}{c}\right )}{c^2 d^2 e (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.69 \[ \int \frac {(e x)^m \left (a+b x^2\right )}{(c+d x)^2} \, dx=-\frac {x (e x)^m \left (-b c^2+2 b c^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d x}{c}\right )-\left (b c^2+a d^2\right ) \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,-\frac {d x}{c}\right )\right )}{c^2 d^2 (1+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {519, 27, 90, 74}

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.

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

\(\Big \downarrow \) 519

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 74

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

Input:

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

Output:

((a + (b*c^2)/d^2)*(e*x)^(1 + m))/(c*e*(c + d*x)) - (-((b*c*(e*x)^(1 + m)) 
/(d*e*(1 + m))) + ((a*d^2*m + b*c^2*(2 + m))*(e*x)^(1 + m)*Hypergeometric2 
F1[1, 1 + m, 2 + m, -((d*x)/c)])/(c*d*e*(1 + m)))/(c*d)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 519

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2)^p, c + d*x, x], R = 
 PolynomialRemainder[(a + b*x^2)^p, c + d*x, x]}, Simp[(-R)*(e*x)^(m + 1)*( 
(c + d*x)^(n + 1)/(c*e*(n + 1))), x] + Simp[1/(c*(n + 1))   Int[(e*x)^m*(c 
+ d*x)^(n + 1)*ExpandToSum[c*(n + 1)*Qx + R*(m + n + 2), x], x], x]] /; Fre 
eQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0] && LtQ[n, -1] &&  !IntegerQ[m]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.34 (sec) , antiderivative size = 678, normalized size of antiderivative = 6.22 \[ \int \frac {(e x)^m \left (a+b x^2\right )}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

a*(-c*e**m*m**2*x**(m + 1)*lerchphi(d*x*exp_polar(I*pi)/c, 1, m + 1)*gamma 
(m + 1)/(c**3*gamma(m + 2) + c**2*d*x*gamma(m + 2)) - c*e**m*m*x**(m + 1)* 
lerchphi(d*x*exp_polar(I*pi)/c, 1, m + 1)*gamma(m + 1)/(c**3*gamma(m + 2) 
+ c**2*d*x*gamma(m + 2)) + c*e**m*m*x**(m + 1)*gamma(m + 1)/(c**3*gamma(m 
+ 2) + c**2*d*x*gamma(m + 2)) + c*e**m*x**(m + 1)*gamma(m + 1)/(c**3*gamma 
(m + 2) + c**2*d*x*gamma(m + 2)) - d*e**m*m**2*x*x**(m + 1)*lerchphi(d*x*e 
xp_polar(I*pi)/c, 1, m + 1)*gamma(m + 1)/(c**3*gamma(m + 2) + c**2*d*x*gam 
ma(m + 2)) - d*e**m*m*x*x**(m + 1)*lerchphi(d*x*exp_polar(I*pi)/c, 1, m + 
1)*gamma(m + 1)/(c**3*gamma(m + 2) + c**2*d*x*gamma(m + 2))) + b*(-c*e**m* 
m**2*x**(m + 3)*lerchphi(d*x*exp_polar(I*pi)/c, 1, m + 3)*gamma(m + 3)/(c* 
*3*gamma(m + 4) + c**2*d*x*gamma(m + 4)) - 5*c*e**m*m*x**(m + 3)*lerchphi( 
d*x*exp_polar(I*pi)/c, 1, m + 3)*gamma(m + 3)/(c**3*gamma(m + 4) + c**2*d* 
x*gamma(m + 4)) + c*e**m*m*x**(m + 3)*gamma(m + 3)/(c**3*gamma(m + 4) + c* 
*2*d*x*gamma(m + 4)) - 6*c*e**m*x**(m + 3)*lerchphi(d*x*exp_polar(I*pi)/c, 
 1, m + 3)*gamma(m + 3)/(c**3*gamma(m + 4) + c**2*d*x*gamma(m + 4)) + 3*c* 
e**m*x**(m + 3)*gamma(m + 3)/(c**3*gamma(m + 4) + c**2*d*x*gamma(m + 4)) - 
 d*e**m*m**2*x*x**(m + 3)*lerchphi(d*x*exp_polar(I*pi)/c, 1, m + 3)*gamma( 
m + 3)/(c**3*gamma(m + 4) + c**2*d*x*gamma(m + 4)) - 5*d*e**m*m*x*x**(m + 
3)*lerchphi(d*x*exp_polar(I*pi)/c, 1, m + 3)*gamma(m + 3)/(c**3*gamma(m + 
4) + c**2*d*x*gamma(m + 4)) - 6*d*e**m*x*x**(m + 3)*lerchphi(d*x*exp_po...
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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