\(\int \frac {(e x)^m (c+d x)}{(a+b x^2)^3} \, dx\) [1722]

Optimal result

Integrand size = 20, antiderivative size = 91 \[ \int \frac {(e x)^m (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {c (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a^3 e (1+m)}+\frac {d (e x)^{2+m} \operatorname {Hypergeometric2F1}\left (3,\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )}{a^3 e^2 (2+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.90 \[ \int \frac {(e x)^m (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {x (e x)^m \left (d (1+m) x \operatorname {Hypergeometric2F1}\left (3,1+\frac {m}{2},2+\frac {m}{2},-\frac {b x^2}{a}\right )+c (2+m) \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )\right )}{a^3 (1+m) (2+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {557, 278}

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.

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

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 557

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[c   Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e   Int[(e*x)^( 
m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 68.11 (sec) , antiderivative size = 2530, normalized size of antiderivative = 27.80 \[ \int \frac {(e x)^m (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

c*(a**2*e**m*m**3*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1 
/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 
+ 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) - 3*a**2*e**m*m**2*x**(m + 1) 
*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a** 
5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*g 
amma(m/2 + 3/2)) - 2*a**2*e**m*m**2*x**(m + 1)*gamma(m/2 + 1/2)/(32*a**5*g 
amma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamm 
a(m/2 + 3/2)) - a**2*e**m*m*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 
1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x**2* 
gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) + 8*a**2*e**m*m*x** 
(m + 1)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma( 
m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) + 3*a**2*e**m*x**(m + 1)* 
lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5 
*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*ga 
mma(m/2 + 3/2)) + 10*a**2*e**m*x**(m + 1)*gamma(m/2 + 1/2)/(32*a**5*gamma( 
m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 
 + 3/2)) + 2*a*b*e**m*m**3*x**2*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi) 
/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x 
**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) - 6*a*b*e**m*m* 
*2*x**2*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gam...
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e x)^m (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {e^{m} \left (x^{m} d +\left (\int \frac {x^{m}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a^{2} b c m -4 \left (\int \frac {x^{m}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a^{2} b c +2 \left (\int \frac {x^{m}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a \,b^{2} c m \,x^{2}-8 \left (\int \frac {x^{m}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a \,b^{2} c \,x^{2}+\left (\int \frac {x^{m}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) b^{3} c m \,x^{4}-4 \left (\int \frac {x^{m}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) b^{3} c \,x^{4}-\left (\int \frac {x^{m}}{b^{3} m \,x^{7}-4 b^{3} x^{7}+3 a \,b^{2} m \,x^{5}-12 a \,b^{2} x^{5}+3 a^{2} b m \,x^{3}-12 a^{2} b \,x^{3}+a^{3} m x -4 a^{3} x}d x \right ) a^{3} d \,m^{2}+4 \left (\int \frac {x^{m}}{b^{3} m \,x^{7}-4 b^{3} x^{7}+3 a \,b^{2} m \,x^{5}-12 a \,b^{2} x^{5}+3 a^{2} b m \,x^{3}-12 a^{2} b \,x^{3}+a^{3} m x -4 a^{3} x}d x \right ) a^{3} d m -2 \left (\int \frac {x^{m}}{b^{3} m \,x^{7}-4 b^{3} x^{7}+3 a \,b^{2} m \,x^{5}-12 a \,b^{2} x^{5}+3 a^{2} b m \,x^{3}-12 a^{2} b \,x^{3}+a^{3} m x -4 a^{3} x}d x \right ) a^{2} b d \,m^{2} x^{2}+8 \left (\int \frac {x^{m}}{b^{3} m \,x^{7}-4 b^{3} x^{7}+3 a \,b^{2} m \,x^{5}-12 a \,b^{2} x^{5}+3 a^{2} b m \,x^{3}-12 a^{2} b \,x^{3}+a^{3} m x -4 a^{3} x}d x \right ) a^{2} b d m \,x^{2}-\left (\int \frac {x^{m}}{b^{3} m \,x^{7}-4 b^{3} x^{7}+3 a \,b^{2} m \,x^{5}-12 a \,b^{2} x^{5}+3 a^{2} b m \,x^{3}-12 a^{2} b \,x^{3}+a^{3} m x -4 a^{3} x}d x \right ) a \,b^{2} d \,m^{2} x^{4}+4 \left (\int \frac {x^{m}}{b^{3} m \,x^{7}-4 b^{3} x^{7}+3 a \,b^{2} m \,x^{5}-12 a \,b^{2} x^{5}+3 a^{2} b m \,x^{3}-12 a^{2} b \,x^{3}+a^{3} m x -4 a^{3} x}d x \right ) a \,b^{2} d m \,x^{4}\right )}{b \left (b^{2} m \,x^{4}-4 b^{2} x^{4}+2 a b m \,x^{2}-8 a b \,x^{2}+a^{2} m -4 a^{2}\right )} \] Input:

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

Output:

(e**m*(x**m*d + int(x**m/(a**3 + 3*a**2*b*x**2 + 3*a*b**2*x**4 + b**3*x**6 
),x)*a**2*b*c*m - 4*int(x**m/(a**3 + 3*a**2*b*x**2 + 3*a*b**2*x**4 + b**3* 
x**6),x)*a**2*b*c + 2*int(x**m/(a**3 + 3*a**2*b*x**2 + 3*a*b**2*x**4 + b** 
3*x**6),x)*a*b**2*c*m*x**2 - 8*int(x**m/(a**3 + 3*a**2*b*x**2 + 3*a*b**2*x 
**4 + b**3*x**6),x)*a*b**2*c*x**2 + int(x**m/(a**3 + 3*a**2*b*x**2 + 3*a*b 
**2*x**4 + b**3*x**6),x)*b**3*c*m*x**4 - 4*int(x**m/(a**3 + 3*a**2*b*x**2 
+ 3*a*b**2*x**4 + b**3*x**6),x)*b**3*c*x**4 - int(x**m/(a**3*m*x - 4*a**3* 
x + 3*a**2*b*m*x**3 - 12*a**2*b*x**3 + 3*a*b**2*m*x**5 - 12*a*b**2*x**5 + 
b**3*m*x**7 - 4*b**3*x**7),x)*a**3*d*m**2 + 4*int(x**m/(a**3*m*x - 4*a**3* 
x + 3*a**2*b*m*x**3 - 12*a**2*b*x**3 + 3*a*b**2*m*x**5 - 12*a*b**2*x**5 + 
b**3*m*x**7 - 4*b**3*x**7),x)*a**3*d*m - 2*int(x**m/(a**3*m*x - 4*a**3*x + 
 3*a**2*b*m*x**3 - 12*a**2*b*x**3 + 3*a*b**2*m*x**5 - 12*a*b**2*x**5 + b** 
3*m*x**7 - 4*b**3*x**7),x)*a**2*b*d*m**2*x**2 + 8*int(x**m/(a**3*m*x - 4*a 
**3*x + 3*a**2*b*m*x**3 - 12*a**2*b*x**3 + 3*a*b**2*m*x**5 - 12*a*b**2*x** 
5 + b**3*m*x**7 - 4*b**3*x**7),x)*a**2*b*d*m*x**2 - int(x**m/(a**3*m*x - 4 
*a**3*x + 3*a**2*b*m*x**3 - 12*a**2*b*x**3 + 3*a*b**2*m*x**5 - 12*a*b**2*x 
**5 + b**3*m*x**7 - 4*b**3*x**7),x)*a*b**2*d*m**2*x**4 + 4*int(x**m/(a**3* 
m*x - 4*a**3*x + 3*a**2*b*m*x**3 - 12*a**2*b*x**3 + 3*a*b**2*m*x**5 - 12*a 
*b**2*x**5 + b**3*m*x**7 - 4*b**3*x**7),x)*a*b**2*d*m*x**4))/(b*(a**2*m - 
4*a**2 + 2*a*b*m*x**2 - 8*a*b*x**2 + b**2*m*x**4 - 4*b**2*x**4))