\(\int \frac {x^m}{(a+b x^3)^2} \, dx\) [443]

Optimal result

Integrand size = 13, antiderivative size = 39 \[ \int \frac {x^m}{\left (a+b x^3\right )^2} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{3},\frac {4+m}{3},-\frac {b x^3}{a}\right )}{a^2 (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.05 \[ \int \frac {x^m}{\left (a+b x^3\right )^2} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{3},1+\frac {1+m}{3},-\frac {b x^3}{a}\right )}{a^2 (1+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {888}

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.

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 888

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(x^m/(b^2*x^6 + 2*a*b*x^3 + a^2), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 27.65 (sec) , antiderivative size = 520, normalized size of antiderivative = 13.33 \[ \int \frac {x^m}{\left (a+b x^3\right )^2} \, dx=- \frac {a m^{2} x^{m + 1} \Phi \left (\frac {b x^{3} e^{i \pi }}{a}, 1, \frac {m}{3} + \frac {1}{3}\right ) \Gamma \left (\frac {m}{3} + \frac {1}{3}\right )}{27 a^{3} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right ) + 27 a^{2} b x^{3} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} + \frac {a m x^{m + 1} \Phi \left (\frac {b x^{3} e^{i \pi }}{a}, 1, \frac {m}{3} + \frac {1}{3}\right ) \Gamma \left (\frac {m}{3} + \frac {1}{3}\right )}{27 a^{3} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right ) + 27 a^{2} b x^{3} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} + \frac {3 a m x^{m + 1} \Gamma \left (\frac {m}{3} + \frac {1}{3}\right )}{27 a^{3} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right ) + 27 a^{2} b x^{3} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} + \frac {2 a x^{m + 1} \Phi \left (\frac {b x^{3} e^{i \pi }}{a}, 1, \frac {m}{3} + \frac {1}{3}\right ) \Gamma \left (\frac {m}{3} + \frac {1}{3}\right )}{27 a^{3} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right ) + 27 a^{2} b x^{3} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} + \frac {3 a x^{m + 1} \Gamma \left (\frac {m}{3} + \frac {1}{3}\right )}{27 a^{3} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right ) + 27 a^{2} b x^{3} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} - \frac {b m^{2} x^{3} x^{m + 1} \Phi \left (\frac {b x^{3} e^{i \pi }}{a}, 1, \frac {m}{3} + \frac {1}{3}\right ) \Gamma \left (\frac {m}{3} + \frac {1}{3}\right )}{27 a^{3} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right ) + 27 a^{2} b x^{3} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} + \frac {b m x^{3} x^{m + 1} \Phi \left (\frac {b x^{3} e^{i \pi }}{a}, 1, \frac {m}{3} + \frac {1}{3}\right ) \Gamma \left (\frac {m}{3} + \frac {1}{3}\right )}{27 a^{3} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right ) + 27 a^{2} b x^{3} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} + \frac {2 b x^{3} x^{m + 1} \Phi \left (\frac {b x^{3} e^{i \pi }}{a}, 1, \frac {m}{3} + \frac {1}{3}\right ) \Gamma \left (\frac {m}{3} + \frac {1}{3}\right )}{27 a^{3} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right ) + 27 a^{2} b x^{3} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} \] Input:

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

Output:

-a*m**2*x**(m + 1)*lerchphi(b*x**3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma( 
m/3 + 1/3)/(27*a**3*gamma(m/3 + 4/3) + 27*a**2*b*x**3*gamma(m/3 + 4/3)) + 
a*m*x**(m + 1)*lerchphi(b*x**3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 
+ 1/3)/(27*a**3*gamma(m/3 + 4/3) + 27*a**2*b*x**3*gamma(m/3 + 4/3)) + 3*a* 
m*x**(m + 1)*gamma(m/3 + 1/3)/(27*a**3*gamma(m/3 + 4/3) + 27*a**2*b*x**3*g 
amma(m/3 + 4/3)) + 2*a*x**(m + 1)*lerchphi(b*x**3*exp_polar(I*pi)/a, 1, m/ 
3 + 1/3)*gamma(m/3 + 1/3)/(27*a**3*gamma(m/3 + 4/3) + 27*a**2*b*x**3*gamma 
(m/3 + 4/3)) + 3*a*x**(m + 1)*gamma(m/3 + 1/3)/(27*a**3*gamma(m/3 + 4/3) + 
 27*a**2*b*x**3*gamma(m/3 + 4/3)) - b*m**2*x**3*x**(m + 1)*lerchphi(b*x**3 
*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(27*a**3*gamma(m/3 + 4/ 
3) + 27*a**2*b*x**3*gamma(m/3 + 4/3)) + b*m*x**3*x**(m + 1)*lerchphi(b*x** 
3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(27*a**3*gamma(m/3 + 4 
/3) + 27*a**2*b*x**3*gamma(m/3 + 4/3)) + 2*b*x**3*x**(m + 1)*lerchphi(b*x* 
*3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(27*a**3*gamma(m/3 + 
4/3) + 27*a**2*b*x**3*gamma(m/3 + 4/3))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(x**m/(a**2 + 2*a*b*x**3 + b**2*x**6),x)