\(\int \frac {c+d x^{n/2}+e x^n}{(a+b x^n)^2} \, dx\) [62]

Optimal result

Integrand size = 26, antiderivative size = 143 \[ \int \frac {c+d x^{n/2}+e x^n}{\left (a+b x^n\right )^2} \, dx=\frac {x \left (b c-a e+b d x^{n/2}\right )}{a b n \left (a+b x^n\right )}-\frac {d (2-n) x^{\frac {2+n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1+\frac {2}{n}\right ),\frac {1}{2} \left (3+\frac {2}{n}\right ),-\frac {b x^n}{a}\right )}{a^2 n (2+n)}+\frac {(a e-b c (1-n)) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^2 b n} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.74 \[ \int \frac {c+d x^{n/2}+e x^n}{\left (a+b x^n\right )^2} \, dx=\frac {x \left (a e (2+n) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )+2 b d x^{n/2} \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2}+\frac {1}{n},\frac {3}{2}+\frac {1}{n},-\frac {b x^n}{a}\right )+(b c-a e) (2+n) \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )\right )}{a^2 b (2+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.20, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {7293, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

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

Input:

int((c+d*x^(1/2*n)+e*x^n)/(a+b*x^n)^2,x)
 

Output:

int((c+d*x^(1/2*n)+e*x^n)/(a+b*x^n)^2,x)
 

Fricas [F]

\[ \int \frac {c+d x^{n/2}+e x^n}{\left (a+b x^n\right )^2} \, dx=\int { \frac {d x^{\frac {1}{2} \, n} + e x^{n} + c}{{\left (b x^{n} + a\right )}^{2}} \,d x } \] Input:

integrate((c+d*x^(1/2*n)+e*x^n)/(a+b*x^n)^2,x, algorithm="fricas")
 

Output:

integral((d*x^(1/2*n) + e*x^n + c)/(b^2*x^(2*n) + 2*a*b*x^n + a^2), x)
 

Sympy [F(-1)]

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

integrate((c+d*x**(1/2*n)+e*x**n)/(a+b*x**n)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {c+d x^{n/2}+e x^n}{\left (a+b x^n\right )^2} \, dx=\int { \frac {d x^{\frac {1}{2} \, n} + e x^{n} + c}{{\left (b x^{n} + a\right )}^{2}} \,d x } \] Input:

integrate((c+d*x^(1/2*n)+e*x^n)/(a+b*x^n)^2,x, algorithm="maxima")
 

Output:

(b*d*x*x^(1/2*n) + (b*c - a*e)*x)/(a*b^2*n*x^n + a^2*b*n) + integrate(1/2* 
(b*d*(n - 2)*x^(1/2*n) + 2*b*c*(n - 1) + 2*a*e)/(a*b^2*n*x^n + a^2*b*n), x 
)
 

Giac [F]

\[ \int \frac {c+d x^{n/2}+e x^n}{\left (a+b x^n\right )^2} \, dx=\int { \frac {d x^{\frac {1}{2} \, n} + e x^{n} + c}{{\left (b x^{n} + a\right )}^{2}} \,d x } \] Input:

integrate((c+d*x^(1/2*n)+e*x^n)/(a+b*x^n)^2,x, algorithm="giac")
 

Output:

integrate((d*x^(1/2*n) + e*x^n + c)/(b*x^n + a)^2, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {c+d x^{n/2}+e x^n}{\left (a+b x^n\right )^2} \, dx =\text {Too large to display} \] Input:

int((c+d*x^(1/2*n)+e*x^n)/(a+b*x^n)^2,x)
 

Output:

(x**n*int(x**(n/2)/(x**(2*n)*b**2*n - x**(2*n)*b**2 + 2*x**n*a*b*n - 2*x** 
n*a*b + a**2*n - a**2),x)*b**2*d*n**2 - 2*x**n*int(x**(n/2)/(x**(2*n)*b**2 
*n - x**(2*n)*b**2 + 2*x**n*a*b*n - 2*x**n*a*b + a**2*n - a**2),x)*b**2*d* 
n + x**n*int(x**(n/2)/(x**(2*n)*b**2*n - x**(2*n)*b**2 + 2*x**n*a*b*n - 2* 
x**n*a*b + a**2*n - a**2),x)*b**2*d + x**n*int(1/(x**(2*n)*b**2*n - x**(2* 
n)*b**2 + 2*x**n*a*b*n - 2*x**n*a*b + a**2*n - a**2),x)*a*b*e*n - x**n*int 
(1/(x**(2*n)*b**2*n - x**(2*n)*b**2 + 2*x**n*a*b*n - 2*x**n*a*b + a**2*n - 
 a**2),x)*a*b*e + x**n*int(1/(x**(2*n)*b**2*n - x**(2*n)*b**2 + 2*x**n*a*b 
*n - 2*x**n*a*b + a**2*n - a**2),x)*b**2*c*n**2 - 2*x**n*int(1/(x**(2*n)*b 
**2*n - x**(2*n)*b**2 + 2*x**n*a*b*n - 2*x**n*a*b + a**2*n - a**2),x)*b**2 
*c*n + x**n*int(1/(x**(2*n)*b**2*n - x**(2*n)*b**2 + 2*x**n*a*b*n - 2*x**n 
*a*b + a**2*n - a**2),x)*b**2*c + int(x**(n/2)/(x**(2*n)*b**2*n - x**(2*n) 
*b**2 + 2*x**n*a*b*n - 2*x**n*a*b + a**2*n - a**2),x)*a*b*d*n**2 - 2*int(x 
**(n/2)/(x**(2*n)*b**2*n - x**(2*n)*b**2 + 2*x**n*a*b*n - 2*x**n*a*b + a** 
2*n - a**2),x)*a*b*d*n + int(x**(n/2)/(x**(2*n)*b**2*n - x**(2*n)*b**2 + 2 
*x**n*a*b*n - 2*x**n*a*b + a**2*n - a**2),x)*a*b*d + int(1/(x**(2*n)*b**2* 
n - x**(2*n)*b**2 + 2*x**n*a*b*n - 2*x**n*a*b + a**2*n - a**2),x)*a**2*e*n 
 - int(1/(x**(2*n)*b**2*n - x**(2*n)*b**2 + 2*x**n*a*b*n - 2*x**n*a*b + a* 
*2*n - a**2),x)*a**2*e + int(1/(x**(2*n)*b**2*n - x**(2*n)*b**2 + 2*x**n*a 
*b*n - 2*x**n*a*b + a**2*n - a**2),x)*a*b*c*n**2 - 2*int(1/(x**(2*n)*b*...