\(\int \frac {a+b x^n}{(c+d x^n)^4} \, dx\) [74]

Optimal result

Integrand size = 17, antiderivative size = 61 \[ \int \frac {a+b x^n}{\left (c+d x^n\right )^4} \, dx=\frac {b x}{d (1-3 n) \left (c+d x^n\right )^3}+\frac {\left (a-\frac {b c}{d-3 d n}\right ) x \operatorname {Hypergeometric2F1}\left (4,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c^4} \] Output:

b*x/d/(1-3*n)/(c+d*x^n)^3+(a-b*c/(-3*d*n+d))*x*hypergeom([4, 1/n],[1+1/n], 
-d*x^n/c)/c^4
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.95 \[ \int \frac {a+b x^n}{\left (c+d x^n\right )^4} \, dx=\frac {x \left (\frac {b}{\left (c+d x^n\right )^3}-\frac {(b c+a d (-1+3 n)) \operatorname {Hypergeometric2F1}\left (4,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c^4}\right )}{d-3 d n} \] Input:

Integrate[(a + b*x^n)/(c + d*x^n)^4,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.28, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {910, 778}

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.

\(\displaystyle \int \frac {a+b x^n}{\left (c+d x^n\right )^4} \, dx\)

\(\Big \downarrow \) 910

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

\(\Big \downarrow \) 778

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

Input:

Int[(a + b*x^n)/(c + d*x^n)^4,x]
 

Output:

-1/3*((b*c - a*d)*x)/(c*d*n*(c + d*x^n)^3) + ((b*c - a*d*(1 - 3*n))*x*Hype 
rgeometric2F1[3, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/(3*c^4*d*n)
 

Defintions of rubi rules used

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 910

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si 
mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - 
 b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1))   Int[(a + b*x^n)^(p + 1), x], x] /; 
FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ 
n + p, 0])
 

Maple [F]

\[\int \frac {a +b \,x^{n}}{\left (c +d \,x^{n}\right )^{4}}d x\]

Input:

int((a+b*x^n)/(c+d*x^n)^4,x)
 

Output:

int((a+b*x^n)/(c+d*x^n)^4,x)
 

Fricas [F]

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

integrate((a+b*x^n)/(c+d*x^n)^4,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b x^n}{\left (c+d x^n\right )^4} \, dx=\text {Timed out} \] Input:

integrate((a+b*x**n)/(c+d*x**n)**4,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((a+b*x^n)/(c+d*x^n)^4,x, algorithm="maxima")
 

Output:

((2*n^2 - 3*n + 1)*b*c + (6*n^3 - 11*n^2 + 6*n - 1)*a*d)*integrate(1/6/(c^ 
3*d^2*n^3*x^n + c^4*d*n^3), x) + 1/6*(((6*n^2 - 5*n + 1)*a*d^3 + b*c*d^2*( 
2*n - 1))*x*x^(2*n) + ((15*n^2 - 11*n + 2)*a*c*d^2 + b*c^2*d*(5*n - 2))*x* 
x^n - ((2*n^2 - 3*n + 1)*b*c^3 - (11*n^2 - 6*n + 1)*a*c^2*d)*x)/(c^3*d^4*n 
^3*x^(3*n) + 3*c^4*d^3*n^3*x^(2*n) + 3*c^5*d^2*n^3*x^n + c^6*d*n^3)
 

Giac [F]

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

integrate((a+b*x^n)/(c+d*x^n)^4,x, algorithm="giac")
 

Output:

integrate((b*x^n + a)/(d*x^n + c)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b x^n}{\left (c+d x^n\right )^4} \, dx=\int \frac {a+b\,x^n}{{\left (c+d\,x^n\right )}^4} \,d x \] Input:

int((a + b*x^n)/(c + d*x^n)^4,x)
 

Output:

int((a + b*x^n)/(c + d*x^n)^4, x)
 

Reduce [F]

\[ \int \frac {a+b x^n}{\left (c+d x^n\right )^4} \, dx=\text {too large to display} \] Input:

int((a+b*x^n)/(c+d*x^n)^4,x)
 

Output:

(6*x**(3*n)*int(x**(2*n)/(x**(4*n)*d**4*n + x**(4*n)*d**4 + 4*x**(3*n)*c*d 
**3*n + 4*x**(3*n)*c*d**3 + 6*x**(2*n)*c**2*d**2*n + 6*x**(2*n)*c**2*d**2 
+ 4*x**n*c**3*d*n + 4*x**n*c**3*d + c**4*n + c**4),x)*a*d**5*n**3 + x**(3* 
n)*int(x**(2*n)/(x**(4*n)*d**4*n + x**(4*n)*d**4 + 4*x**(3*n)*c*d**3*n + 4 
*x**(3*n)*c*d**3 + 6*x**(2*n)*c**2*d**2*n + 6*x**(2*n)*c**2*d**2 + 4*x**n* 
c**3*d*n + 4*x**n*c**3*d + c**4*n + c**4),x)*a*d**5*n**2 - 4*x**(3*n)*int( 
x**(2*n)/(x**(4*n)*d**4*n + x**(4*n)*d**4 + 4*x**(3*n)*c*d**3*n + 4*x**(3* 
n)*c*d**3 + 6*x**(2*n)*c**2*d**2*n + 6*x**(2*n)*c**2*d**2 + 4*x**n*c**3*d* 
n + 4*x**n*c**3*d + c**4*n + c**4),x)*a*d**5*n + x**(3*n)*int(x**(2*n)/(x* 
*(4*n)*d**4*n + x**(4*n)*d**4 + 4*x**(3*n)*c*d**3*n + 4*x**(3*n)*c*d**3 + 
6*x**(2*n)*c**2*d**2*n + 6*x**(2*n)*c**2*d**2 + 4*x**n*c**3*d*n + 4*x**n*c 
**3*d + c**4*n + c**4),x)*a*d**5 + 2*x**(3*n)*int(x**(2*n)/(x**(4*n)*d**4* 
n + x**(4*n)*d**4 + 4*x**(3*n)*c*d**3*n + 4*x**(3*n)*c*d**3 + 6*x**(2*n)*c 
**2*d**2*n + 6*x**(2*n)*c**2*d**2 + 4*x**n*c**3*d*n + 4*x**n*c**3*d + c**4 
*n + c**4),x)*b*c*d**4*n**2 + x**(3*n)*int(x**(2*n)/(x**(4*n)*d**4*n + x** 
(4*n)*d**4 + 4*x**(3*n)*c*d**3*n + 4*x**(3*n)*c*d**3 + 6*x**(2*n)*c**2*d** 
2*n + 6*x**(2*n)*c**2*d**2 + 4*x**n*c**3*d*n + 4*x**n*c**3*d + c**4*n + c* 
*4),x)*b*c*d**4*n - x**(3*n)*int(x**(2*n)/(x**(4*n)*d**4*n + x**(4*n)*d**4 
 + 4*x**(3*n)*c*d**3*n + 4*x**(3*n)*c*d**3 + 6*x**(2*n)*c**2*d**2*n + 6*x* 
*(2*n)*c**2*d**2 + 4*x**n*c**3*d*n + 4*x**n*c**3*d + c**4*n + c**4),x)*...