\(\int \frac {c+d x^{n/2}+e x^n+f x^{3 n/2}}{(a+b x^n)^3} \, dx\) [66]

Optimal result

Integrand size = 35, antiderivative size = 244 \[ \int \frac {c+d x^{n/2}+e x^n+f x^{3 n/2}}{\left (a+b x^n\right )^3} \, dx=\frac {x \left (b c-a e+(b d-a f) x^{n/2}\right )}{2 a b n \left (a+b x^n\right )^2}+\frac {x \left (2 (a e-b c (1-2 n))-(b d (2-3 n)-a f (2+n)) x^{n/2}\right )}{4 a^2 b n^2 \left (a+b x^n\right )}+\frac {(2-n) (b d (2-3 n)-a f (2+n)) x^{\frac {2+n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1+\frac {2}{n}\right ),\frac {1}{2} \left (3+\frac {2}{n}\right ),-\frac {b x^n}{a}\right )}{4 a^3 b n^2 (2+n)}-\frac {(a e-b c (1-2 n)) (1-n) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{2 a^3 b n^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.60 \[ \int \frac {c+d x^{n/2}+e x^n+f x^{3 n/2}}{\left (a+b x^n\right )^3} \, dx=\frac {x \left (\frac {2 a f x^{n/2} \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2}+\frac {1}{n},\frac {3}{2}+\frac {1}{n},-\frac {b x^n}{a}\right )}{2+n}+a e \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )+\frac {2 (b d-a f) x^{n/2} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2}+\frac {1}{n},\frac {3}{2}+\frac {1}{n},-\frac {b x^n}{a}\right )}{2+n}+(b c-a e) \operatorname {Hypergeometric2F1}\left (3,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )\right )}{a^3 b} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2431, 1761, 27, 1748, 778, 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.

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

\(\Big \downarrow \) 2431

\(\displaystyle \frac {\int \frac {2 (a e-b c (1-2 n))-(b d (2-3 n)-a f (n+2)) x^{n/2}}{\left (b x^n+a\right )^2}dx}{4 a b n}+\frac {x \left (x^{n/2} (b d-a f)-a e+b c\right )}{2 a b n \left (a+b x^n\right )^2}\)

\(\Big \downarrow \) 1761

\(\displaystyle \frac {\frac {x \left (2 (a e-b c (1-2 n))-x^{n/2} (b d (2-3 n)-a f (n+2))\right )}{a n \left (a+b x^n\right )}-\frac {\int \frac {4 (a e-b c (1-2 n)) (1-n)-(2-n) (b d (2-3 n)-a f (n+2)) x^{n/2}}{2 \left (b x^n+a\right )}dx}{a n}}{4 a b n}+\frac {x \left (x^{n/2} (b d-a f)-a e+b c\right )}{2 a b n \left (a+b x^n\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {x \left (2 (a e-b c (1-2 n))-x^{n/2} (b d (2-3 n)-a f (n+2))\right )}{a n \left (a+b x^n\right )}-\frac {\int \frac {4 (a e-b c (1-2 n)) (1-n)-(2-n) (b d (2-3 n)-a f (n+2)) x^{n/2}}{b x^n+a}dx}{2 a n}}{4 a b n}+\frac {x \left (x^{n/2} (b d-a f)-a e+b c\right )}{2 a b n \left (a+b x^n\right )^2}\)

\(\Big \downarrow \) 1748

\(\displaystyle \frac {\frac {x \left (2 (a e-b c (1-2 n))-x^{n/2} (b d (2-3 n)-a f (n+2))\right )}{a n \left (a+b x^n\right )}-\frac {4 (1-n) (a e-b c (1-2 n)) \int \frac {1}{b x^n+a}dx-(2-n) (b d (2-3 n)-a f (n+2)) \int \frac {x^{n/2}}{b x^n+a}dx}{2 a n}}{4 a b n}+\frac {x \left (x^{n/2} (b d-a f)-a e+b c\right )}{2 a b n \left (a+b x^n\right )^2}\)

\(\Big \downarrow \) 778

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

(x*(b*c - a*e + (b*d - a*f)*x^(n/2)))/(2*a*b*n*(a + b*x^n)^2) + ((x*(2*(a* 
e - b*c*(1 - 2*n)) - (b*d*(2 - 3*n) - a*f*(2 + n))*x^(n/2)))/(a*n*(a + b*x 
^n)) - ((-2*(2 - n)*(b*d*(2 - 3*n) - a*f*(2 + n))*x^((2 + n)/2)*Hypergeome 
tric2F1[1, (1 + 2/n)/2, (3 + 2/n)/2, -((b*x^n)/a)])/(a*(2 + n)) + (4*(a*e 
- b*c*(1 - 2*n))*(1 - n)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x 
^n)/a)])/a)/(2*a*n))/(4*a*b*n)
 

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 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 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])
 

rule 1748

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Simp[d 
   Int[1/(a + c*x^(2*n)), x], x] + Simp[e   Int[x^n/(a + c*x^(2*n)), x], x] 
 /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && ( 
PosQ[a*c] ||  !IntegerQ[n])
 

rule 1761

Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> S 
imp[(-x)*(d + e*x^n)*((a + c*x^(2*n))^(p + 1)/(2*a*n*(p + 1))), x] + Simp[1 
/(2*a*n*(p + 1))   Int[(d*(2*n*p + 2*n + 1) + e*(2*n*p + 3*n + 1)*x^n)*(a + 
 c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && 
 ILtQ[p, -1]
 

rule 2431

Int[(P3_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{A = Coeff[P3, x 
^(n/2), 0], B = Coeff[P3, x^(n/2), 1], C = Coeff[P3, x^(n/2), 2], D = Coeff 
[P3, x^(n/2), 3]}, Simp[-(x*(b*A - a*C + (b*B - a*D)*x^(n/2))*(a + b*x^n)^( 
p + 1))/(a*b*n*(p + 1)), x] - Simp[1/(2*a*b*n*(p + 1))   Int[(a + b*x^n)^(p 
 + 1)*Simp[2*a*C - 2*b*A*(n*(p + 1) + 1) + (a*D*(n + 2) - b*B*(n*(2*p + 3) 
+ 2))*x^(n/2), x], x], x]] /; FreeQ[{a, b, n}, x] && PolyQ[P3, x^(n/2), 3] 
&& ILtQ[p, -1]
 

Maple [F]

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

Input:

int((c+d*x^(1/2*n)+e*x^n+f*x^(3/2*n))/(a+b*x^n)^3,x)
 

Output:

int((c+d*x^(1/2*n)+e*x^n+f*x^(3/2*n))/(a+b*x^n)^3,x)
 

Fricas [F]

\[ \int \frac {c+d x^{n/2}+e x^n+f x^{3 n/2}}{\left (a+b x^n\right )^3} \, dx=\int { \frac {f x^{\frac {3}{2} \, n} + d x^{\frac {1}{2} \, n} + e x^{n} + c}{{\left (b x^{n} + a\right )}^{3}} \,d x } \] Input:

integrate((c+d*x^(1/2*n)+e*x^n+f*x^(3/2*n))/(a+b*x^n)^3,x, algorithm="fric 
as")
 

Output:

integral((f*x^(3/2*n) + d*x^(1/2*n) + e*x^n + c)/(b^3*x^(3*n) + 3*a*b^2*x^ 
(2*n) + 3*a^2*b*x^n + a^3), x)
 

Sympy [F(-1)]

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

integrate((c+d*x**(1/2*n)+e*x**n+f*x**(3/2*n))/(a+b*x**n)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {c+d x^{n/2}+e x^n+f x^{3 n/2}}{\left (a+b x^n\right )^3} \, dx=\int { \frac {f x^{\frac {3}{2} \, n} + d x^{\frac {1}{2} \, n} + e x^{n} + c}{{\left (b x^{n} + a\right )}^{3}} \,d x } \] Input:

integrate((c+d*x^(1/2*n)+e*x^n+f*x^(3/2*n))/(a+b*x^n)^3,x, algorithm="maxi 
ma")
 

Output:

1/4*((b^2*d*(3*n - 2) + a*b*f*(n + 2))*x*x^(3/2*n) + (a*b*d*(5*n - 2) - a^ 
2*f*(n - 2))*x*x^(1/2*n) + 2*(b^2*c*(2*n - 1) + a*b*e)*x*x^n + 2*(a*b*c*(3 
*n - 1) - a^2*e*(n - 1))*x)/(a^2*b^3*n^2*x^(2*n) + 2*a^3*b^2*n^2*x^n + a^4 
*b*n^2) + integrate(1/8*(4*(2*n^2 - 3*n + 1)*b*c + 4*a*e*(n - 1) + ((3*n^2 
 - 8*n + 4)*b*d + (n^2 - 4)*a*f)*x^(1/2*n))/(a^2*b^2*n^2*x^n + a^3*b*n^2), 
 x)
 

Giac [F]

\[ \int \frac {c+d x^{n/2}+e x^n+f x^{3 n/2}}{\left (a+b x^n\right )^3} \, dx=\int { \frac {f x^{\frac {3}{2} \, n} + d x^{\frac {1}{2} \, n} + e x^{n} + c}{{\left (b x^{n} + a\right )}^{3}} \,d x } \] Input:

integrate((c+d*x^(1/2*n)+e*x^n+f*x^(3/2*n))/(a+b*x^n)^3,x, algorithm="giac 
")
 

Output:

integrate((f*x^(3/2*n) + d*x^(1/2*n) + e*x^n + c)/(b*x^n + a)^3, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {c+d x^{n/2}+e x^n+f x^{3 n/2}}{\left (a+b x^n\right )^3} \, dx=\text {too large to display} \] Input:

int((c+d*x^(1/2*n)+e*x^n+f*x^(3/2*n))/(a+b*x^n)^3,x)
                                                                                    
                                                                                    
 

Output:

(4*x**(2*n)*int(x**((3*n)/2)/(2*x**(3*n)*b**3*n - x**(3*n)*b**3 + 6*x**(2* 
n)*a*b**2*n - 3*x**(2*n)*a*b**2 + 6*x**n*a**2*b*n - 3*x**n*a**2*b + 2*a**3 
*n - a**3),x)*b**3*f*n**2 - 4*x**(2*n)*int(x**((3*n)/2)/(2*x**(3*n)*b**3*n 
 - x**(3*n)*b**3 + 6*x**(2*n)*a*b**2*n - 3*x**(2*n)*a*b**2 + 6*x**n*a**2*b 
*n - 3*x**n*a**2*b + 2*a**3*n - a**3),x)*b**3*f*n + x**(2*n)*int(x**((3*n) 
/2)/(2*x**(3*n)*b**3*n - x**(3*n)*b**3 + 6*x**(2*n)*a*b**2*n - 3*x**(2*n)* 
a*b**2 + 6*x**n*a**2*b*n - 3*x**n*a**2*b + 2*a**3*n - a**3),x)*b**3*f + 4* 
x**(2*n)*int(x**(n/2)/(2*x**(3*n)*b**3*n - x**(3*n)*b**3 + 6*x**(2*n)*a*b* 
*2*n - 3*x**(2*n)*a*b**2 + 6*x**n*a**2*b*n - 3*x**n*a**2*b + 2*a**3*n - a* 
*3),x)*b**3*d*n**2 - 4*x**(2*n)*int(x**(n/2)/(2*x**(3*n)*b**3*n - x**(3*n) 
*b**3 + 6*x**(2*n)*a*b**2*n - 3*x**(2*n)*a*b**2 + 6*x**n*a**2*b*n - 3*x**n 
*a**2*b + 2*a**3*n - a**3),x)*b**3*d*n + x**(2*n)*int(x**(n/2)/(2*x**(3*n) 
*b**3*n - x**(3*n)*b**3 + 6*x**(2*n)*a*b**2*n - 3*x**(2*n)*a*b**2 + 6*x**n 
*a**2*b*n - 3*x**n*a**2*b + 2*a**3*n - a**3),x)*b**3*d + 2*x**(2*n)*int(1/ 
(2*x**(3*n)*b**3*n - x**(3*n)*b**3 + 6*x**(2*n)*a*b**2*n - 3*x**(2*n)*a*b* 
*2 + 6*x**n*a**2*b*n - 3*x**n*a**2*b + 2*a**3*n - a**3),x)*a*b**2*e*n - x* 
*(2*n)*int(1/(2*x**(3*n)*b**3*n - x**(3*n)*b**3 + 6*x**(2*n)*a*b**2*n - 3* 
x**(2*n)*a*b**2 + 6*x**n*a**2*b*n - 3*x**n*a**2*b + 2*a**3*n - a**3),x)*a* 
b**2*e + 4*x**(2*n)*int(1/(2*x**(3*n)*b**3*n - x**(3*n)*b**3 + 6*x**(2*n)* 
a*b**2*n - 3*x**(2*n)*a*b**2 + 6*x**n*a**2*b*n - 3*x**n*a**2*b + 2*a**3...