3.6.88 \(\int \frac {c+d x^{n/2}+e x^n+f x^{3 n/2}}{(a+b x^n)^2} \, dx\) [588]

3.6.88.1 Optimal result

Integrand size = 35, antiderivative size = 162 \[ \int \frac {c+d x^{n/2}+e x^n+f x^{3 n/2}}{\left (a+b x^n\right )^2} \, dx=\frac {x \left (b c-a e+(b d-a f) x^{n/2}\right )}{a b n \left (a+b x^n\right )}-\frac {(b d (2-n)-a f (2+n)) x^{\frac {2+n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1+\frac {2}{n}\right ),\frac {1}{2} \left (3+\frac {2}{n}\right ),-\frac {b x^n}{a}\right )}{a^2 b n (2+n)}+\frac {(a e-b c (1-n)) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^2 b n} \]

output

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

3.6.88.2 Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.91 \[ \int \frac {c+d x^{n/2}+e x^n+f x^{3 n/2}}{\left (a+b x^n\right )^2} \, dx=\frac {x \left (\frac {2 a f x^{n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+\frac {1}{n},\frac {3}{2}+\frac {1}{n},-\frac {b x^n}{a}\right )}{2+n}+a e \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )+\frac {2 (b d-a f) x^{n/2} \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2}+\frac {1}{n},\frac {3}{2}+\frac {1}{n},-\frac {b x^n}{a}\right )}{2+n}+(b c-a e) \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )\right )}{a^2 b} \]

input

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

output

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

3.6.88.3 Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2431, 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.

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

\(\Big \downarrow \) 2431

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

\(\Big \downarrow \) 1748

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

\(\Big \downarrow \) 778

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

\(\Big \downarrow \) 888

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

input

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

output

(x*(b*c - a*e + (b*d - a*f)*x^(n/2)))/(a*b*n*(a + b*x^n)) + ((-2*(b*d*(2 - 
 n) - a*f*(2 + n))*x^((2 + n)/2)*Hypergeometric2F1[1, (1 + 2/n)/2, (3 + 2/ 
n)/2, -((b*x^n)/a)])/(a*(2 + n)) + (2*(a*e - b*c*(1 - n))*x*Hypergeometric 
2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a)/(2*a*b*n)
 

3.6.88.3.1 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 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 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]
 

3.6.88.4 Maple [F]

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

input

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

output

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

3.6.88.5 Fricas [F]

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

input

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

output

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

3.6.88.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.6.88.7 Maxima [F]

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

input

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

output

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

3.6.88.8 Giac [F]

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

input

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

output

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

3.6.88.9 Mupad [F(-1)]

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

input

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

output

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