\(\int \frac {x^8 (A+B x^4)}{(a+b x^4)^2} \, dx\) [17]

Optimal result

Integrand size = 20, antiderivative size = 211 \[ \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right )^2} \, dx=\frac {(A b-2 a B) x}{b^3}+\frac {B x^5}{5 b^2}+\frac {a (A b-a B) x}{4 b^3 \left (a+b x^4\right )}+\frac {\sqrt [4]{a} (5 A b-9 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} b^{13/4}}-\frac {\sqrt [4]{a} (5 A b-9 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} b^{13/4}}-\frac {\sqrt [4]{a} (5 A b-9 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a}+\sqrt {b} x^2}\right )}{8 \sqrt {2} b^{13/4}} \] Output:

(A*b-2*B*a)*x/b^3+1/5*B*x^5/b^2+1/4*a*(A*b-B*a)*x/b^3/(b*x^4+a)-1/16*a^(1/ 
4)*(5*A*b-9*B*a)*arctan(-1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/b^(13/4)-1/1 
6*a^(1/4)*(5*A*b-9*B*a)*arctan(1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/b^(13/ 
4)-1/16*a^(1/4)*(5*A*b-9*B*a)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*x/(a^(1/2)+b 
^(1/2)*x^2))*2^(1/2)/b^(13/4)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.21 \[ \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right )^2} \, dx=\frac {160 \sqrt [4]{b} (A b-2 a B) x+32 b^{5/4} B x^5+\frac {40 a \sqrt [4]{b} (A b-a B) x}{a+b x^4}-10 \sqrt {2} \sqrt [4]{a} (-5 A b+9 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+10 \sqrt {2} \sqrt [4]{a} (-5 A b+9 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )-5 \sqrt {2} \sqrt [4]{a} (-5 A b+9 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+5 \sqrt {2} \sqrt [4]{a} (-5 A b+9 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{160 b^{13/4}} \] Input:

Integrate[(x^8*(A + B*x^4))/(a + b*x^4)^2,x]
 

Output:

(160*b^(1/4)*(A*b - 2*a*B)*x + 32*b^(5/4)*B*x^5 + (40*a*b^(1/4)*(A*b - a*B 
)*x)/(a + b*x^4) - 10*Sqrt[2]*a^(1/4)*(-5*A*b + 9*a*B)*ArcTan[1 - (Sqrt[2] 
*b^(1/4)*x)/a^(1/4)] + 10*Sqrt[2]*a^(1/4)*(-5*A*b + 9*a*B)*ArcTan[1 + (Sqr 
t[2]*b^(1/4)*x)/a^(1/4)] - 5*Sqrt[2]*a^(1/4)*(-5*A*b + 9*a*B)*Log[Sqrt[a] 
- Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 5*Sqrt[2]*a^(1/4)*(-5*A*b + 9 
*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(160*b^(13/4 
))
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.19, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {957, 831, 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.

\(\displaystyle \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right )^2} \, dx\)

\(\Big \downarrow \) 957

\(\displaystyle \frac {x^9 (A b-a B)}{4 a b \left (a+b x^4\right )}-\frac {(5 A b-9 a B) \int \frac {x^8}{b x^4+a}dx}{4 a b}\)

\(\Big \downarrow \) 831

\(\displaystyle \frac {x^9 (A b-a B)}{4 a b \left (a+b x^4\right )}-\frac {(5 A b-9 a B) \int \left (\frac {x^4}{b}+\frac {a^2}{b^2 \left (b x^4+a\right )}-\frac {a}{b^2}\right )dx}{4 a b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^9 (A b-a B)}{4 a b \left (a+b x^4\right )}-\frac {(5 A b-9 a B) \left (-\frac {a^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{9/4}}+\frac {a^{5/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} b^{9/4}}-\frac {a^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} b^{9/4}}+\frac {a^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} b^{9/4}}-\frac {a x}{b^2}+\frac {x^5}{5 b}\right )}{4 a b}\)

Input:

Int[(x^8*(A + B*x^4))/(a + b*x^4)^2,x]
 

Output:

((A*b - a*B)*x^9)/(4*a*b*(a + b*x^4)) - ((5*A*b - 9*a*B)*(-((a*x)/b^2) + x 
^5/(5*b) - (a^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*b^ 
(9/4)) + (a^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*b^(9 
/4)) - (a^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4 
*Sqrt[2]*b^(9/4)) + (a^(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqr 
t[b]*x^2])/(4*Sqrt[2]*b^(9/4))))/(4*a*b)
 

Defintions of rubi rules used

rule 831

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x 
^m, a + b*x^n, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && Gt 
Q[m, 2*n - 1]
 

rule 957

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a 
*b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* 
(p + 1))   Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, 
 m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N 
eQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 
, m, (-n)*(p + 1)]))
 

rule 2009

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.41

method	result	size
risch	\(\frac {B \,x^{5}}{5 b^{2}}+\frac {A x}{b^{2}}-\frac {2 B a x}{b^{3}}+\frac {\left (\frac {1}{4} a b A -\frac {1}{4} a^{2} B \right ) x}{b^{3} \left (b \,x^{4}+a \right )}+\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (-5 A b +9 B a \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{16 b^{4}}\)	\(87\)
default	\(\frac {\frac {1}{5} b B \,x^{5}+A b x -2 B a x}{b^{3}}-\frac {a \left (\frac {\left (-\frac {A b}{4}+\frac {B a}{4}\right ) x}{b \,x^{4}+a}+\frac {\left (5 A b -9 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{b^{3}}\)	\(160\)

Input:

int(x^8*(B*x^4+A)/(b*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/5*B*x^5/b^2+1/b^2*A*x-2/b^3*B*a*x+(1/4*a*b*A-1/4*a^2*B)*x/b^3/(b*x^4+a)+ 
1/16/b^4*a*sum((-5*A*b+9*B*a)/_R^3*ln(x-_R),_R=RootOf(_Z^4*b+a))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 686, normalized size of antiderivative = 3.25 \[ \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(x^8*(B*x^4+A)/(b*x^4+a)^2,x, algorithm="fricas")
 

Output:

1/80*(16*B*b^2*x^9 - 16*(9*B*a*b - 5*A*b^2)*x^5 - 5*(b^4*x^4 + a*b^3)*(-(6 
561*B^4*a^5 - 14580*A*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2*b 
^3 + 625*A^4*a*b^4)/b^13)^(1/4)*log(b^3*(-(6561*B^4*a^5 - 14580*A*B^3*a^4* 
b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^4*a*b^4)/b^13)^(1/4 
) - (9*B*a - 5*A*b)*x) - 5*(I*b^4*x^4 + I*a*b^3)*(-(6561*B^4*a^5 - 14580*A 
*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^4*a*b^4)/b 
^13)^(1/4)*log(I*b^3*(-(6561*B^4*a^5 - 14580*A*B^3*a^4*b + 12150*A^2*B^2*a 
^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^4*a*b^4)/b^13)^(1/4) - (9*B*a - 5*A*b) 
*x) - 5*(-I*b^4*x^4 - I*a*b^3)*(-(6561*B^4*a^5 - 14580*A*B^3*a^4*b + 12150 
*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^4*a*b^4)/b^13)^(1/4)*log(-I* 
b^3*(-(6561*B^4*a^5 - 14580*A*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3 
*B*a^2*b^3 + 625*A^4*a*b^4)/b^13)^(1/4) - (9*B*a - 5*A*b)*x) + 5*(b^4*x^4 
+ a*b^3)*(-(6561*B^4*a^5 - 14580*A*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 450 
0*A^3*B*a^2*b^3 + 625*A^4*a*b^4)/b^13)^(1/4)*log(-b^3*(-(6561*B^4*a^5 - 14 
580*A*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^4*a*b 
^4)/b^13)^(1/4) - (9*B*a - 5*A*b)*x) - 20*(9*B*a^2 - 5*A*a*b)*x)/(b^4*x^4 
+ a*b^3)
 

Sympy [A] (verification not implemented)

Time = 0.79 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.67 \[ \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right )^2} \, dx=\frac {B x^{5}}{5 b^{2}} + x \left (\frac {A}{b^{2}} - \frac {2 B a}{b^{3}}\right ) + \frac {x \left (A a b - B a^{2}\right )}{4 a b^{3} + 4 b^{4} x^{4}} + \operatorname {RootSum} {\left (65536 t^{4} b^{13} + 625 A^{4} a b^{4} - 4500 A^{3} B a^{2} b^{3} + 12150 A^{2} B^{2} a^{3} b^{2} - 14580 A B^{3} a^{4} b + 6561 B^{4} a^{5}, \left ( t \mapsto t \log {\left (\frac {16 t b^{3}}{- 5 A b + 9 B a} + x \right )} \right )\right )} \] Input:

integrate(x**8*(B*x**4+A)/(b*x**4+a)**2,x)
 

Output:

B*x**5/(5*b**2) + x*(A/b**2 - 2*B*a/b**3) + x*(A*a*b - B*a**2)/(4*a*b**3 + 
 4*b**4*x**4) + RootSum(65536*_t**4*b**13 + 625*A**4*a*b**4 - 4500*A**3*B* 
a**2*b**3 + 12150*A**2*B**2*a**3*b**2 - 14580*A*B**3*a**4*b + 6561*B**4*a* 
*5, Lambda(_t, _t*log(16*_t*b**3/(-5*A*b + 9*B*a) + x)))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.25 \[ \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right )^2} \, dx=-\frac {{\left (B a^{2} - A a b\right )} x}{4 \, {\left (b^{4} x^{4} + a b^{3}\right )}} + \frac {{\left (\frac {2 \, \sqrt {2} {\left (9 \, B a - 5 \, A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (9 \, B a - 5 \, A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (9 \, B a - 5 \, A b\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (9 \, B a - 5 \, A b\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} a}{32 \, b^{3}} + \frac {B b x^{5} - 5 \, {\left (2 \, B a - A b\right )} x}{5 \, b^{3}} \] Input:

integrate(x^8*(B*x^4+A)/(b*x^4+a)^2,x, algorithm="maxima")
 

Output:

-1/4*(B*a^2 - A*a*b)*x/(b^4*x^4 + a*b^3) + 1/32*(2*sqrt(2)*(9*B*a - 5*A*b) 
*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*s 
qrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(9*B*a - 5*A*b)*arcta 
n(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b) 
))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(9*B*a - 5*A*b)*log(sqrt(b)*x 
^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(9*B 
*a - 5*A*b)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4 
)*b^(1/4)))*a/b^3 + 1/5*(B*b*x^5 - 5*(2*B*a - A*b)*x)/b^3
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.36 \[ \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right )^2} \, dx=\frac {\sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, b^{4}} + \frac {\sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, b^{4}} + \frac {\sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, b^{4}} - \frac {\sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, b^{4}} - \frac {B a^{2} x - A a b x}{4 \, {\left (b x^{4} + a\right )} b^{3}} + \frac {B b^{8} x^{5} - 10 \, B a b^{7} x + 5 \, A b^{8} x}{5 \, b^{10}} \] Input:

integrate(x^8*(B*x^4+A)/(b*x^4+a)^2,x, algorithm="giac")
 

Output:

1/16*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - 5*(a*b^3)^(1/4)*A*b)*arctan(1/2*sqrt(2 
)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/b^4 + 1/16*sqrt(2)*(9*(a*b^3)^( 
1/4)*B*a - 5*(a*b^3)^(1/4)*A*b)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1 
/4))/(a/b)^(1/4))/b^4 + 1/32*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - 5*(a*b^3)^(1/4 
)*A*b)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/b^4 - 1/32*sqrt(2)*(9* 
(a*b^3)^(1/4)*B*a - 5*(a*b^3)^(1/4)*A*b)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + 
 sqrt(a/b))/b^4 - 1/4*(B*a^2*x - A*a*b*x)/((b*x^4 + a)*b^3) + 1/5*(B*b^8*x 
^5 - 10*B*a*b^7*x + 5*A*b^8*x)/b^10
 

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 810, normalized size of antiderivative = 3.84 \[ \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right )^2} \, dx =\text {Too large to display} \] Input:

int((x^8*(A + B*x^4))/(a + b*x^4)^2,x)
 

Output:

x*(A/b^2 - (2*B*a)/b^3) - (x*((B*a^2)/4 - (A*a*b)/4))/(a*b^3 + b^4*x^4) + 
(B*x^5)/(5*b^2) + ((-a)^(1/4)*atan((((-a)^(1/4)*(5*A*b - 9*B*a)*((x*(81*B^ 
2*a^4 + 25*A^2*a^2*b^2 - 90*A*B*a^3*b))/(4*b^3) - ((-a)^(1/4)*(5*A*b - 9*B 
*a)*(36*B*a^3 - 20*A*a^2*b))/(16*b^(13/4)))*1i)/(16*b^(13/4)) + ((-a)^(1/4 
)*(5*A*b - 9*B*a)*((x*(81*B^2*a^4 + 25*A^2*a^2*b^2 - 90*A*B*a^3*b))/(4*b^3 
) + ((-a)^(1/4)*(5*A*b - 9*B*a)*(36*B*a^3 - 20*A*a^2*b))/(16*b^(13/4)))*1i 
)/(16*b^(13/4)))/(((-a)^(1/4)*(5*A*b - 9*B*a)*((x*(81*B^2*a^4 + 25*A^2*a^2 
*b^2 - 90*A*B*a^3*b))/(4*b^3) - ((-a)^(1/4)*(5*A*b - 9*B*a)*(36*B*a^3 - 20 
*A*a^2*b))/(16*b^(13/4))))/(16*b^(13/4)) - ((-a)^(1/4)*(5*A*b - 9*B*a)*((x 
*(81*B^2*a^4 + 25*A^2*a^2*b^2 - 90*A*B*a^3*b))/(4*b^3) + ((-a)^(1/4)*(5*A* 
b - 9*B*a)*(36*B*a^3 - 20*A*a^2*b))/(16*b^(13/4))))/(16*b^(13/4))))*(5*A*b 
 - 9*B*a)*1i)/(8*b^(13/4)) + ((-a)^(1/4)*atan((((-a)^(1/4)*(5*A*b - 9*B*a) 
*((x*(81*B^2*a^4 + 25*A^2*a^2*b^2 - 90*A*B*a^3*b))/(4*b^3) - ((-a)^(1/4)*( 
5*A*b - 9*B*a)*(36*B*a^3 - 20*A*a^2*b)*1i)/(16*b^(13/4))))/(16*b^(13/4)) + 
 ((-a)^(1/4)*(5*A*b - 9*B*a)*((x*(81*B^2*a^4 + 25*A^2*a^2*b^2 - 90*A*B*a^3 
*b))/(4*b^3) + ((-a)^(1/4)*(5*A*b - 9*B*a)*(36*B*a^3 - 20*A*a^2*b)*1i)/(16 
*b^(13/4))))/(16*b^(13/4)))/(((-a)^(1/4)*(5*A*b - 9*B*a)*((x*(81*B^2*a^4 + 
 25*A^2*a^2*b^2 - 90*A*B*a^3*b))/(4*b^3) - ((-a)^(1/4)*(5*A*b - 9*B*a)*(36 
*B*a^3 - 20*A*a^2*b)*1i)/(16*b^(13/4)))*1i)/(16*b^(13/4)) - ((-a)^(1/4)*(5 
*A*b - 9*B*a)*((x*(81*B^2*a^4 + 25*A^2*a^2*b^2 - 90*A*B*a^3*b))/(4*b^3)...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.73 \[ \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right )^2} \, dx=\frac {-10 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )+10 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-5 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {b}\, x^{2}\right )+5 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {b}\, x^{2}\right )-40 a b x +8 b^{2} x^{5}}{40 b^{3}} \] Input:

int(x^8*(B*x^4+A)/(b*x^4+a)^2,x)
 

Output:

( - 10*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt( 
b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a + 10*b**(3/4)*a**(1/4)*sqrt(2)*atan(( 
b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a - 
5*b**(3/4)*a**(1/4)*sqrt(2)*log( - b**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + 
 sqrt(b)*x**2)*a + 5*b**(3/4)*a**(1/4)*sqrt(2)*log(b**(1/4)*a**(1/4)*sqrt( 
2)*x + sqrt(a) + sqrt(b)*x**2)*a - 40*a*b*x + 8*b**2*x**5)/(40*b**3)