\(\int \frac {x^9 (A+B x^4)}{(a+b x^4) (c+d x^4)} \, dx\) [6]

Optimal result

Integrand size = 29, antiderivative size = 136 \[ \int \frac {x^9 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {(b B c-A b d+a B d) x^2}{2 b^2 d^2}+\frac {B x^6}{6 b d}+\frac {a^{3/2} (A b-a B) \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 b^{5/2} (b c-a d)}+\frac {c^{3/2} (B c-A d) \arctan \left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{2 d^{5/2} (b c-a d)} \] Output:

-1/2*(-A*b*d+B*a*d+B*b*c)*x^2/b^2/d^2+1/6*B*x^6/b/d+1/2*a^(3/2)*(A*b-B*a)* 
arctan(b^(1/2)*x^2/a^(1/2))/b^(5/2)/(-a*d+b*c)+1/2*c^(3/2)*(-A*d+B*c)*arct 
an(d^(1/2)*x^2/c^(1/2))/d^(5/2)/(-a*d+b*c)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.96 \[ \int \frac {x^9 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {1}{6} \left (-\frac {3 (b B c-A b d+a B d) x^2}{b^2 d^2}+\frac {B x^6}{b d}+\frac {3 a^{3/2} (-A b+a B) \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{b^{5/2} (-b c+a d)}+\frac {3 c^{3/2} (B c-A d) \arctan \left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{d^{5/2} (b c-a d)}\right ) \] Input:

Integrate[(x^9*(A + B*x^4))/((a + b*x^4)*(c + d*x^4)),x]
 

Output:

((-3*(b*B*c - A*b*d + a*B*d)*x^2)/(b^2*d^2) + (B*x^6)/(b*d) + (3*a^(3/2)*( 
-(A*b) + a*B)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(b^(5/2)*(-(b*c) + a*d)) + (3 
*c^(3/2)*(B*c - A*d)*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/(d^(5/2)*(b*c - a*d))) 
/6
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1045, 444, 27, 444, 397, 218}

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.

\(\displaystyle \int \frac {x^9 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx\)

\(\Big \downarrow \) 1045

\(\displaystyle \frac {1}{2} \int \frac {x^8 \left (B x^4+A\right )}{\left (b x^4+a\right ) \left (d x^4+c\right )}dx^2\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {1}{2} \left (\frac {B x^6}{3 b d}-\frac {\int \frac {3 x^4 \left ((b B c-A b d+a B d) x^4+a B c\right )}{\left (b x^4+a\right ) \left (d x^4+c\right )}dx^2}{3 b d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {B x^6}{3 b d}-\frac {\int \frac {x^4 \left ((b B c-A b d+a B d) x^4+a B c\right )}{\left (b x^4+a\right ) \left (d x^4+c\right )}dx^2}{b d}\right )\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {1}{2} \left (\frac {B x^6}{3 b d}-\frac {\frac {x^2 (a B d-A b d+b B c)}{b d}-\frac {\int \frac {\left (c (B c-A d) b^2+a d (B c-A d) b+a^2 B d^2\right ) x^4+a c (b B c-A b d+a B d)}{\left (b x^4+a\right ) \left (d x^4+c\right )}dx^2}{b d}}{b d}\right )\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {1}{2} \left (\frac {B x^6}{3 b d}-\frac {\frac {x^2 (a B d-A b d+b B c)}{b d}-\frac {\frac {a^2 d^2 (A b-a B) \int \frac {1}{b x^4+a}dx^2}{b c-a d}+\frac {b^2 c^2 (B c-A d) \int \frac {1}{d x^4+c}dx^2}{b c-a d}}{b d}}{b d}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{2} \left (\frac {B x^6}{3 b d}-\frac {\frac {x^2 (a B d-A b d+b B c)}{b d}-\frac {\frac {a^{3/2} d^2 (A b-a B) \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {b} (b c-a d)}+\frac {b^2 c^{3/2} (B c-A d) \arctan \left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{\sqrt {d} (b c-a d)}}{b d}}{b d}\right )\)

Input:

Int[(x^9*(A + B*x^4))/((a + b*x^4)*(c + d*x^4)),x]
 

Output:

((B*x^6)/(3*b*d) - (((b*B*c - A*b*d + a*B*d)*x^2)/(b*d) - ((a^(3/2)*(A*b - 
 a*B)*d^2*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt[b]*(b*c - a*d)) + (b^2*c^(3 
/2)*(B*c - A*d)*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/(Sqrt[d]*(b*c - a*d)))/(b*d 
))/(b*d))/2
 

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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 444

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ 
(p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ 
(b*d*(m + 2*(p + q + 1) + 1))   Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) 
^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( 
m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, 
q}, x] && GtQ[m, 1]
 

rule 1045

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
)*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :> With[{k = GCD[m + 1, n]}, Si 
mp[1/k   Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q*(e 
 + f*x^(n/k))^r, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, d, e, f, p, 
q, r}, x] && IGtQ[n, 0] && IntegerQ[m]
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.90

method	result	size
default	\(\frac {\frac {B \,x^{6} b d}{6}+\frac {\left (A b d -B a d -B b c \right ) x^{2}}{2}}{b^{2} d^{2}}-\frac {a^{2} \left (A b -B a \right ) \arctan \left (\frac {b \,x^{2}}{\sqrt {a b}}\right )}{2 b^{2} \left (a d -c b \right ) \sqrt {a b}}+\frac {c^{2} \left (A d -B c \right ) \arctan \left (\frac {d \,x^{2}}{\sqrt {c d}}\right )}{2 d^{2} \left (a d -c b \right ) \sqrt {c d}}\)	\(122\)
risch	\(\text {Expression too large to display}\)	\(2692\)

Input:

int(x^9*(B*x^4+A)/(b*x^4+a)/(d*x^4+c),x,method=_RETURNVERBOSE)
 

Output:

1/b^2/d^2*(1/6*B*x^6*b*d+1/2*(A*b*d-B*a*d-B*b*c)*x^2)-1/2*a^2*(A*b-B*a)/b^ 
2/(a*d-b*c)/(a*b)^(1/2)*arctan(b*x^2/(a*b)^(1/2))+1/2*c^2*(A*d-B*c)/d^2/(a 
*d-b*c)/(c*d)^(1/2)*arctan(d*x^2/(c*d)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 71.71 (sec) , antiderivative size = 728, normalized size of antiderivative = 5.35 \[ \int \frac {x^9 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\left [\frac {2 \, {\left (B b^{2} c d - B a b d^{2}\right )} x^{6} + 3 \, {\left (B a^{2} - A a b\right )} d^{2} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{4} - 2 \, b x^{2} \sqrt {-\frac {a}{b}} - a}{b x^{4} + a}\right ) - 6 \, {\left (B b^{2} c^{2} - A b^{2} c d - {\left (B a^{2} - A a b\right )} d^{2}\right )} x^{2} + 3 \, {\left (B b^{2} c^{2} - A b^{2} c d\right )} \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{4} + 2 \, d x^{2} \sqrt {-\frac {c}{d}} - c}{d x^{4} + c}\right )}{12 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}}, \frac {2 \, {\left (B b^{2} c d - B a b d^{2}\right )} x^{6} - 6 \, {\left (B a^{2} - A a b\right )} d^{2} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x^{2} \sqrt {\frac {a}{b}}}{a}\right ) - 6 \, {\left (B b^{2} c^{2} - A b^{2} c d - {\left (B a^{2} - A a b\right )} d^{2}\right )} x^{2} + 3 \, {\left (B b^{2} c^{2} - A b^{2} c d\right )} \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{4} + 2 \, d x^{2} \sqrt {-\frac {c}{d}} - c}{d x^{4} + c}\right )}{12 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}}, \frac {2 \, {\left (B b^{2} c d - B a b d^{2}\right )} x^{6} + 3 \, {\left (B a^{2} - A a b\right )} d^{2} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{4} - 2 \, b x^{2} \sqrt {-\frac {a}{b}} - a}{b x^{4} + a}\right ) - 6 \, {\left (B b^{2} c^{2} - A b^{2} c d - {\left (B a^{2} - A a b\right )} d^{2}\right )} x^{2} + 6 \, {\left (B b^{2} c^{2} - A b^{2} c d\right )} \sqrt {\frac {c}{d}} \arctan \left (\frac {d x^{2} \sqrt {\frac {c}{d}}}{c}\right )}{12 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}}, \frac {{\left (B b^{2} c d - B a b d^{2}\right )} x^{6} - 3 \, {\left (B a^{2} - A a b\right )} d^{2} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x^{2} \sqrt {\frac {a}{b}}}{a}\right ) - 3 \, {\left (B b^{2} c^{2} - A b^{2} c d - {\left (B a^{2} - A a b\right )} d^{2}\right )} x^{2} + 3 \, {\left (B b^{2} c^{2} - A b^{2} c d\right )} \sqrt {\frac {c}{d}} \arctan \left (\frac {d x^{2} \sqrt {\frac {c}{d}}}{c}\right )}{6 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}}\right ] \] Input:

integrate(x^9*(B*x^4+A)/(b*x^4+a)/(d*x^4+c),x, algorithm="fricas")
 

Output:

[1/12*(2*(B*b^2*c*d - B*a*b*d^2)*x^6 + 3*(B*a^2 - A*a*b)*d^2*sqrt(-a/b)*lo 
g((b*x^4 - 2*b*x^2*sqrt(-a/b) - a)/(b*x^4 + a)) - 6*(B*b^2*c^2 - A*b^2*c*d 
 - (B*a^2 - A*a*b)*d^2)*x^2 + 3*(B*b^2*c^2 - A*b^2*c*d)*sqrt(-c/d)*log((d* 
x^4 + 2*d*x^2*sqrt(-c/d) - c)/(d*x^4 + c)))/(b^3*c*d^2 - a*b^2*d^3), 1/12* 
(2*(B*b^2*c*d - B*a*b*d^2)*x^6 - 6*(B*a^2 - A*a*b)*d^2*sqrt(a/b)*arctan(b* 
x^2*sqrt(a/b)/a) - 6*(B*b^2*c^2 - A*b^2*c*d - (B*a^2 - A*a*b)*d^2)*x^2 + 3 
*(B*b^2*c^2 - A*b^2*c*d)*sqrt(-c/d)*log((d*x^4 + 2*d*x^2*sqrt(-c/d) - c)/( 
d*x^4 + c)))/(b^3*c*d^2 - a*b^2*d^3), 1/12*(2*(B*b^2*c*d - B*a*b*d^2)*x^6 
+ 3*(B*a^2 - A*a*b)*d^2*sqrt(-a/b)*log((b*x^4 - 2*b*x^2*sqrt(-a/b) - a)/(b 
*x^4 + a)) - 6*(B*b^2*c^2 - A*b^2*c*d - (B*a^2 - A*a*b)*d^2)*x^2 + 6*(B*b^ 
2*c^2 - A*b^2*c*d)*sqrt(c/d)*arctan(d*x^2*sqrt(c/d)/c))/(b^3*c*d^2 - a*b^2 
*d^3), 1/6*((B*b^2*c*d - B*a*b*d^2)*x^6 - 3*(B*a^2 - A*a*b)*d^2*sqrt(a/b)* 
arctan(b*x^2*sqrt(a/b)/a) - 3*(B*b^2*c^2 - A*b^2*c*d - (B*a^2 - A*a*b)*d^2 
)*x^2 + 3*(B*b^2*c^2 - A*b^2*c*d)*sqrt(c/d)*arctan(d*x^2*sqrt(c/d)/c))/(b^ 
3*c*d^2 - a*b^2*d^3)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^9 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Timed out} \] Input:

integrate(x**9*(B*x**4+A)/(b*x**4+a)/(d*x**4+c),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.95 \[ \int \frac {x^9 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {{\left (B a^{3} - A a^{2} b\right )} \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, {\left (b^{3} c - a b^{2} d\right )} \sqrt {a b}} + \frac {{\left (B c^{3} - A c^{2} d\right )} \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c d^{2} - a d^{3}\right )} \sqrt {c d}} + \frac {B b d x^{6} - 3 \, {\left (B b c + {\left (B a - A b\right )} d\right )} x^{2}}{6 \, b^{2} d^{2}} \] Input:

integrate(x^9*(B*x^4+A)/(b*x^4+a)/(d*x^4+c),x, algorithm="maxima")
 

Output:

-1/2*(B*a^3 - A*a^2*b)*arctan(b*x^2/sqrt(a*b))/((b^3*c - a*b^2*d)*sqrt(a*b 
)) + 1/2*(B*c^3 - A*c^2*d)*arctan(d*x^2/sqrt(c*d))/((b*c*d^2 - a*d^3)*sqrt 
(c*d)) + 1/6*(B*b*d*x^6 - 3*(B*b*c + (B*a - A*b)*d)*x^2)/(b^2*d^2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.08 \[ \int \frac {x^9 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {{\left (B a^{3} - A a^{2} b\right )} \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, {\left (b^{3} c - a b^{2} d\right )} \sqrt {a b}} + \frac {{\left (B c^{3} - A c^{2} d\right )} \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c d^{2} - a d^{3}\right )} \sqrt {c d}} + \frac {B b^{2} d^{2} x^{6} - 3 \, B b^{2} c d x^{2} - 3 \, B a b d^{2} x^{2} + 3 \, A b^{2} d^{2} x^{2}}{6 \, b^{3} d^{3}} \] Input:

integrate(x^9*(B*x^4+A)/(b*x^4+a)/(d*x^4+c),x, algorithm="giac")
 

Output:

-1/2*(B*a^3 - A*a^2*b)*arctan(b*x^2/sqrt(a*b))/((b^3*c - a*b^2*d)*sqrt(a*b 
)) + 1/2*(B*c^3 - A*c^2*d)*arctan(d*x^2/sqrt(c*d))/((b*c*d^2 - a*d^3)*sqrt 
(c*d)) + 1/6*(B*b^2*d^2*x^6 - 3*B*b^2*c*d*x^2 - 3*B*a*b*d^2*x^2 + 3*A*b^2* 
d^2*x^2)/(b^3*d^3)
 

Mupad [B] (verification not implemented)

Time = 7.19 (sec) , antiderivative size = 20507, normalized size of antiderivative = 150.79 \[ \int \frac {x^9 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Too large to display} \] Input:

int((x^9*(A + B*x^4))/((a + b*x^4)*(c + d*x^4)),x)
 

Output:

x^2*(A/(2*b*d) - (B*(a*d + b*c))/(2*b^2*d^2)) - (atan((((A*d - B*c)*((4*x^ 
2*(B^5*a^3*b^9*c^12 + B^5*a^12*c^3*d^9 - A^5*a^2*b^10*c^8*d^4 - A^5*a^3*b^ 
9*c^7*d^5 - A^5*a^4*b^8*c^6*d^6 - A^5*a^6*b^6*c^4*d^8 - A^5*a^7*b^5*c^3*d^ 
9 - A^5*a^8*b^4*c^2*d^10 + B^5*a^5*b^7*c^10*d^2 + B^5*a^6*b^6*c^9*d^3 + B^ 
5*a^9*b^3*c^6*d^6 + B^5*a^10*b^2*c^5*d^7 - A*B^4*a^2*b^10*c^12 - A*B^4*a^1 
2*c^2*d^10 + B^5*a^4*b^8*c^11*d + B^5*a^11*b*c^4*d^8 - 5*A*B^4*a^4*b^8*c^1 
0*d^2 - 5*A*B^4*a^5*b^7*c^9*d^3 - 3*A*B^4*a^6*b^6*c^8*d^4 - 3*A*B^4*a^8*b^ 
4*c^6*d^6 - 5*A*B^4*a^9*b^3*c^5*d^7 - 5*A*B^4*a^10*b^2*c^4*d^8 + 4*A^2*B^3 
*a^2*b^10*c^11*d + 4*A^2*B^3*a^11*b*c^2*d^10 + 4*A^4*B*a^2*b^10*c^9*d^3 + 
5*A^4*B*a^3*b^9*c^8*d^4 + 5*A^4*B*a^4*b^8*c^7*d^5 + 2*A^4*B*a^5*b^7*c^6*d^ 
6 + 2*A^4*B*a^6*b^6*c^5*d^7 + 5*A^4*B*a^7*b^5*c^4*d^8 + 5*A^4*B*a^8*b^4*c^ 
3*d^9 + 4*A^4*B*a^9*b^3*c^2*d^10 + 10*A^2*B^3*a^3*b^9*c^10*d^2 + 10*A^2*B^ 
3*a^4*b^8*c^9*d^3 + 9*A^2*B^3*a^5*b^7*c^8*d^4 + 3*A^2*B^3*a^6*b^6*c^7*d^5 
+ 3*A^2*B^3*a^7*b^5*c^6*d^6 + 9*A^2*B^3*a^8*b^4*c^5*d^7 + 10*A^2*B^3*a^9*b 
^3*c^4*d^8 + 10*A^2*B^3*a^10*b^2*c^3*d^9 - 6*A^3*B^2*a^2*b^10*c^10*d^2 - 1 
0*A^3*B^2*a^3*b^9*c^9*d^3 - 10*A^3*B^2*a^4*b^8*c^8*d^4 - 7*A^3*B^2*a^5*b^7 
*c^7*d^5 - 2*A^3*B^2*a^6*b^6*c^6*d^6 - 7*A^3*B^2*a^7*b^5*c^5*d^7 - 10*A^3* 
B^2*a^8*b^4*c^4*d^8 - 10*A^3*B^2*a^9*b^3*c^3*d^9 - 6*A^3*B^2*a^10*b^2*c^2* 
d^10 - 5*A*B^4*a^3*b^9*c^11*d - 5*A*B^4*a^11*b*c^3*d^9))/(b^6*d^6) - (((16 
*(B^4*a*b^12*c^13 + B^4*a^13*c*d^12 + 2*A^4*a^5*b^8*c^5*d^8 + 2*B^4*a^7...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.65 \[ \int \frac {x^9 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {-3 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) c -3 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) c -3 c d \,x^{2}+d^{2} x^{6}}{6 d^{3}} \] Input:

int(x^9*(B*x^4+A)/(b*x^4+a)/(d*x^4+c),x)
 

Output:

( - 3*sqrt(d)*sqrt(c)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(d)*x)/(d**( 
1/4)*c**(1/4)*sqrt(2)))*c - 3*sqrt(d)*sqrt(c)*atan((d**(1/4)*c**(1/4)*sqrt 
(2) + 2*sqrt(d)*x)/(d**(1/4)*c**(1/4)*sqrt(2)))*c - 3*c*d*x**2 + d**2*x**6 
)/(6*d**3)