\(\int \frac {A+B x^4}{x^7 (a+b x^4) (c+d x^4)} \, dx\) [10]

Optimal result

Integrand size = 29, antiderivative size = 136 \[ \int \frac {A+B x^4}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {A}{6 a c x^6}+\frac {A b c-a B c+a A d}{2 a^2 c^2 x^2}+\frac {b^{3/2} (A b-a B) \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 a^{5/2} (b c-a d)}+\frac {d^{3/2} (B c-A d) \arctan \left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{2 c^{5/2} (b c-a d)} \] Output:

-1/6*A/a/c/x^6+1/2*(A*a*d+A*b*c-B*a*c)/a^2/c^2/x^2+1/2*b^(3/2)*(A*b-B*a)*a 
rctan(b^(1/2)*x^2/a^(1/2))/a^(5/2)/(-a*d+b*c)+1/2*d^(3/2)*(-A*d+B*c)*arcta 
n(d^(1/2)*x^2/c^(1/2))/c^(5/2)/(-a*d+b*c)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.82 \[ \int \frac {A+B x^4}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {3 b^{3/2} (A b-a B) c^{5/2} x^6 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+3 b^{3/2} (A b-a B) c^{5/2} x^6 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt {a} \left (-\sqrt {c} (-b c+a d) \left (-3 A b c x^4+3 a B c x^4+a A \left (c-3 d x^4\right )\right )+3 a^2 d^{3/2} (B c-A d) x^6 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+3 a^2 d^{3/2} (B c-A d) x^6 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )\right )}{6 a^{5/2} c^{5/2} (-b c+a d) x^6} \] Input:

Integrate[(A + B*x^4)/(x^7*(a + b*x^4)*(c + d*x^4)),x]
 

Output:

(3*b^(3/2)*(A*b - a*B)*c^(5/2)*x^6*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] 
 + 3*b^(3/2)*(A*b - a*B)*c^(5/2)*x^6*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4 
)] + Sqrt[a]*(-(Sqrt[c]*(-(b*c) + a*d)*(-3*A*b*c*x^4 + 3*a*B*c*x^4 + a*A*( 
c - 3*d*x^4))) + 3*a^2*d^(3/2)*(B*c - A*d)*x^6*ArcTan[1 - (Sqrt[2]*d^(1/4) 
*x)/c^(1/4)] + 3*a^2*d^(3/2)*(B*c - A*d)*x^6*ArcTan[1 + (Sqrt[2]*d^(1/4)*x 
)/c^(1/4)]))/(6*a^(5/2)*c^(5/2)*(-(b*c) + a*d)*x^6)
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1045, 445, 27, 445, 25, 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.

\(\displaystyle \int \frac {A+B x^4}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx\)

\(\Big \downarrow \) 1045

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

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

Input:

Int[(A + B*x^4)/(x^7*(a + b*x^4)*(c + d*x^4)),x]
 

Output:

(-1/3*A/(a*c*x^6) - (-((A*b*c - a*B*c + a*A*d)/(a*c*x^2)) + (-((b^(3/2)*(A 
*b - a*B)*c^2*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt[a]*(b*c - a*d))) - (a^2 
*d^(3/2)*(B*c - A*d)*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d))) 
/(a*c))/(a*c))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[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 = 0.23 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.91

method	result	size
default	\(-\frac {b^{2} \left (A b -B a \right ) \arctan \left (\frac {b \,x^{2}}{\sqrt {a b}}\right )}{2 a^{2} \left (a d -c b \right ) \sqrt {a b}}-\frac {A}{6 a c \,x^{6}}-\frac {-A a d -A b c +a B c}{2 a^{2} c^{2} x^{2}}+\frac {d^{2} \left (A d -B c \right ) \arctan \left (\frac {d \,x^{2}}{\sqrt {c d}}\right )}{2 c^{2} \left (a d -c b \right ) \sqrt {c d}}\)	\(124\)
risch	\(\text {Expression too large to display}\)	\(1453\)

Input:

int((B*x^4+A)/x^7/(b*x^4+a)/(d*x^4+c),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 88.73 (sec) , antiderivative size = 728, normalized size of antiderivative = 5.35 \[ \int \frac {A+B x^4}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\left [\frac {3 \, {\left (B a b - A b^{2}\right )} c^{2} x^{6} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{4} - 2 \, a x^{2} \sqrt {-\frac {b}{a}} - a}{b x^{4} + a}\right ) + 3 \, {\left (B a^{2} c d - A a^{2} d^{2}\right )} x^{6} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{4} + 2 \, c x^{2} \sqrt {-\frac {d}{c}} - c}{d x^{4} + c}\right ) - 2 \, A a b c^{2} + 2 \, A a^{2} c d + 6 \, {\left (B a^{2} c d - A a^{2} d^{2} - {\left (B a b - A b^{2}\right )} c^{2}\right )} x^{4}}{12 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}, \frac {3 \, {\left (B a b - A b^{2}\right )} c^{2} x^{6} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{4} - 2 \, a x^{2} \sqrt {-\frac {b}{a}} - a}{b x^{4} + a}\right ) + 6 \, {\left (B a^{2} c d - A a^{2} d^{2}\right )} x^{6} \sqrt {\frac {d}{c}} \arctan \left (x^{2} \sqrt {\frac {d}{c}}\right ) - 2 \, A a b c^{2} + 2 \, A a^{2} c d + 6 \, {\left (B a^{2} c d - A a^{2} d^{2} - {\left (B a b - A b^{2}\right )} c^{2}\right )} x^{4}}{12 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}, -\frac {6 \, {\left (B a b - A b^{2}\right )} c^{2} x^{6} \sqrt {\frac {b}{a}} \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right ) - 3 \, {\left (B a^{2} c d - A a^{2} d^{2}\right )} x^{6} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{4} + 2 \, c x^{2} \sqrt {-\frac {d}{c}} - c}{d x^{4} + c}\right ) + 2 \, A a b c^{2} - 2 \, A a^{2} c d - 6 \, {\left (B a^{2} c d - A a^{2} d^{2} - {\left (B a b - A b^{2}\right )} c^{2}\right )} x^{4}}{12 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}, -\frac {3 \, {\left (B a b - A b^{2}\right )} c^{2} x^{6} \sqrt {\frac {b}{a}} \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right ) - 3 \, {\left (B a^{2} c d - A a^{2} d^{2}\right )} x^{6} \sqrt {\frac {d}{c}} \arctan \left (x^{2} \sqrt {\frac {d}{c}}\right ) + A a b c^{2} - A a^{2} c d - 3 \, {\left (B a^{2} c d - A a^{2} d^{2} - {\left (B a b - A b^{2}\right )} c^{2}\right )} x^{4}}{6 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}\right ] \] Input:

integrate((B*x^4+A)/x^7/(b*x^4+a)/(d*x^4+c),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x^4}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Timed out} \] Input:

integrate((B*x**4+A)/x**7/(b*x**4+a)/(d*x**4+c),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x^4}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {{\left (B a b^{2} - A b^{3}\right )} \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b c - a^{3} d\right )} \sqrt {a b}} + \frac {{\left (B c d^{2} - A d^{3}\right )} \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c^{3} - a c^{2} d\right )} \sqrt {c d}} + \frac {3 \, {\left (A a d - {\left (B a - A b\right )} c\right )} x^{4} - A a c}{6 \, a^{2} c^{2} x^{6}} \] Input:

integrate((B*x^4+A)/x^7/(b*x^4+a)/(d*x^4+c),x, algorithm="maxima")
 

Output:

-1/2*(B*a*b^2 - A*b^3)*arctan(b*x^2/sqrt(a*b))/((a^2*b*c - a^3*d)*sqrt(a*b 
)) + 1/2*(B*c*d^2 - A*d^3)*arctan(d*x^2/sqrt(c*d))/((b*c^3 - a*c^2*d)*sqrt 
(c*d)) + 1/6*(3*(A*a*d - (B*a - A*b)*c)*x^4 - A*a*c)/(a^2*c^2*x^6)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x^4}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {{\left (B a b^{2} - A b^{3}\right )} \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b c - a^{3} d\right )} \sqrt {a b}} + \frac {{\left (B c d^{2} - A d^{3}\right )} \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c^{3} - a c^{2} d\right )} \sqrt {c d}} - \frac {3 \, B a c x^{4} - 3 \, A b c x^{4} - 3 \, A a d x^{4} + A a c}{6 \, a^{2} c^{2} x^{6}} \] Input:

integrate((B*x^4+A)/x^7/(b*x^4+a)/(d*x^4+c),x, algorithm="giac")
 

Output:

-1/2*(B*a*b^2 - A*b^3)*arctan(b*x^2/sqrt(a*b))/((a^2*b*c - a^3*d)*sqrt(a*b 
)) + 1/2*(B*c*d^2 - A*d^3)*arctan(d*x^2/sqrt(c*d))/((b*c^3 - a*c^2*d)*sqrt 
(c*d)) - 1/6*(3*B*a*c*x^4 - 3*A*b*c*x^4 - 3*A*a*d*x^4 + A*a*c)/(a^2*c^2*x^ 
6)
 

Mupad [B] (verification not implemented)

Time = 8.11 (sec) , antiderivative size = 15821, normalized size of antiderivative = 116.33 \[ \int \frac {A+B x^4}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Too large to display} \] Input:

int((A + B*x^4)/(x^7*(a + b*x^4)*(c + d*x^4)),x)
 

Output:

(atan((((x^2*(4*A^5*a^15*b^13*c^19*d^9 + 4*A^5*a^19*b^9*c^15*d^13 + 4*A*B^ 
4*a^18*b^10*c^20*d^8 + 4*A*B^4*a^20*b^8*c^18*d^10 - 4*A^4*B*a^15*b^13*c^20 
*d^8 - 12*A^4*B*a^16*b^12*c^19*d^9 - 12*A^4*B*a^19*b^9*c^16*d^12 - 4*A^4*B 
*a^20*b^8*c^15*d^13 - 12*A^2*B^3*a^17*b^11*c^20*d^8 - 4*A^2*B^3*a^18*b^10* 
c^19*d^9 - 4*A^2*B^3*a^19*b^9*c^18*d^10 - 12*A^2*B^3*a^20*b^8*c^17*d^11 + 
12*A^3*B^2*a^16*b^12*c^20*d^8 + 12*A^3*B^2*a^17*b^11*c^19*d^9 + 12*A^3*B^2 
*a^19*b^9*c^17*d^11 + 12*A^3*B^2*a^20*b^8*c^16*d^12) + ((A*d - B*c)*(-c^5* 
d^3)^(1/2)*(((x^2*(64*A^3*a^17*b^13*c^24*d^6 - 64*A^3*a^16*b^14*c^25*d^5 + 
 64*A^3*a^19*b^11*c^22*d^8 - 64*A^3*a^20*b^10*c^21*d^9 - 64*A^3*a^21*b^9*c 
^20*d^10 + 64*A^3*a^22*b^8*c^19*d^11 + 64*A^3*a^24*b^6*c^17*d^13 - 64*A^3* 
a^25*b^5*c^16*d^14 + 64*B^3*a^19*b^11*c^25*d^5 - 64*B^3*a^20*b^10*c^24*d^6 
 + 64*B^3*a^21*b^9*c^23*d^7 - 128*B^3*a^22*b^8*c^22*d^8 + 64*B^3*a^23*b^7* 
c^21*d^9 - 64*B^3*a^24*b^6*c^20*d^10 + 64*B^3*a^25*b^5*c^19*d^11 - 192*A*B 
^2*a^18*b^12*c^25*d^5 + 192*A*B^2*a^19*b^11*c^24*d^6 + 192*A*B^2*a^24*b^6* 
c^19*d^11 - 192*A*B^2*a^25*b^5*c^18*d^12 + 192*A^2*B*a^17*b^13*c^25*d^5 - 
192*A^2*B*a^18*b^12*c^24*d^6 - 64*A^2*B*a^19*b^11*c^23*d^7 + 64*A^2*B*a^20 
*b^10*c^22*d^8 + 64*A^2*B*a^22*b^8*c^20*d^10 - 64*A^2*B*a^23*b^7*c^19*d^11 
 - 192*A^2*B*a^24*b^6*c^18*d^12 + 192*A^2*B*a^25*b^5*c^17*d^13) + ((A*d - 
B*c)*(-c^5*d^3)^(1/2)*(((A*d - B*c)*(-c^5*d^3)^(1/2)*(x^2*(2048*A*a^21*b^1 
1*c^27*d^5 - 1024*A*a^20*b^12*c^28*d^4 - 2048*A*a^23*b^9*c^25*d^7 + 204...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.70 \[ \int \frac {A+B x^4}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {-3 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) d \,x^{6}-3 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) d \,x^{6}-c^{2}+3 c d \,x^{4}}{6 c^{3} x^{6}} \] Input:

int((B*x^4+A)/x^7/(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)))*d*x**6 - 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)))*d*x**6 - c**2 + 3*c*d 
*x**4)/(6*c**3*x**6)