\(\int \frac {A+B x^4}{x^4 (a+b x^4)} \, dx\) [15]

Optimal result

Integrand size = 20, antiderivative size = 169 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right )} \, dx=-\frac {A}{3 a x^3}+\frac {(A b-a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {(A b-a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {(A b-a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a}+\sqrt {b} x^2}\right )}{2 \sqrt {2} a^{7/4} \sqrt [4]{b}} \] Output:

-1/3*A/a/x^3-1/4*(A*b-B*a)*arctan(-1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/a^ 
(7/4)/b^(1/4)-1/4*(A*b-B*a)*arctan(1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/a^ 
(7/4)/b^(1/4)-1/4*(A*b-B*a)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*x/(a^(1/2)+b^( 
1/2)*x^2))*2^(1/2)/a^(7/4)/b^(1/4)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.24 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right )} \, dx=\frac {-\frac {8 a^{3/4} A}{x^3}+\frac {6 \sqrt {2} (A b-a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}-\frac {6 \sqrt {2} (A b-a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac {3 \sqrt {2} (A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{\sqrt [4]{b}}+\frac {3 \sqrt {2} (-A b+a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{\sqrt [4]{b}}}{24 a^{7/4}} \] Input:

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

Output:

((-8*a^(3/4)*A)/x^3 + (6*Sqrt[2]*(A*b - a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x 
)/a^(1/4)])/b^(1/4) - (6*Sqrt[2]*(A*b - a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x 
)/a^(1/4)])/b^(1/4) + (3*Sqrt[2]*(A*b - a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4) 
*b^(1/4)*x + Sqrt[b]*x^2])/b^(1/4) + (3*Sqrt[2]*(-(A*b) + a*B)*Log[Sqrt[a] 
 + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(1/4))/(24*a^(7/4))
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.33, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {955, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

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

\(\Big \downarrow \) 955

\(\displaystyle -\frac {(A b-a B) \int \frac {1}{b x^4+a}dx}{a}-\frac {A}{3 a x^3}\)

\(\Big \downarrow \) 755

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

\(\Big \downarrow \) 1476

\(\displaystyle -\frac {(A b-a B) \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {b}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {b}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}\right )}{a}-\frac {A}{3 a x^3}\)

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

\(\displaystyle -\frac {(A b-a B) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {A}{3 a x^3}\)

\(\Big \downarrow \) 1479

\(\displaystyle -\frac {(A b-a B) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{\sqrt [4]{b} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {A}{3 a x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {(A b-a B) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{\sqrt [4]{b} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {A}{3 a x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(A b-a B) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {A}{3 a x^3}\)

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {(A b-a B) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {A}{3 a x^3}\)

Input:

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

Output:

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

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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 955

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

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.71

method	result	size
default	\(\frac {\left (-A b +B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a^{2}}-\frac {A}{3 a \,x^{3}}\)	\(120\)
risch	\(-\frac {A}{3 a \,x^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,a^{7} \textit {\_Z}^{4}+A^{4} b^{4}-4 A^{3} B a \,b^{3}+6 A^{2} B^{2} a^{2} b^{2}-4 A \,B^{3} a^{3} b +B^{4} a^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (-5 \textit {\_R}^{4} a^{7} b -4 A^{4} b^{4}+16 A^{3} B a \,b^{3}-24 A^{2} B^{2} a^{2} b^{2}+16 A \,B^{3} a^{3} b -4 B^{4} a^{4}\right ) x +\left (-A^{3} a^{2} b^{3}+3 A^{2} B \,a^{3} b^{2}-3 A \,B^{2} a^{4} b +B^{3} a^{5}\right ) \textit {\_R} \right )\right )}{4}\)	\(183\)

Input:

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

Output:

1/8*(-A*b+B*a)/a^2*(a/b)^(1/4)*2^(1/2)*(ln((x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b 
)^(1/2))/(x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^( 
1/4)*x+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x-1))-1/3*A/a/x^3
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 579, normalized size of antiderivative = 3.43 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right )} \, dx=-\frac {3 \, a x^{3} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} \log \left (a^{2} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} x\right ) + 3 i \, a x^{3} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} \log \left (i \, a^{2} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} x\right ) - 3 i \, a x^{3} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} \log \left (-i \, a^{2} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} x\right ) - 3 \, a x^{3} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} \log \left (-a^{2} \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{7} b}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} x\right ) + 4 \, A}{12 \, a x^{3}} \] Input:

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

Output:

-1/12*(3*a*x^3*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a* 
b^3 + A^4*b^4)/(a^7*b))^(1/4)*log(a^2*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B 
^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^7*b))^(1/4) - (B*a - A*b)*x) + 3* 
I*a*x^3*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A 
^4*b^4)/(a^7*b))^(1/4)*log(I*a^2*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^ 
2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^7*b))^(1/4) - (B*a - A*b)*x) - 3*I*a*x 
^3*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^ 
4)/(a^7*b))^(1/4)*log(-I*a^2*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^ 
2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^7*b))^(1/4) - (B*a - A*b)*x) - 3*a*x^3*(-( 
B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^ 
7*b))^(1/4)*log(-a^2*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^ 
3*B*a*b^3 + A^4*b^4)/(a^7*b))^(1/4) - (B*a - A*b)*x) + 4*A)/(a*x^3)
 

Sympy [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.52 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right )} \, dx=- \frac {A}{3 a x^{3}} + \operatorname {RootSum} {\left (256 t^{4} a^{7} b + A^{4} b^{4} - 4 A^{3} B a b^{3} + 6 A^{2} B^{2} a^{2} b^{2} - 4 A B^{3} a^{3} b + B^{4} a^{4}, \left ( t \mapsto t \log {\left (\frac {4 t a^{2}}{- A b + B a} + x \right )} \right )\right )} \] Input:

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

Output:

-A/(3*a*x**3) + RootSum(256*_t**4*a**7*b + A**4*b**4 - 4*A**3*B*a*b**3 + 6 
*A**2*B**2*a**2*b**2 - 4*A*B**3*a**3*b + B**4*a**4, Lambda(_t, _t*log(4*_t 
*a**2/(-A*b + B*a) + x)))
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.27 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right )} \, dx=\frac {\frac {2 \, \sqrt {2} {\left (B a - A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (B a - A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (B a - A b\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (B a - A b\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{8 \, a} - \frac {A}{3 \, a x^{3}} \] Input:

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

Output:

1/8*(2*sqrt(2)*(B*a - A*b)*arctan(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))) + 2*sqr 
t(2)*(B*a - A*b)*arctan(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)*(B*a - 
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)*(B*a - A*b)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sq 
rt(a))/(a^(3/4)*b^(1/4)))/a - 1/3*A/(a*x^3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (120) = 240\).

Time = 0.13 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.47 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right )} \, dx=\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a^{2} b} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a^{2} b} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a^{2} b} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a^{2} b} - \frac {A}{3 \, a x^{3}} \] Input:

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

Output:

1/4*sqrt(2)*((a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*arctan(1/2*sqrt(2)*(2* 
x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b) + 1/4*sqrt(2)*((a*b^3)^(1/4) 
*B*a - (a*b^3)^(1/4)*A*b)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/( 
a/b)^(1/4))/(a^2*b) + 1/8*sqrt(2)*((a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)* 
log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b) - 1/8*sqrt(2)*((a*b^3 
)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/ 
b))/(a^2*b) - 1/3*A/(a*x^3)
 

Mupad [B] (verification not implemented)

Time = 3.37 (sec) , antiderivative size = 795, normalized size of antiderivative = 4.70 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

- A/(3*a*x^3) - (atan((((x*(4*A^2*a^3*b^5 + 4*B^2*a^5*b^3 - 8*A*B*a^4*b^4) 
 - ((A*b - B*a)*(16*A*a^5*b^4 - 16*B*a^6*b^3))/(4*(-a)^(7/4)*b^(1/4)))*(A* 
b - B*a)*1i)/(4*(-a)^(7/4)*b^(1/4)) + ((x*(4*A^2*a^3*b^5 + 4*B^2*a^5*b^3 - 
 8*A*B*a^4*b^4) + ((A*b - B*a)*(16*A*a^5*b^4 - 16*B*a^6*b^3))/(4*(-a)^(7/4 
)*b^(1/4)))*(A*b - B*a)*1i)/(4*(-a)^(7/4)*b^(1/4)))/(((x*(4*A^2*a^3*b^5 + 
4*B^2*a^5*b^3 - 8*A*B*a^4*b^4) - ((A*b - B*a)*(16*A*a^5*b^4 - 16*B*a^6*b^3 
))/(4*(-a)^(7/4)*b^(1/4)))*(A*b - B*a))/(4*(-a)^(7/4)*b^(1/4)) - ((x*(4*A^ 
2*a^3*b^5 + 4*B^2*a^5*b^3 - 8*A*B*a^4*b^4) + ((A*b - B*a)*(16*A*a^5*b^4 - 
16*B*a^6*b^3))/(4*(-a)^(7/4)*b^(1/4)))*(A*b - B*a))/(4*(-a)^(7/4)*b^(1/4)) 
))*(A*b - B*a)*1i)/(2*(-a)^(7/4)*b^(1/4)) - (atan((((x*(4*A^2*a^3*b^5 + 4* 
B^2*a^5*b^3 - 8*A*B*a^4*b^4) - ((A*b - B*a)*(16*A*a^5*b^4 - 16*B*a^6*b^3)* 
1i)/(4*(-a)^(7/4)*b^(1/4)))*(A*b - B*a))/(4*(-a)^(7/4)*b^(1/4)) + ((x*(4*A 
^2*a^3*b^5 + 4*B^2*a^5*b^3 - 8*A*B*a^4*b^4) + ((A*b - B*a)*(16*A*a^5*b^4 - 
 16*B*a^6*b^3)*1i)/(4*(-a)^(7/4)*b^(1/4)))*(A*b - B*a))/(4*(-a)^(7/4)*b^(1 
/4)))/(((x*(4*A^2*a^3*b^5 + 4*B^2*a^5*b^3 - 8*A*B*a^4*b^4) - ((A*b - B*a)* 
(16*A*a^5*b^4 - 16*B*a^6*b^3)*1i)/(4*(-a)^(7/4)*b^(1/4)))*(A*b - B*a)*1i)/ 
(4*(-a)^(7/4)*b^(1/4)) - ((x*(4*A^2*a^3*b^5 + 4*B^2*a^5*b^3 - 8*A*B*a^4*b^ 
4) + ((A*b - B*a)*(16*A*a^5*b^4 - 16*B*a^6*b^3)*1i)/(4*(-a)^(7/4)*b^(1/4)) 
)*(A*b - B*a)*1i)/(4*(-a)^(7/4)*b^(1/4))))*(A*b - B*a))/(2*(-a)^(7/4)*b^(1 
/4))
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.03 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right )} \, dx=-\frac {1}{3 x^{3}} \] Input:

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

Output:

( - 1)/(3*x**3)