\(\int \frac {A+B x^4}{x^4 (a+b x^4)^2} \, dx\) [23]

Optimal result

Integrand size = 20, antiderivative size = 197 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right )^2} \, dx=-\frac {A}{3 a^2 x^3}-\frac {(A b-a B) x}{4 a^2 \left (a+b x^4\right )}+\frac {(7 A b-3 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a}+\sqrt {b} x^2}\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}} \] Output:

-1/3*A/a^2/x^3-1/4*(A*b-B*a)*x/a^2/(b*x^4+a)-1/16*(7*A*b-3*B*a)*arctan(-1+ 
2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/a^(11/4)/b^(1/4)-1/16*(7*A*b-3*B*a)*arc 
tan(1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/a^(11/4)/b^(1/4)-1/16*(7*A*b-3*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^(11/ 
4)/b^(1/4)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.21 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right )^2} \, dx=\frac {-\frac {32 a^{3/4} A}{x^3}+\frac {24 a^{3/4} (-A b+a B) x}{a+b x^4}+\frac {6 \sqrt {2} (7 A b-3 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}-\frac {6 \sqrt {2} (7 A b-3 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac {3 \sqrt {2} (7 A b-3 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{\sqrt [4]{b}}+\frac {3 \sqrt {2} (-7 A b+3 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{\sqrt [4]{b}}}{96 a^{11/4}} \] Input:

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

Output:

((-32*a^(3/4)*A)/x^3 + (24*a^(3/4)*(-(A*b) + a*B)*x)/(a + b*x^4) + (6*Sqrt 
[2]*(7*A*b - 3*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(1/4) - (6* 
Sqrt[2]*(7*A*b - 3*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(1/4) + 
 (3*Sqrt[2]*(7*A*b - 3*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt 
[b]*x^2])/b^(1/4) + (3*Sqrt[2]*(-7*A*b + 3*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1 
/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(1/4))/(96*a^(11/4))
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.36, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {957, 847, 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.

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

\(\Big \downarrow \) 957

\(\displaystyle \frac {(7 A b-3 a B) \int \frac {1}{x^4 \left (b x^4+a\right )}dx}{4 a b}+\frac {A b-a B}{4 a b x^3 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {(7 A b-3 a B) \left (-\frac {b \int \frac {1}{b x^4+a}dx}{a}-\frac {1}{3 a x^3}\right )}{4 a b}+\frac {A b-a B}{4 a b x^3 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {(7 A b-3 a B) \left (-\frac {b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x^2+\sqrt {a}}{b x^4+a}dx}{2 \sqrt {a}}\right )}{a}-\frac {1}{3 a x^3}\right )}{4 a b}+\frac {A b-a B}{4 a b x^3 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {(7 A b-3 a B) \left (-\frac {b \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {b}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {b}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}\right )}{a}-\frac {1}{3 a x^3}\right )}{4 a b}+\frac {A b-a B}{4 a b x^3 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {(7 A b-3 a B) \left (-\frac {b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {1}{3 a x^3}\right )}{4 a b}+\frac {A b-a B}{4 a b x^3 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(7 A b-3 a B) \left (-\frac {b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {1}{3 a x^3}\right )}{4 a b}+\frac {A b-a B}{4 a b x^3 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {(7 A b-3 a B) \left (-\frac {b \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{\sqrt [4]{b} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {1}{3 a x^3}\right )}{4 a b}+\frac {A b-a B}{4 a b x^3 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(7 A b-3 a B) \left (-\frac {b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{\sqrt [4]{b} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {1}{3 a x^3}\right )}{4 a b}+\frac {A b-a B}{4 a b x^3 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(7 A b-3 a B) \left (-\frac {b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {1}{3 a x^3}\right )}{4 a b}+\frac {A b-a B}{4 a b x^3 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {(7 A b-3 a B) \left (-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {1}{3 a x^3}\right )}{4 a b}+\frac {A b-a B}{4 a b x^3 \left (a+b x^4\right )}\)

Input:

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

Output:

(A*b - a*B)/(4*a*b*x^3*(a + b*x^4)) + ((7*A*b - 3*a*B)*(-1/3*1/(a*x^3) - ( 
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*Sqr 
t[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))/(4*a*b)
 

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 847

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x 
)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) 
+ 1)/(a*c^n*(m + 1)))   Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a 
, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

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 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 = 0.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.75

method	result	size
default	\(-\frac {\frac {\left (\frac {A b}{4}-\frac {B a}{4}\right ) x}{b \,x^{4}+a}+\frac {\left (7 A b -3 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}}{a^{2}}-\frac {A}{3 a^{2} x^{3}}\)	\(147\)
risch	\(\frac {-\frac {\left (7 A b -3 B a \right ) x^{4}}{12 a^{2}}-\frac {A}{3 a}}{x^{3} \left (b \,x^{4}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{11} b \,\textit {\_Z}^{4}+2401 A^{4} b^{4}-4116 A^{3} B a \,b^{3}+2646 A^{2} B^{2} a^{2} b^{2}-756 A \,B^{3} a^{3} b +81 B^{4} a^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (-5 \textit {\_R}^{4} a^{11} b -9604 A^{4} b^{4}+16464 A^{3} B a \,b^{3}-10584 A^{2} B^{2} a^{2} b^{2}+3024 A \,B^{3} a^{3} b -324 B^{4} a^{4}\right ) x +\left (-343 A^{3} a^{3} b^{3}+441 A^{2} B \,a^{4} b^{2}-189 A \,B^{2} a^{5} b +27 B^{3} a^{6}\right ) \textit {\_R} \right )\right )}{16}\)	\(214\)

Input:

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

Output:

-1/a^2*((1/4*A*b-1/4*B*a)*x/(b*x^4+a)+1/32*(7*A*b-3*B*a)*(a/b)^(1/4)/a*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^2/x^3
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 678, normalized size of antiderivative = 3.44 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right )^2} \, dx=\frac {4 \, {\left (3 \, B a - 7 \, A b\right )} x^{4} - 3 \, {\left (a^{2} b x^{7} + a^{3} x^{3}\right )} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} \log \left (a^{3} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} - {\left (3 \, B a - 7 \, A b\right )} x\right ) - 3 \, {\left (i \, a^{2} b x^{7} + i \, a^{3} x^{3}\right )} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} \log \left (i \, a^{3} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} - {\left (3 \, B a - 7 \, A b\right )} x\right ) - 3 \, {\left (-i \, a^{2} b x^{7} - i \, a^{3} x^{3}\right )} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} \log \left (-i \, a^{3} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} - {\left (3 \, B a - 7 \, A b\right )} x\right ) + 3 \, {\left (a^{2} b x^{7} + a^{3} x^{3}\right )} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} \log \left (-a^{3} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} - {\left (3 \, B a - 7 \, A b\right )} x\right ) - 16 \, A a}{48 \, {\left (a^{2} b x^{7} + a^{3} x^{3}\right )}} \] Input:

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

Output:

1/48*(4*(3*B*a - 7*A*b)*x^4 - 3*(a^2*b*x^7 + a^3*x^3)*(-(81*B^4*a^4 - 756* 
A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^1 
1*b))^(1/4)*log(a^3*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 
 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))^(1/4) - (3*B*a - 7*A*b)*x) - 
 3*(I*a^2*b*x^7 + I*a^3*x^3)*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^ 
2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))^(1/4)*log(I*a^3*(-( 
81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2 
401*A^4*b^4)/(a^11*b))^(1/4) - (3*B*a - 7*A*b)*x) - 3*(-I*a^2*b*x^7 - I*a^ 
3*x^3)*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B 
*a*b^3 + 2401*A^4*b^4)/(a^11*b))^(1/4)*log(-I*a^3*(-(81*B^4*a^4 - 756*A*B^ 
3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b) 
)^(1/4) - (3*B*a - 7*A*b)*x) + 3*(a^2*b*x^7 + a^3*x^3)*(-(81*B^4*a^4 - 756 
*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^ 
11*b))^(1/4)*log(-a^3*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b 
^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))^(1/4) - (3*B*a - 7*A*b)*x) 
 - 16*A*a)/(a^2*b*x^7 + a^3*x^3)
 

Sympy [A] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.63 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right )^2} \, dx=\frac {- 4 A a + x^{4} \left (- 7 A b + 3 B a\right )}{12 a^{3} x^{3} + 12 a^{2} b x^{7}} + \operatorname {RootSum} {\left (65536 t^{4} a^{11} b + 2401 A^{4} b^{4} - 4116 A^{3} B a b^{3} + 2646 A^{2} B^{2} a^{2} b^{2} - 756 A B^{3} a^{3} b + 81 B^{4} a^{4}, \left ( t \mapsto t \log {\left (\frac {16 t a^{3}}{- 7 A b + 3 B a} + x \right )} \right )\right )} \] Input:

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

Output:

(-4*A*a + x**4*(-7*A*b + 3*B*a))/(12*a**3*x**3 + 12*a**2*b*x**7) + RootSum 
(65536*_t**4*a**11*b + 2401*A**4*b**4 - 4116*A**3*B*a*b**3 + 2646*A**2*B** 
2*a**2*b**2 - 756*A*B**3*a**3*b + 81*B**4*a**4, Lambda(_t, _t*log(16*_t*a* 
*3/(-7*A*b + 3*B*a) + x)))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.26 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right )^2} \, dx=\frac {{\left (3 \, B a - 7 \, A b\right )} x^{4} - 4 \, A a}{12 \, {\left (a^{2} b x^{7} + a^{3} x^{3}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, B a - 7 \, A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (3 \, B a - 7 \, A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (3 \, B a - 7 \, A b\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (3 \, B a - 7 \, A b\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{32 \, a^{2}} \] Input:

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

Output:

1/12*((3*B*a - 7*A*b)*x^4 - 4*A*a)/(a^2*b*x^7 + a^3*x^3) + 1/32*(2*sqrt(2) 
*(3*B*a - 7*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*sqrt(2)*(3*B* 
a - 7*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)*(3*B*a - 7*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)*(3*B*a - 7*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^2
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.40 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right )^2} \, dx=\frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 7 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{3} b} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 7 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{3} b} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 7 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{3} b} - \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 7 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{3} b} + \frac {B a x - A b x}{4 \, {\left (b x^{4} + a\right )} a^{2}} - \frac {A}{3 \, a^{2} x^{3}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 3.64 (sec) , antiderivative size = 843, normalized size of antiderivative = 4.28 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

- (A/(3*a) + (x^4*(7*A*b - 3*B*a))/(12*a^2))/(a*x^3 + b*x^7) - (atan((((x* 
(12544*A^2*a^6*b^5 + 2304*B^2*a^8*b^3 - 10752*A*B*a^7*b^4) - ((7*A*b - 3*B 
*a)*(28672*A*a^9*b^4 - 12288*B*a^10*b^3))/(16*(-a)^(11/4)*b^(1/4)))*(7*A*b 
 - 3*B*a)*1i)/(16*(-a)^(11/4)*b^(1/4)) + ((x*(12544*A^2*a^6*b^5 + 2304*B^2 
*a^8*b^3 - 10752*A*B*a^7*b^4) + ((7*A*b - 3*B*a)*(28672*A*a^9*b^4 - 12288* 
B*a^10*b^3))/(16*(-a)^(11/4)*b^(1/4)))*(7*A*b - 3*B*a)*1i)/(16*(-a)^(11/4) 
*b^(1/4)))/(((x*(12544*A^2*a^6*b^5 + 2304*B^2*a^8*b^3 - 10752*A*B*a^7*b^4) 
 - ((7*A*b - 3*B*a)*(28672*A*a^9*b^4 - 12288*B*a^10*b^3))/(16*(-a)^(11/4)* 
b^(1/4)))*(7*A*b - 3*B*a))/(16*(-a)^(11/4)*b^(1/4)) - ((x*(12544*A^2*a^6*b 
^5 + 2304*B^2*a^8*b^3 - 10752*A*B*a^7*b^4) + ((7*A*b - 3*B*a)*(28672*A*a^9 
*b^4 - 12288*B*a^10*b^3))/(16*(-a)^(11/4)*b^(1/4)))*(7*A*b - 3*B*a))/(16*( 
-a)^(11/4)*b^(1/4))))*(7*A*b - 3*B*a)*1i)/(8*(-a)^(11/4)*b^(1/4)) - (atan( 
(((x*(12544*A^2*a^6*b^5 + 2304*B^2*a^8*b^3 - 10752*A*B*a^7*b^4) - ((7*A*b 
- 3*B*a)*(28672*A*a^9*b^4 - 12288*B*a^10*b^3)*1i)/(16*(-a)^(11/4)*b^(1/4)) 
)*(7*A*b - 3*B*a))/(16*(-a)^(11/4)*b^(1/4)) + ((x*(12544*A^2*a^6*b^5 + 230 
4*B^2*a^8*b^3 - 10752*A*B*a^7*b^4) + ((7*A*b - 3*B*a)*(28672*A*a^9*b^4 - 1 
2288*B*a^10*b^3)*1i)/(16*(-a)^(11/4)*b^(1/4)))*(7*A*b - 3*B*a))/(16*(-a)^( 
11/4)*b^(1/4)))/(((x*(12544*A^2*a^6*b^5 + 2304*B^2*a^8*b^3 - 10752*A*B*a^7 
*b^4) - ((7*A*b - 3*B*a)*(28672*A*a^9*b^4 - 12288*B*a^10*b^3)*1i)/(16*(-a) 
^(11/4)*b^(1/4)))*(7*A*b - 3*B*a)*1i)/(16*(-a)^(11/4)*b^(1/4)) - ((x*(1...
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right )^2} \, dx=\frac {6 b^{\frac {3}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{3}-6 b^{\frac {3}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{3}+3 b^{\frac {3}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {b}\, x^{2}\right ) x^{3}-3 b^{\frac {3}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {b}\, x^{2}\right ) x^{3}-8 a}{24 a^{2} x^{3}} \] Input:

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

Output:

(6*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)))*x**3 - 6*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)))*x**3 + 
 3*b**(3/4)*a**(1/4)*sqrt(2)*log( - b**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) 
+ sqrt(b)*x**2)*x**3 - 3*b**(3/4)*a**(1/4)*sqrt(2)*log(b**(1/4)*a**(1/4)*s 
qrt(2)*x + sqrt(a) + sqrt(b)*x**2)*x**3 - 8*a)/(24*a**2*x**3)