\(\int \frac {x^4 (A+B x^4)}{a+b x^4} \, dx\) [11]

Optimal result

Integrand size = 20, antiderivative size = 182 \[ \int \frac {x^4 \left (A+B x^4\right )}{a+b x^4} \, dx=\frac {(A b-a B) x}{b^2}+\frac {B x^5}{5 b}+\frac {\sqrt [4]{a} (A b-a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{9/4}}-\frac {\sqrt [4]{a} (A b-a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{9/4}}-\frac {\sqrt [4]{a} (A b-a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a}+\sqrt {b} x^2}\right )}{2 \sqrt {2} b^{9/4}} \] Output:

(A*b-B*a)*x/b^2+1/5*B*x^5/b-1/4*a^(1/4)*(A*b-B*a)*arctan(-1+2^(1/2)*b^(1/4 
)*x/a^(1/4))*2^(1/2)/b^(9/4)-1/4*a^(1/4)*(A*b-B*a)*arctan(1+2^(1/2)*b^(1/4 
)*x/a^(1/4))*2^(1/2)/b^(9/4)-1/4*a^(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)/b^(9/4)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.24 \[ \int \frac {x^4 \left (A+B x^4\right )}{a+b x^4} \, dx=\frac {40 \sqrt [4]{b} (A b-a B) x+8 b^{5/4} B x^5-10 \sqrt {2} \sqrt [4]{a} (-A b+a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+10 \sqrt {2} \sqrt [4]{a} (-A b+a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )-5 \sqrt {2} \sqrt [4]{a} (-A b+a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+5 \sqrt {2} \sqrt [4]{a} (-A b+a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{40 b^{9/4}} \] Input:

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

Output:

(40*b^(1/4)*(A*b - a*B)*x + 8*b^(5/4)*B*x^5 - 10*Sqrt[2]*a^(1/4)*(-(A*b) + 
 a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 10*Sqrt[2]*a^(1/4)*(-(A*b) 
 + a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 5*Sqrt[2]*a^(1/4)*(-(A*b 
) + a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 5*Sqrt[2 
]*a^(1/4)*(-(A*b) + a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b] 
*x^2])/(40*b^(9/4))
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.30, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {959, 843, 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.

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 843

\(\displaystyle \frac {(A b-a B) \left (\frac {x}{b}-\frac {a \int \frac {1}{b x^4+a}dx}{b}\right )}{b}+\frac {B x^5}{5 b}\)

\(\Big \downarrow \) 755

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

\(\displaystyle \frac {(A b-a B) \left (\frac {x}{b}-\frac {a \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{b}+\frac {B x^5}{5 b}\)

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(B*x^5)/(5*b) + ((A*b - a*B)*(x/b - (a*((-(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*S 
qrt[a])))/b))/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 843

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

rule 959

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

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 [C] (verified)

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

Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.33

method	result	size
risch	\(\frac {B \,x^{5}}{5 b}+\frac {A x}{b}-\frac {B a x}{b^{2}}+\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (-A b +B a \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{4 b^{3}}\)	\(60\)
default	\(\frac {\frac {1}{5} b B \,x^{5}+A b x -B a x}{b^{2}}-\frac {\left (A b -B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b^{2}}\)	\(132\)

Input:

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

Output:

1/5*B*x^5/b+1/b*A*x-1/b^2*B*a*x+1/4/b^3*a*sum((-A*b+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 = 0.09 (sec) , antiderivative size = 587, normalized size of antiderivative = 3.23 \[ \int \frac {x^4 \left (A+B x^4\right )}{a+b x^4} \, dx=\frac {4 \, B b x^{5} - 5 \, b^{2} \left (-\frac {B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac {1}{4}} \log \left (b^{2} \left (-\frac {B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} x\right ) - 5 i \, b^{2} \left (-\frac {B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac {1}{4}} \log \left (i \, b^{2} \left (-\frac {B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} x\right ) + 5 i \, b^{2} \left (-\frac {B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac {1}{4}} \log \left (-i \, b^{2} \left (-\frac {B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} x\right ) + 5 \, b^{2} \left (-\frac {B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac {1}{4}} \log \left (-b^{2} \left (-\frac {B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} x\right ) - 20 \, {\left (B a - A b\right )} x}{20 \, b^{2}} \] Input:

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

Output:

1/20*(4*B*b*x^5 - 5*b^2*(-(B^4*a^5 - 4*A*B^3*a^4*b + 6*A^2*B^2*a^3*b^2 - 4 
*A^3*B*a^2*b^3 + A^4*a*b^4)/b^9)^(1/4)*log(b^2*(-(B^4*a^5 - 4*A*B^3*a^4*b 
+ 6*A^2*B^2*a^3*b^2 - 4*A^3*B*a^2*b^3 + A^4*a*b^4)/b^9)^(1/4) - (B*a - A*b 
)*x) - 5*I*b^2*(-(B^4*a^5 - 4*A*B^3*a^4*b + 6*A^2*B^2*a^3*b^2 - 4*A^3*B*a^ 
2*b^3 + A^4*a*b^4)/b^9)^(1/4)*log(I*b^2*(-(B^4*a^5 - 4*A*B^3*a^4*b + 6*A^2 
*B^2*a^3*b^2 - 4*A^3*B*a^2*b^3 + A^4*a*b^4)/b^9)^(1/4) - (B*a - A*b)*x) + 
5*I*b^2*(-(B^4*a^5 - 4*A*B^3*a^4*b + 6*A^2*B^2*a^3*b^2 - 4*A^3*B*a^2*b^3 + 
 A^4*a*b^4)/b^9)^(1/4)*log(-I*b^2*(-(B^4*a^5 - 4*A*B^3*a^4*b + 6*A^2*B^2*a 
^3*b^2 - 4*A^3*B*a^2*b^3 + A^4*a*b^4)/b^9)^(1/4) - (B*a - A*b)*x) + 5*b^2* 
(-(B^4*a^5 - 4*A*B^3*a^4*b + 6*A^2*B^2*a^3*b^2 - 4*A^3*B*a^2*b^3 + A^4*a*b 
^4)/b^9)^(1/4)*log(-b^2*(-(B^4*a^5 - 4*A*B^3*a^4*b + 6*A^2*B^2*a^3*b^2 - 4 
*A^3*B*a^2*b^3 + A^4*a*b^4)/b^9)^(1/4) - (B*a - A*b)*x) - 20*(B*a - A*b)*x 
)/b^2
 

Sympy [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.56 \[ \int \frac {x^4 \left (A+B x^4\right )}{a+b x^4} \, dx=\frac {B x^{5}}{5 b} + x \left (\frac {A}{b} - \frac {B a}{b^{2}}\right ) + \operatorname {RootSum} {\left (256 t^{4} b^{9} + A^{4} a b^{4} - 4 A^{3} B a^{2} b^{3} + 6 A^{2} B^{2} a^{3} b^{2} - 4 A B^{3} a^{4} b + B^{4} a^{5}, \left ( t \mapsto t \log {\left (\frac {4 t b^{2}}{- A b + B a} + x \right )} \right )\right )} \] Input:

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

Output:

B*x**5/(5*b) + x*(A/b - B*a/b**2) + RootSum(256*_t**4*b**9 + A**4*a*b**4 - 
 4*A**3*B*a**2*b**3 + 6*A**2*B**2*a**3*b**2 - 4*A*B**3*a**4*b + B**4*a**5, 
 Lambda(_t, _t*log(4*_t*b**2/(-A*b + B*a) + x)))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.26 \[ \int \frac {x^4 \left (A+B x^4\right )}{a+b x^4} \, dx=\frac {{\left (\frac {2 \, \sqrt {2} {\left (B a - A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (B a - A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (B a - A b\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (B a - A b\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} a}{8 \, b^{2}} + \frac {B b x^{5} - 5 \, {\left (B a - A b\right )} x}{5 \, b^{2}} \] Input:

integrate(x^4*(B*x^4+A)/(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/b^2 + 1/5*(B*b*x^5 - 5*(B*a - A*b)*x)/b^2
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.41 \[ \int \frac {x^4 \left (A+B x^4\right )}{a+b x^4} \, dx=\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, b^{3}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, b^{3}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, b^{3}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, b^{3}} + \frac {B b^{4} x^{5} - 5 \, B a b^{3} x + 5 \, A b^{4} x}{5 \, b^{5}} \] Input:

integrate(x^4*(B*x^4+A)/(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))/b^3 + 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))/b^3 + 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))/b^3 - 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))/b^3 + 1/ 
5*(B*b^4*x^5 - 5*B*a*b^3*x + 5*A*b^4*x)/b^5
 

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 770, normalized size of antiderivative = 4.23 \[ \int \frac {x^4 \left (A+B x^4\right )}{a+b x^4} \, dx =\text {Too large to display} \] Input:

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

Output:

x*(A/b - (B*a)/b^2) + (B*x^5)/(5*b) - ((-a)^(1/4)*atan((((-a)^(1/4)*(A*b - 
 B*a)*((4*x*(B^2*a^4 + A^2*a^2*b^2 - 2*A*B*a^3*b))/b - ((-a)^(1/4)*(16*A*a 
^2*b^2 - 16*B*a^3*b)*(A*b - B*a))/(4*b^(9/4)))*1i)/(4*b^(9/4)) + ((-a)^(1/ 
4)*(A*b - B*a)*((4*x*(B^2*a^4 + A^2*a^2*b^2 - 2*A*B*a^3*b))/b + ((-a)^(1/4 
)*(16*A*a^2*b^2 - 16*B*a^3*b)*(A*b - B*a))/(4*b^(9/4)))*1i)/(4*b^(9/4)))/( 
((-a)^(1/4)*(A*b - B*a)*((4*x*(B^2*a^4 + A^2*a^2*b^2 - 2*A*B*a^3*b))/b - ( 
(-a)^(1/4)*(16*A*a^2*b^2 - 16*B*a^3*b)*(A*b - B*a))/(4*b^(9/4))))/(4*b^(9/ 
4)) - ((-a)^(1/4)*(A*b - B*a)*((4*x*(B^2*a^4 + A^2*a^2*b^2 - 2*A*B*a^3*b)) 
/b + ((-a)^(1/4)*(16*A*a^2*b^2 - 16*B*a^3*b)*(A*b - B*a))/(4*b^(9/4))))/(4 
*b^(9/4))))*(A*b - B*a)*1i)/(2*b^(9/4)) - ((-a)^(1/4)*atan((((-a)^(1/4)*(A 
*b - B*a)*((4*x*(B^2*a^4 + A^2*a^2*b^2 - 2*A*B*a^3*b))/b - ((-a)^(1/4)*(16 
*A*a^2*b^2 - 16*B*a^3*b)*(A*b - B*a)*1i)/(4*b^(9/4))))/(4*b^(9/4)) + ((-a) 
^(1/4)*(A*b - B*a)*((4*x*(B^2*a^4 + A^2*a^2*b^2 - 2*A*B*a^3*b))/b + ((-a)^ 
(1/4)*(16*A*a^2*b^2 - 16*B*a^3*b)*(A*b - B*a)*1i)/(4*b^(9/4))))/(4*b^(9/4) 
))/(((-a)^(1/4)*(A*b - B*a)*((4*x*(B^2*a^4 + A^2*a^2*b^2 - 2*A*B*a^3*b))/b 
 - ((-a)^(1/4)*(16*A*a^2*b^2 - 16*B*a^3*b)*(A*b - B*a)*1i)/(4*b^(9/4)))*1i 
)/(4*b^(9/4)) - ((-a)^(1/4)*(A*b - B*a)*((4*x*(B^2*a^4 + A^2*a^2*b^2 - 2*A 
*B*a^3*b))/b + ((-a)^(1/4)*(16*A*a^2*b^2 - 16*B*a^3*b)*(A*b - B*a)*1i)/(4* 
b^(9/4)))*1i)/(4*b^(9/4))))*(A*b - B*a))/(2*b^(9/4))
 

Reduce [B] (verification not implemented)

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

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

Output:

x**5/5