\(\int \frac {A+B x^4}{x^6 (a+b x^4)^2} \, dx\) [24]

Optimal result

Integrand size = 20, antiderivative size = 216 \[ \int \frac {A+B x^4}{x^6 \left (a+b x^4\right )^2} \, dx=-\frac {A}{5 a^2 x^5}+\frac {2 A b-a B}{a^3 x}+\frac {b (A b-a B) x^3}{4 a^3 \left (a+b x^4\right )}-\frac {\sqrt [4]{b} (9 A b-5 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{13/4}}+\frac {\sqrt [4]{b} (9 A b-5 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{13/4}}-\frac {\sqrt [4]{b} (9 A b-5 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a}+\sqrt {b} x^2}\right )}{8 \sqrt {2} a^{13/4}} \] Output:

-1/5*A/a^2/x^5+(2*A*b-B*a)/a^3/x+1/4*b*(A*b-B*a)*x^3/a^3/(b*x^4+a)+1/16*b^ 
(1/4)*(9*A*b-5*B*a)*arctan(-1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/a^(13/4)+ 
1/16*b^(1/4)*(9*A*b-5*B*a)*arctan(1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/a^( 
13/4)-1/16*b^(1/4)*(9*A*b-5*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^(13/4)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.20 \[ \int \frac {A+B x^4}{x^6 \left (a+b x^4\right )^2} \, dx=\frac {-\frac {32 a^{5/4} A}{x^5}-\frac {160 \sqrt [4]{a} (-2 A b+a B)}{x}-\frac {40 \sqrt [4]{a} b (-A b+a B) x^3}{a+b x^4}-10 \sqrt {2} \sqrt [4]{b} (9 A b-5 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+10 \sqrt {2} \sqrt [4]{b} (9 A b-5 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+5 \sqrt {2} \sqrt [4]{b} (9 A b-5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+5 \sqrt {2} \sqrt [4]{b} (-9 A b+5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{160 a^{13/4}} \] Input:

Integrate[(A + B*x^4)/(x^6*(a + b*x^4)^2),x]
 

Output:

((-32*a^(5/4)*A)/x^5 - (160*a^(1/4)*(-2*A*b + a*B))/x - (40*a^(1/4)*b*(-(A 
*b) + a*B)*x^3)/(a + b*x^4) - 10*Sqrt[2]*b^(1/4)*(9*A*b - 5*a*B)*ArcTan[1 
- (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 10*Sqrt[2]*b^(1/4)*(9*A*b - 5*a*B)*ArcTan 
[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 5*Sqrt[2]*b^(1/4)*(9*A*b - 5*a*B)*Log[ 
Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 5*Sqrt[2]*b^(1/4)*(-9 
*A*b + 5*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(160 
*a^(13/4))
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.31, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {957, 847, 847, 826, 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.

\(\displaystyle \int \frac {A+B x^4}{x^6 \left (a+b x^4\right )^2} \, dx\)

\(\Big \downarrow \) 957

\(\displaystyle \frac {(9 A b-5 a B) \int \frac {1}{x^6 \left (b x^4+a\right )}dx}{4 a b}+\frac {A b-a B}{4 a b x^5 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \int \frac {1}{x^2 \left (b x^4+a\right )}dx}{a}-\frac {1}{5 a x^5}\right )}{4 a b}+\frac {A b-a B}{4 a b x^5 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {b \int \frac {x^2}{b x^4+a}dx}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{5 a x^5}\right )}{4 a b}+\frac {A b-a B}{4 a b x^5 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (\frac {\int \frac {\sqrt {b} x^2+\sqrt {a}}{b x^4+a}dx}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {b}}\right )}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{5 a x^5}\right )}{4 a b}+\frac {A b-a B}{4 a b x^5 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {b}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {b}}\right )}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{5 a x^5}\right )}{4 a b}+\frac {A b-a B}{4 a b x^5 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {b}}\right )}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{5 a x^5}\right )}{4 a b}+\frac {A b-a B}{4 a b x^5 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {b}}\right )}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{5 a x^5}\right )}{4 a b}+\frac {A b-a B}{4 a b x^5 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{\sqrt [4]{b} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{5 a x^5}\right )}{4 a b}+\frac {A b-a B}{4 a b x^5 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{\sqrt [4]{b} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{5 a x^5}\right )}{4 a b}+\frac {A b-a B}{4 a b x^5 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {b}}\right )}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{5 a x^5}\right )}{4 a b}+\frac {A b-a B}{4 a b x^5 \left (a+b x^4\right )}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{5 a x^5}\right )}{4 a b}+\frac {A b-a B}{4 a b x^5 \left (a+b x^4\right )}\)

Input:

Int[(A + B*x^4)/(x^6*(a + b*x^4)^2),x]
 

Output:

(A*b - a*B)/(4*a*b*x^5*(a + b*x^4)) + ((9*A*b - 5*a*B)*(-1/5*1/(a*x^5) - ( 
b*(-(1/(a*x)) - (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[b]) - (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + S 
qrt[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[b])))/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 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   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 = 165, normalized size of antiderivative = 0.76

method	result	size
default	\(\frac {b \left (\frac {\left (\frac {A b}{4}-\frac {B a}{4}\right ) x^{3}}{b \,x^{4}+a}+\frac {\left (\frac {9 A b}{4}-\frac {5 B a}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{3}}-\frac {A}{5 a^{2} x^{5}}-\frac {-2 A b +B a}{a^{3} x}\)	\(165\)
risch	\(\frac {\frac {b \left (9 A b -5 B a \right ) x^{8}}{4 a^{3}}+\frac {\left (9 A b -5 B a \right ) x^{4}}{5 a^{2}}-\frac {A}{5 a}}{x^{5} \left (b \,x^{4}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{13} \textit {\_Z}^{4}+6561 A^{4} b^{5}-14580 A^{3} B a \,b^{4}+12150 A^{2} B^{2} a^{2} b^{3}-4500 A \,B^{3} a^{3} b^{2}+625 B^{4} a^{4} b \right )}{\sum }\textit {\_R} \ln \left (\left (5 \textit {\_R}^{4} a^{13}+26244 A^{4} b^{5}-58320 A^{3} B a \,b^{4}+48600 A^{2} B^{2} a^{2} b^{3}-18000 A \,B^{3} a^{3} b^{2}+2500 B^{4} a^{4} b \right ) x +\left (-9 A \,a^{10} b +5 B \,a^{11}\right ) \textit {\_R}^{3}\right )\right )}{16}\)	\(210\)

Input:

int((B*x^4+A)/x^6/(b*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/a^3*b*((1/4*A*b-1/4*B*a)*x^3/(b*x^4+a)+1/8*(9/4*A*b-5/4*B*a)/b/(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)))-1/5*A/a^2/x^5-(-2*A*b+B*a)/a^3/x
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 824, normalized size of antiderivative = 3.81 \[ \int \frac {A+B x^4}{x^6 \left (a+b x^4\right )^2} \, dx =\text {Too large to display} \] Input:

integrate((B*x^4+A)/x^6/(b*x^4+a)^2,x, algorithm="fricas")
 

Output:

-1/80*(20*(5*B*a*b - 9*A*b^2)*x^8 + 16*(5*B*a^2 - 9*A*a*b)*x^4 + 16*A*a^2 
- 5*(a^3*b*x^9 + a^4*x^5)*(-(625*B^4*a^4*b - 4500*A*B^3*a^3*b^2 + 12150*A^ 
2*B^2*a^2*b^3 - 14580*A^3*B*a*b^4 + 6561*A^4*b^5)/a^13)^(1/4)*log(a^10*(-( 
625*B^4*a^4*b - 4500*A*B^3*a^3*b^2 + 12150*A^2*B^2*a^2*b^3 - 14580*A^3*B*a 
*b^4 + 6561*A^4*b^5)/a^13)^(3/4) - (125*B^3*a^3*b - 675*A*B^2*a^2*b^2 + 12 
15*A^2*B*a*b^3 - 729*A^3*b^4)*x) + 5*(I*a^3*b*x^9 + I*a^4*x^5)*(-(625*B^4* 
a^4*b - 4500*A*B^3*a^3*b^2 + 12150*A^2*B^2*a^2*b^3 - 14580*A^3*B*a*b^4 + 6 
561*A^4*b^5)/a^13)^(1/4)*log(I*a^10*(-(625*B^4*a^4*b - 4500*A*B^3*a^3*b^2 
+ 12150*A^2*B^2*a^2*b^3 - 14580*A^3*B*a*b^4 + 6561*A^4*b^5)/a^13)^(3/4) - 
(125*B^3*a^3*b - 675*A*B^2*a^2*b^2 + 1215*A^2*B*a*b^3 - 729*A^3*b^4)*x) + 
5*(-I*a^3*b*x^9 - I*a^4*x^5)*(-(625*B^4*a^4*b - 4500*A*B^3*a^3*b^2 + 12150 
*A^2*B^2*a^2*b^3 - 14580*A^3*B*a*b^4 + 6561*A^4*b^5)/a^13)^(1/4)*log(-I*a^ 
10*(-(625*B^4*a^4*b - 4500*A*B^3*a^3*b^2 + 12150*A^2*B^2*a^2*b^3 - 14580*A 
^3*B*a*b^4 + 6561*A^4*b^5)/a^13)^(3/4) - (125*B^3*a^3*b - 675*A*B^2*a^2*b^ 
2 + 1215*A^2*B*a*b^3 - 729*A^3*b^4)*x) + 5*(a^3*b*x^9 + a^4*x^5)*(-(625*B^ 
4*a^4*b - 4500*A*B^3*a^3*b^2 + 12150*A^2*B^2*a^2*b^3 - 14580*A^3*B*a*b^4 + 
 6561*A^4*b^5)/a^13)^(1/4)*log(-a^10*(-(625*B^4*a^4*b - 4500*A*B^3*a^3*b^2 
 + 12150*A^2*B^2*a^2*b^3 - 14580*A^3*B*a*b^4 + 6561*A^4*b^5)/a^13)^(3/4) - 
 (125*B^3*a^3*b - 675*A*B^2*a^2*b^2 + 1215*A^2*B*a*b^3 - 729*A^3*b^4)*x))/ 
(a^3*b*x^9 + a^4*x^5)
 

Sympy [A] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.85 \[ \int \frac {A+B x^4}{x^6 \left (a+b x^4\right )^2} \, dx=\operatorname {RootSum} {\left (65536 t^{4} a^{13} + 6561 A^{4} b^{5} - 14580 A^{3} B a b^{4} + 12150 A^{2} B^{2} a^{2} b^{3} - 4500 A B^{3} a^{3} b^{2} + 625 B^{4} a^{4} b, \left ( t \mapsto t \log {\left (- \frac {4096 t^{3} a^{10}}{- 729 A^{3} b^{4} + 1215 A^{2} B a b^{3} - 675 A B^{2} a^{2} b^{2} + 125 B^{3} a^{3} b} + x \right )} \right )\right )} + \frac {- 4 A a^{2} + x^{8} \cdot \left (45 A b^{2} - 25 B a b\right ) + x^{4} \cdot \left (36 A a b - 20 B a^{2}\right )}{20 a^{4} x^{5} + 20 a^{3} b x^{9}} \] Input:

integrate((B*x**4+A)/x**6/(b*x**4+a)**2,x)
 

Output:

RootSum(65536*_t**4*a**13 + 6561*A**4*b**5 - 14580*A**3*B*a*b**4 + 12150*A 
**2*B**2*a**2*b**3 - 4500*A*B**3*a**3*b**2 + 625*B**4*a**4*b, Lambda(_t, _ 
t*log(-4096*_t**3*a**10/(-729*A**3*b**4 + 1215*A**2*B*a*b**3 - 675*A*B**2* 
a**2*b**2 + 125*B**3*a**3*b) + x))) + (-4*A*a**2 + x**8*(45*A*b**2 - 25*B* 
a*b) + x**4*(36*A*a*b - 20*B*a**2))/(20*a**4*x**5 + 20*a**3*b*x**9)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x^4}{x^6 \left (a+b x^4\right )^2} \, dx=-\frac {5 \, {\left (5 \, B a b - 9 \, A b^{2}\right )} x^{8} + 4 \, {\left (5 \, B a^{2} - 9 \, A a b\right )} x^{4} + 4 \, A a^{2}}{20 \, {\left (a^{3} b x^{9} + a^{4} x^{5}\right )}} - \frac {{\left (5 \, B a b - 9 \, A b^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{32 \, a^{3}} \] Input:

integrate((B*x^4+A)/x^6/(b*x^4+a)^2,x, algorithm="maxima")
 

Output:

-1/20*(5*(5*B*a*b - 9*A*b^2)*x^8 + 4*(5*B*a^2 - 9*A*a*b)*x^4 + 4*A*a^2)/(a 
^3*b*x^9 + a^4*x^5) - 1/32*(5*B*a*b - 9*A*b^2)*(2*sqrt(2)*arctan(1/2*sqrt( 
2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sq 
rt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt 
(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b) 
) - sqrt(2)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(1/4 
)*b^(3/4)) + sqrt(2)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a) 
)/(a^(1/4)*b^(3/4)))/a^3
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.39 \[ \int \frac {A+B x^4}{x^6 \left (a+b x^4\right )^2} \, dx=-\frac {B a b x^{3} - A b^{2} x^{3}}{4 \, {\left (b x^{4} + a\right )} a^{3}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{4} b^{2}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{4} b^{2}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{4} b^{2}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{4} b^{2}} - \frac {5 \, B a x^{4} - 10 \, A b x^{4} + A a}{5 \, a^{3} x^{5}} \] Input:

integrate((B*x^4+A)/x^6/(b*x^4+a)^2,x, algorithm="giac")
 

Output:

-1/4*(B*a*b*x^3 - A*b^2*x^3)/((b*x^4 + a)*a^3) - 1/16*sqrt(2)*(5*(a*b^3)^( 
3/4)*B*a - 9*(a*b^3)^(3/4)*A*b)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1 
/4))/(a/b)^(1/4))/(a^4*b^2) - 1/16*sqrt(2)*(5*(a*b^3)^(3/4)*B*a - 9*(a*b^3 
)^(3/4)*A*b)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/( 
a^4*b^2) + 1/32*sqrt(2)*(5*(a*b^3)^(3/4)*B*a - 9*(a*b^3)^(3/4)*A*b)*log(x^ 
2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^4*b^2) - 1/32*sqrt(2)*(5*(a*b^3) 
^(3/4)*B*a - 9*(a*b^3)^(3/4)*A*b)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a 
/b))/(a^4*b^2) - 1/5*(5*B*a*x^4 - 10*A*b*x^4 + A*a)/(a^3*x^5)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 3.55 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.54 \[ \int \frac {A+B x^4}{x^6 \left (a+b x^4\right )^2} \, dx=\frac {\frac {x^4\,\left (9\,A\,b-5\,B\,a\right )}{5\,a^2}-\frac {A}{5\,a}+\frac {b\,x^8\,\left (9\,A\,b-5\,B\,a\right )}{4\,a^3}}{b\,x^9+a\,x^5}+\frac {{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,x}{a^{1/4}}\right )\,\left (9\,A\,b-5\,B\,a\right )}{8\,a^{13/4}}-\frac {{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,x}{a^{1/4}}\right )\,\left (9\,A\,b-5\,B\,a\right )}{8\,a^{13/4}} \] Input:

int((A + B*x^4)/(x^6*(a + b*x^4)^2),x)
 

Output:

((x^4*(9*A*b - 5*B*a))/(5*a^2) - A/(5*a) + (b*x^8*(9*A*b - 5*B*a))/(4*a^3) 
)/(a*x^5 + b*x^9) + ((-b)^(1/4)*atan(((-b)^(1/4)*x)/a^(1/4))*(9*A*b - 5*B* 
a))/(8*a^(13/4)) - ((-b)^(1/4)*atanh(((-b)^(1/4)*x)/a^(1/4))*(9*A*b - 5*B* 
a))/(8*a^(13/4))
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.78 \[ \int \frac {A+B x^4}{x^6 \left (a+b x^4\right )^2} \, dx=\frac {-10 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{5}+10 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{5}+5 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {b}\, x^{2}\right ) x^{5}-5 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {b}\, x^{2}\right ) x^{5}-8 a^{2}+40 a b \,x^{4}}{40 a^{3} x^{5}} \] Input:

int((B*x^4+A)/x^6/(b*x^4+a)^2,x)
 

Output:

( - 10*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt( 
b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*b*x**5 + 10*b**(1/4)*a**(3/4)*sqrt(2)*a 
tan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2))) 
*b*x**5 + 5*b**(1/4)*a**(3/4)*sqrt(2)*log( - b**(1/4)*a**(1/4)*sqrt(2)*x + 
 sqrt(a) + sqrt(b)*x**2)*b*x**5 - 5*b**(1/4)*a**(3/4)*sqrt(2)*log(b**(1/4) 
*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(b)*x**2)*b*x**5 - 8*a**2 + 40*a*b*x** 
4)/(40*a**3*x**5)