\(\int \frac {(a+b x^4)^2}{(c+d x^4)^3} \, dx\) [14]

Optimal result

Integrand size = 19, antiderivative size = 270 \[ \int \frac {\left (a+b x^4\right )^2}{\left (c+d x^4\right )^3} \, dx=\frac {(b c-a d)^2 x}{8 c d^2 \left (c+d x^4\right )^2}-\frac {(b c-a d) (9 b c+7 a d) x}{32 c^2 d^2 \left (c+d x^4\right )}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x}{\sqrt {c}+\sqrt {d} x^2}\right )}{64 \sqrt {2} c^{11/4} d^{9/4}} \] Output:

1/8*(-a*d+b*c)^2*x/c/d^2/(d*x^4+c)^2-1/32*(-a*d+b*c)*(7*a*d+9*b*c)*x/c^2/d 
^2/(d*x^4+c)+1/128*(21*a^2*d^2+6*a*b*c*d+5*b^2*c^2)*arctan(-1+2^(1/2)*d^(1 
/4)*x/c^(1/4))*2^(1/2)/c^(11/4)/d^(9/4)+1/128*(21*a^2*d^2+6*a*b*c*d+5*b^2* 
c^2)*arctan(1+2^(1/2)*d^(1/4)*x/c^(1/4))*2^(1/2)/c^(11/4)/d^(9/4)+1/128*(2 
1*a^2*d^2+6*a*b*c*d+5*b^2*c^2)*arctanh(2^(1/2)*c^(1/4)*d^(1/4)*x/(c^(1/2)+ 
d^(1/2)*x^2))*2^(1/2)/c^(11/4)/d^(9/4)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b x^4\right )^2}{\left (c+d x^4\right )^3} \, dx=\frac {\frac {32 c^{7/4} \sqrt [4]{d} (b c-a d)^2 x}{\left (c+d x^4\right )^2}-\frac {8 c^{3/4} \sqrt [4]{d} \left (9 b^2 c^2-2 a b c d-7 a^2 d^2\right ) x}{c+d x^4}-2 \sqrt {2} \left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+2 \sqrt {2} \left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )-\sqrt {2} \left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )+\sqrt {2} \left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{256 c^{11/4} d^{9/4}} \] Input:

Integrate[(a + b*x^4)^2/(c + d*x^4)^3,x]
 

Output:

((32*c^(7/4)*d^(1/4)*(b*c - a*d)^2*x)/(c + d*x^4)^2 - (8*c^(3/4)*d^(1/4)*( 
9*b^2*c^2 - 2*a*b*c*d - 7*a^2*d^2)*x)/(c + d*x^4) - 2*Sqrt[2]*(5*b^2*c^2 + 
 6*a*b*c*d + 21*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 2*Sqrt[ 
2]*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^( 
1/4)] - Sqrt[2]*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] - Sqrt[2] 
*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] + Sqrt[2]*(5*b^2*c^2 + 6*a*b*c*d + 21*a^ 
2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(256*c^(11/ 
4)*d^(9/4))
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {930, 910, 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.

\(\displaystyle \int \frac {\left (a+b x^4\right )^2}{\left (c+d x^4\right )^3} \, dx\)

\(\Big \downarrow \) 930

\(\displaystyle \frac {\int \frac {b (5 b c+3 a d) x^4+a (b c+7 a d)}{\left (d x^4+c\right )^2}dx}{8 c d}-\frac {x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2}\)

\(\Big \downarrow \) 910

\(\displaystyle \frac {\frac {1}{4} \left (\frac {21 a^2 d}{c}+6 a b+\frac {5 b^2 c}{d}\right ) \int \frac {1}{d x^4+c}dx-\frac {x \left (-\frac {7 a^2 d}{c}+2 a b+\frac {5 b^2 c}{d}\right )}{4 \left (c+d x^4\right )}}{8 c d}-\frac {x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {\frac {1}{4} \left (\frac {21 a^2 d}{c}+6 a b+\frac {5 b^2 c}{d}\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{d x^4+c}dx}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {d} x^2+\sqrt {c}}{d x^4+c}dx}{2 \sqrt {c}}\right )-\frac {x \left (-\frac {7 a^2 d}{c}+2 a b+\frac {5 b^2 c}{d}\right )}{4 \left (c+d x^4\right )}}{8 c d}-\frac {x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\frac {1}{4} \left (\frac {21 a^2 d}{c}+6 a b+\frac {5 b^2 c}{d}\right ) \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}dx}{2 \sqrt {d}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}dx}{2 \sqrt {d}}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{d x^4+c}dx}{2 \sqrt {c}}\right )-\frac {x \left (-\frac {7 a^2 d}{c}+2 a b+\frac {5 b^2 c}{d}\right )}{4 \left (c+d x^4\right )}}{8 c d}-\frac {x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {1}{4} \left (\frac {21 a^2 d}{c}+6 a b+\frac {5 b^2 c}{d}\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{d x^4+c}dx}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )-\frac {x \left (-\frac {7 a^2 d}{c}+2 a b+\frac {5 b^2 c}{d}\right )}{4 \left (c+d x^4\right )}}{8 c d}-\frac {x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {1}{4} \left (\frac {21 a^2 d}{c}+6 a b+\frac {5 b^2 c}{d}\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{d x^4+c}dx}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )-\frac {x \left (-\frac {7 a^2 d}{c}+2 a b+\frac {5 b^2 c}{d}\right )}{4 \left (c+d x^4\right )}}{8 c d}-\frac {x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\frac {1}{4} \left (\frac {21 a^2 d}{c}+6 a b+\frac {5 b^2 c}{d}\right ) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} x}{\sqrt [4]{d} \left (x^2-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} x+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x^2+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )-\frac {x \left (-\frac {7 a^2 d}{c}+2 a b+\frac {5 b^2 c}{d}\right )}{4 \left (c+d x^4\right )}}{8 c d}-\frac {x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{4} \left (\frac {21 a^2 d}{c}+6 a b+\frac {5 b^2 c}{d}\right ) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} x}{\sqrt [4]{d} \left (x^2-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} x+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x^2+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )-\frac {x \left (-\frac {7 a^2 d}{c}+2 a b+\frac {5 b^2 c}{d}\right )}{4 \left (c+d x^4\right )}}{8 c d}-\frac {x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{4} \left (\frac {21 a^2 d}{c}+6 a b+\frac {5 b^2 c}{d}\right ) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} x}{x^2-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} x+\sqrt [4]{c}}{x^2+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}dx}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )-\frac {x \left (-\frac {7 a^2 d}{c}+2 a b+\frac {5 b^2 c}{d}\right )}{4 \left (c+d x^4\right )}}{8 c d}-\frac {x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {1}{4} \left (\frac {21 a^2 d}{c}+6 a b+\frac {5 b^2 c}{d}\right ) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )-\frac {x \left (-\frac {7 a^2 d}{c}+2 a b+\frac {5 b^2 c}{d}\right )}{4 \left (c+d x^4\right )}}{8 c d}-\frac {x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2}\)

Input:

Int[(a + b*x^4)^2/(c + d*x^4)^3,x]
 

Output:

-1/8*((b*c - a*d)*x*(a + b*x^4))/(c*d*(c + d*x^4)^2) + (-1/4*((2*a*b + (5* 
b^2*c)/d - (7*a^2*d)/c)*x)/(c + d*x^4) + ((6*a*b + (5*b^2*c)/d + (21*a^2*d 
)/c)*((-(ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)]/(Sqrt[2]*c^(1/4)*d^(1/4)) 
) + ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)]/(Sqrt[2]*c^(1/4)*d^(1/4)))/(2* 
Sqrt[c]) + (-1/2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2]/(S 
qrt[2]*c^(1/4)*d^(1/4)) + Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d 
]*x^2]/(2*Sqrt[2]*c^(1/4)*d^(1/4)))/(2*Sqrt[c])))/4)/(8*c*d)
 

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 910

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

rule 930

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

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 = 0.90 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.49

method	result	size
risch	\(\frac {\frac {\left (7 a^{2} d^{2}+2 a b c d -9 b^{2} c^{2}\right ) x^{5}}{32 c^{2} d}+\frac {\left (11 a^{2} d^{2}-6 a b c d -5 b^{2} c^{2}\right ) x}{32 c \,d^{2}}}{\left (d \,x^{4}+c \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+c \right )}{\sum }\frac {\left (21 a^{2} d^{2}+6 a b c d +5 b^{2} c^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 c^{2} d^{3}}\)	\(131\)
default	\(\frac {\frac {\left (7 a^{2} d^{2}+2 a b c d -9 b^{2} c^{2}\right ) x^{5}}{32 c^{2} d}+\frac {\left (11 a^{2} d^{2}-6 a b c d -5 b^{2} c^{2}\right ) x}{32 c \,d^{2}}}{\left (d \,x^{4}+c \right )^{2}}+\frac {\left (21 a^{2} d^{2}+6 a b c d +5 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {c}{d}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {c}{d}}}{x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{3} d^{2}}\)	\(206\)

Input:

int((b*x^4+a)^2/(d*x^4+c)^3,x,method=_RETURNVERBOSE)
 

Output:

(1/32*(7*a^2*d^2+2*a*b*c*d-9*b^2*c^2)/c^2/d*x^5+1/32*(11*a^2*d^2-6*a*b*c*d 
-5*b^2*c^2)/c/d^2*x)/(d*x^4+c)^2+1/128/c^2/d^3*sum((21*a^2*d^2+6*a*b*c*d+5 
*b^2*c^2)/_R^3*ln(x-_R),_R=RootOf(_Z^4*d+c))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 1294, normalized size of antiderivative = 4.79 \[ \int \frac {\left (a+b x^4\right )^2}{\left (c+d x^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((b*x^4+a)^2/(d*x^4+c)^3,x, algorithm="fricas")
 

Output:

-1/128*(4*(9*b^2*c^2*d - 2*a*b*c*d^2 - 7*a^2*d^3)*x^5 - (c^2*d^4*x^8 + 2*c 
^3*d^3*x^4 + c^4*d^2)*(-(625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^ 
6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^ 
3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(c^1 
1*d^9))^(1/4)*log(c^3*d^2*(-(625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^ 
6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^ 
3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/ 
(c^11*d^9))^(1/4) + (5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*x) + (-I*c^2*d^4* 
x^8 - 2*I*c^3*d^3*x^4 - I*c^4*d^2)*(-(625*b^8*c^8 + 3000*a*b^7*c^7*d + 159 
00*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 1769 
04*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481* 
a^8*d^8)/(c^11*d^9))^(1/4)*log(I*c^3*d^2*(-(625*b^8*c^8 + 3000*a*b^7*c^7*d 
 + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 
+ 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 1 
94481*a^8*d^8)/(c^11*d^9))^(1/4) + (5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*x) 
 + (I*c^2*d^4*x^8 + 2*I*c^3*d^3*x^4 + I*c^4*d^2)*(-(625*b^8*c^8 + 3000*a*b 
^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4* 
c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c 
*d^7 + 194481*a^8*d^8)/(c^11*d^9))^(1/4)*log(-I*c^3*d^2*(-(625*b^8*c^8 + 3 
000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 11280...
 

Sympy [A] (verification not implemented)

Time = 3.32 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^4\right )^2}{\left (c+d x^4\right )^3} \, dx=\frac {x^{5} \cdot \left (7 a^{2} d^{3} + 2 a b c d^{2} - 9 b^{2} c^{2} d\right ) + x \left (11 a^{2} c d^{2} - 6 a b c^{2} d - 5 b^{2} c^{3}\right )}{32 c^{4} d^{2} + 64 c^{3} d^{3} x^{4} + 32 c^{2} d^{4} x^{8}} + \operatorname {RootSum} {\left (268435456 t^{4} c^{11} d^{9} + 194481 a^{8} d^{8} + 222264 a^{7} b c d^{7} + 280476 a^{6} b^{2} c^{2} d^{6} + 176904 a^{5} b^{3} c^{3} d^{5} + 112806 a^{4} b^{4} c^{4} d^{4} + 42120 a^{3} b^{5} c^{5} d^{3} + 15900 a^{2} b^{6} c^{6} d^{2} + 3000 a b^{7} c^{7} d + 625 b^{8} c^{8}, \left ( t \mapsto t \log {\left (\frac {128 t c^{3} d^{2}}{21 a^{2} d^{2} + 6 a b c d + 5 b^{2} c^{2}} + x \right )} \right )\right )} \] Input:

integrate((b*x**4+a)**2/(d*x**4+c)**3,x)
                                                                                    
                                                                                    
 

Output:

(x**5*(7*a**2*d**3 + 2*a*b*c*d**2 - 9*b**2*c**2*d) + x*(11*a**2*c*d**2 - 6 
*a*b*c**2*d - 5*b**2*c**3))/(32*c**4*d**2 + 64*c**3*d**3*x**4 + 32*c**2*d* 
*4*x**8) + RootSum(268435456*_t**4*c**11*d**9 + 194481*a**8*d**8 + 222264* 
a**7*b*c*d**7 + 280476*a**6*b**2*c**2*d**6 + 176904*a**5*b**3*c**3*d**5 + 
112806*a**4*b**4*c**4*d**4 + 42120*a**3*b**5*c**5*d**3 + 15900*a**2*b**6*c 
**6*d**2 + 3000*a*b**7*c**7*d + 625*b**8*c**8, Lambda(_t, _t*log(128*_t*c* 
*3*d**2/(21*a**2*d**2 + 6*a*b*c*d + 5*b**2*c**2) + x)))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b x^4\right )^2}{\left (c+d x^4\right )^3} \, dx=-\frac {{\left (9 \, b^{2} c^{2} d - 2 \, a b c d^{2} - 7 \, a^{2} d^{3}\right )} x^{5} + {\left (5 \, b^{2} c^{3} + 6 \, a b c^{2} d - 11 \, a^{2} c d^{2}\right )} x}{32 \, {\left (c^{2} d^{4} x^{8} + 2 \, c^{3} d^{3} x^{4} + c^{4} d^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 21 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 21 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 21 \, a^{2} d^{2}\right )} \log \left (\sqrt {d} x^{2} + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 21 \, a^{2} d^{2}\right )} \log \left (\sqrt {d} x^{2} - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{256 \, c^{2} d^{2}} \] Input:

integrate((b*x^4+a)^2/(d*x^4+c)^3,x, algorithm="maxima")
 

Output:

-1/32*((9*b^2*c^2*d - 2*a*b*c*d^2 - 7*a^2*d^3)*x^5 + (5*b^2*c^3 + 6*a*b*c^ 
2*d - 11*a^2*c*d^2)*x)/(c^2*d^4*x^8 + 2*c^3*d^3*x^4 + c^4*d^2) + 1/256*(2* 
sqrt(2)*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*arctan(1/2*sqrt(2)*(2*sqrt(d) 
*x + sqrt(2)*c^(1/4)*d^(1/4))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c) 
*sqrt(d))) + 2*sqrt(2)*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*arctan(1/2*sqr 
t(2)*(2*sqrt(d)*x - sqrt(2)*c^(1/4)*d^(1/4))/sqrt(sqrt(c)*sqrt(d)))/(sqrt( 
c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*l 
og(sqrt(d)*x^2 + sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - 
sqrt(2)*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*log(sqrt(d)*x^2 - sqrt(2)*c^( 
1/4)*d^(1/4)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/(c^2*d^2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a+b x^4\right )^2}{\left (c+d x^4\right )^3} \, dx=\frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{128 \, c^{3} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{128 \, c^{3} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{256 \, c^{3} d^{3}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{256 \, c^{3} d^{3}} - \frac {9 \, b^{2} c^{2} d x^{5} - 2 \, a b c d^{2} x^{5} - 7 \, a^{2} d^{3} x^{5} + 5 \, b^{2} c^{3} x + 6 \, a b c^{2} d x - 11 \, a^{2} c d^{2} x}{32 \, {\left (d x^{4} + c\right )}^{2} c^{2} d^{2}} \] Input:

integrate((b*x^4+a)^2/(d*x^4+c)^3,x, algorithm="giac")
 

Output:

1/128*sqrt(2)*(5*(c*d^3)^(1/4)*b^2*c^2 + 6*(c*d^3)^(1/4)*a*b*c*d + 21*(c*d 
^3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(c/d)^(1/4))/(c/d)^(1 
/4))/(c^3*d^3) + 1/128*sqrt(2)*(5*(c*d^3)^(1/4)*b^2*c^2 + 6*(c*d^3)^(1/4)* 
a*b*c*d + 21*(c*d^3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(c/d 
)^(1/4))/(c/d)^(1/4))/(c^3*d^3) + 1/256*sqrt(2)*(5*(c*d^3)^(1/4)*b^2*c^2 + 
 6*(c*d^3)^(1/4)*a*b*c*d + 21*(c*d^3)^(1/4)*a^2*d^2)*log(x^2 + sqrt(2)*x*( 
c/d)^(1/4) + sqrt(c/d))/(c^3*d^3) - 1/256*sqrt(2)*(5*(c*d^3)^(1/4)*b^2*c^2 
 + 6*(c*d^3)^(1/4)*a*b*c*d + 21*(c*d^3)^(1/4)*a^2*d^2)*log(x^2 - sqrt(2)*x 
*(c/d)^(1/4) + sqrt(c/d))/(c^3*d^3) - 1/32*(9*b^2*c^2*d*x^5 - 2*a*b*c*d^2* 
x^5 - 7*a^2*d^3*x^5 + 5*b^2*c^3*x + 6*a*b*c^2*d*x - 11*a^2*c*d^2*x)/((d*x^ 
4 + c)^2*c^2*d^2)
 

Mupad [B] (verification not implemented)

Time = 1.01 (sec) , antiderivative size = 1401, normalized size of antiderivative = 5.19 \[ \int \frac {\left (a+b x^4\right )^2}{\left (c+d x^4\right )^3} \, dx=\text {Too large to display} \] Input:

int((a + b*x^4)^2/(c + d*x^4)^3,x)
 

Output:

- ((x*(5*b^2*c^2 - 11*a^2*d^2 + 6*a*b*c*d))/(32*c*d^2) - (x^5*(7*a^2*d^2 - 
 9*b^2*c^2 + 2*a*b*c*d))/(32*c^2*d))/(c^2 + d^2*x^8 + 2*c*d*x^4) - (atan(( 
((((21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d)*(21*a^2*d^3 + 5*b^2*c^2*d + 6*a*b* 
c*d^2))/(256*(-c)^(15/4)*d^(9/4)) - (x*(441*a^4*d^4 + 25*b^4*c^4 + 246*a^2 
*b^2*c^2*d^2 + 60*a*b^3*c^3*d + 252*a^3*b*c*d^3))/(256*c^4*d))*(21*a^2*d^2 
 + 5*b^2*c^2 + 6*a*b*c*d)*1i)/(128*(-c)^(11/4)*d^(9/4)) - ((((21*a^2*d^2 + 
 5*b^2*c^2 + 6*a*b*c*d)*(21*a^2*d^3 + 5*b^2*c^2*d + 6*a*b*c*d^2))/(256*(-c 
)^(15/4)*d^(9/4)) + (x*(441*a^4*d^4 + 25*b^4*c^4 + 246*a^2*b^2*c^2*d^2 + 6 
0*a*b^3*c^3*d + 252*a^3*b*c*d^3))/(256*c^4*d))*(21*a^2*d^2 + 5*b^2*c^2 + 6 
*a*b*c*d)*1i)/(128*(-c)^(11/4)*d^(9/4)))/(((((21*a^2*d^2 + 5*b^2*c^2 + 6*a 
*b*c*d)*(21*a^2*d^3 + 5*b^2*c^2*d + 6*a*b*c*d^2))/(256*(-c)^(15/4)*d^(9/4) 
) - (x*(441*a^4*d^4 + 25*b^4*c^4 + 246*a^2*b^2*c^2*d^2 + 60*a*b^3*c^3*d + 
252*a^3*b*c*d^3))/(256*c^4*d))*(21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d))/(128* 
(-c)^(11/4)*d^(9/4)) + ((((21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d)*(21*a^2*d^3 
 + 5*b^2*c^2*d + 6*a*b*c*d^2))/(256*(-c)^(15/4)*d^(9/4)) + (x*(441*a^4*d^4 
 + 25*b^4*c^4 + 246*a^2*b^2*c^2*d^2 + 60*a*b^3*c^3*d + 252*a^3*b*c*d^3))/( 
256*c^4*d))*(21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d))/(128*(-c)^(11/4)*d^(9/4) 
)))*(21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d)*1i)/(64*(-c)^(11/4)*d^(9/4)) - (a 
tan((((((21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d)*(21*a^2*d^3 + 5*b^2*c^2*d + 6 
*a*b*c*d^2)*1i)/(256*(-c)^(15/4)*d^(9/4)) - (x*(441*a^4*d^4 + 25*b^4*c^...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1484, normalized size of antiderivative = 5.50 \[ \int \frac {\left (a+b x^4\right )^2}{\left (c+d x^4\right )^3} \, dx =\text {Too large to display} \] Input:

int((b*x^4+a)^2/(d*x^4+c)^3,x)
 

Output:

( - 42*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt( 
d)*x)/(d**(1/4)*c**(1/4)*sqrt(2)))*a**2*c**2*d**2 - 84*d**(3/4)*c**(1/4)*s 
qrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(d)*x)/(d**(1/4)*c**(1/4)*s 
qrt(2)))*a**2*c*d**3*x**4 - 42*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c* 
*(1/4)*sqrt(2) - 2*sqrt(d)*x)/(d**(1/4)*c**(1/4)*sqrt(2)))*a**2*d**4*x**8 
- 12*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(d) 
*x)/(d**(1/4)*c**(1/4)*sqrt(2)))*a*b*c**3*d - 24*d**(3/4)*c**(1/4)*sqrt(2) 
*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(d)*x)/(d**(1/4)*c**(1/4)*sqrt(2) 
))*a*b*c**2*d**2*x**4 - 12*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/ 
4)*sqrt(2) - 2*sqrt(d)*x)/(d**(1/4)*c**(1/4)*sqrt(2)))*a*b*c*d**3*x**8 - 1 
0*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(d)*x) 
/(d**(1/4)*c**(1/4)*sqrt(2)))*b**2*c**4 - 20*d**(3/4)*c**(1/4)*sqrt(2)*ata 
n((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(d)*x)/(d**(1/4)*c**(1/4)*sqrt(2)))*b 
**2*c**3*d*x**4 - 10*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqr 
t(2) - 2*sqrt(d)*x)/(d**(1/4)*c**(1/4)*sqrt(2)))*b**2*c**2*d**2*x**8 + 42* 
d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) + 2*sqrt(d)*x)/( 
d**(1/4)*c**(1/4)*sqrt(2)))*a**2*c**2*d**2 + 84*d**(3/4)*c**(1/4)*sqrt(2)* 
atan((d**(1/4)*c**(1/4)*sqrt(2) + 2*sqrt(d)*x)/(d**(1/4)*c**(1/4)*sqrt(2)) 
)*a**2*c*d**3*x**4 + 42*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)* 
sqrt(2) + 2*sqrt(d)*x)/(d**(1/4)*c**(1/4)*sqrt(2)))*a**2*d**4*x**8 + 12...