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

Optimal result

Integrand size = 29, antiderivative size = 375 \[ \int \frac {x^2 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {(A b-a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4} (b c-a d)}+\frac {(A b-a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4} (b c-a d)}-\frac {(B c-A d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)}+\frac {(B c-A d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)}-\frac {(A b-a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a}+\sqrt {b} x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4} (b c-a d)}-\frac {(B c-A d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x}{\sqrt {c}+\sqrt {d} x^2}\right )}{2 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.10 \[ \int \frac {x^2 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {-2 (A b-a B) \sqrt [4]{c} d^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 (A b-a B) \sqrt [4]{c} d^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )-2 \sqrt [4]{a} b^{3/4} (B c-A d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+2 \sqrt [4]{a} b^{3/4} (B c-A d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+(A b-a B) \sqrt [4]{c} d^{3/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )-(A b-a B) \sqrt [4]{c} d^{3/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+\sqrt [4]{a} b^{3/4} (B c-A d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )-\sqrt [4]{a} b^{3/4} (B c-A d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} b^{3/4} \sqrt [4]{c} d^{3/4} (b c-a d)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.37, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1054, 2009}

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.

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

\(\Big \downarrow \) 1054

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*((A*b - a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(Sqrt[2]*a^(1/4 
)*b^(3/4)*(b*c - a*d)) + ((A*b - a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/ 
4)])/(2*Sqrt[2]*a^(1/4)*b^(3/4)*(b*c - a*d)) - ((B*c - A*d)*ArcTan[1 - (Sq 
rt[2]*d^(1/4)*x)/c^(1/4)])/(2*Sqrt[2]*c^(1/4)*d^(3/4)*(b*c - a*d)) + ((B*c 
 - A*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(2*Sqrt[2]*c^(1/4)*d^(3/4 
)*(b*c - a*d)) + ((A*b - a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sq 
rt[b]*x^2])/(4*Sqrt[2]*a^(1/4)*b^(3/4)*(b*c - a*d)) - ((A*b - a*B)*Log[Sqr 
t[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(1/4)*b^(3/4 
)*(b*c - a*d)) + ((B*c - A*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sq 
rt[d]*x^2])/(4*Sqrt[2]*c^(1/4)*d^(3/4)*(b*c - a*d)) - ((B*c - A*d)*Log[Sqr 
t[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*c^(1/4)*d^(3/4 
)*(b*c - a*d))
 

Defintions of rubi rules used

rule 1054

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n 
_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a 
+ b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, p}, x] && IGtQ[n, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.64

method	result	size
default	\(-\frac {\left (A b -B a \right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (a d -c b \right ) b \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {\left (A d -B c \right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {c}{d}}}{x^{2}+\left (\frac {c}{d}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (a d -c b \right ) d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\)	\(240\)
risch	\(\text {Expression too large to display}\)	\(2906\)

Input:

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

Output:

-1/8*(A*b-B*a)/(a*d-b*c)/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))+1/8*(A*d-B*c)/(a*d-b* 
c)/d/(c/d)^(1/4)*2^(1/2)*(ln((x^2-(c/d)^(1/4)*x*2^(1/2)+(c/d)^(1/2))/(x^2+ 
(c/d)^(1/4)*x*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x+1)+2*ar 
ctan(2^(1/2)/(c/d)^(1/4)*x-1))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 17.30 (sec) , antiderivative size = 2507, normalized size of antiderivative = 6.69 \[ \int \frac {x^2 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b 
^4)/(a*b^7*c^4 - 4*a^2*b^6*c^3*d + 6*a^3*b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + a 
^5*b^3*d^4))^(1/4)*log(-(B^3*a^3 - 3*A*B^2*a^2*b + 3*A^2*B*a*b^2 - A^3*b^3 
)*x + (a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3)*(-(B^4 
*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a*b^7 
*c^4 - 4*a^2*b^6*c^3*d + 6*a^3*b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + a^5*b^3*d^4 
))^(3/4)) - 1/4*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a 
*b^3 + A^4*b^4)/(a*b^7*c^4 - 4*a^2*b^6*c^3*d + 6*a^3*b^5*c^2*d^2 - 4*a^4*b 
^4*c*d^3 + a^5*b^3*d^4))^(1/4)*log(-(B^3*a^3 - 3*A*B^2*a^2*b + 3*A^2*B*a*b 
^2 - A^3*b^3)*x - (a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2 
*d^3)*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4 
*b^4)/(a*b^7*c^4 - 4*a^2*b^6*c^3*d + 6*a^3*b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + 
 a^5*b^3*d^4))^(3/4)) + 1/4*I*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b 
^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a*b^7*c^4 - 4*a^2*b^6*c^3*d + 6*a^3*b^5*c^2 
*d^2 - 4*a^4*b^4*c*d^3 + a^5*b^3*d^4))^(1/4)*log(-(B^3*a^3 - 3*A*B^2*a^2*b 
 + 3*A^2*B*a*b^2 - A^3*b^3)*x - (I*a*b^5*c^3 - 3*I*a^2*b^4*c^2*d + 3*I*a^3 
*b^3*c*d^2 - I*a^4*b^2*d^3)*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 
 - 4*A^3*B*a*b^3 + A^4*b^4)/(a*b^7*c^4 - 4*a^2*b^6*c^3*d + 6*a^3*b^5*c^2*d 
^2 - 4*a^4*b^4*c*d^3 + a^5*b^3*d^4))^(3/4)) - 1/4*I*(-(B^4*a^4 - 4*A*B^3*a 
^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a*b^7*c^4 - 4*a^2*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.01 \[ \int \frac {x^2 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {{\left (B a - A b\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{8 \, {\left (b c - a d\right )}} + \frac {{\left (B c - A d\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {d} x^{2} + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {d} x^{2} - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{8 \, {\left (b c - a d\right )}} \] Input:

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

Output:

-1/8*(B*a - A*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)) + 2*sq 
rt(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)))/(b*c - a* 
d) + 1/8*(B*c - A*d)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*sqrt(d)*x + sqrt(2)* 
c^(1/4)*d^(1/4))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) + 
2*sqrt(2)*arctan(1/2*sqrt(2)*(2*sqrt(d)*x - sqrt(2)*c^(1/4)*d^(1/4))/sqrt( 
sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) - sqrt(2)*log(sqrt(d)*x^ 
2 + sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(c))/(c^(1/4)*d^(3/4)) + sqrt(2)*log(s 
qrt(d)*x^2 - sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(c))/(c^(1/4)*d^(3/4)))/(b*c 
- a*d)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (281) = 562\).

Time = 0.14 (sec) , antiderivative size = 597, normalized size of antiderivative = 1.59 \[ \int \frac {x^2 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 9.81 (sec) , antiderivative size = 24377, normalized size of antiderivative = 65.01 \[ \int \frac {x^2 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((x*(4*A^6*a*b^6*c^2*d^5 + 4*A^6*a^2*b^5*c*d^6 + 4*B^6*a^3*b^4*c^6*d 
+ 4*B^6*a^6*b*c^3*d^4 - 16*A*B^5*a^3*b^4*c^5*d^2 - 16*A*B^5*a^5*b^2*c^3*d^ 
4 - 16*A^3*B^3*a*b^6*c^5*d^2 - 16*A^3*B^3*a^5*b^2*c*d^6 + 24*A^4*B^2*a*b^6 
*c^4*d^3 + 24*A^4*B^2*a^4*b^3*c*d^6 - 16*A^5*B*a^2*b^5*c^2*d^5 + 32*A^2*B^ 
4*a^2*b^5*c^5*d^2 + 24*A^2*B^4*a^3*b^4*c^4*d^3 + 24*A^2*B^4*a^4*b^3*c^3*d^ 
4 + 32*A^2*B^4*a^5*b^2*c^2*d^5 - 48*A^3*B^3*a^2*b^5*c^4*d^3 - 32*A^3*B^3*a 
^3*b^4*c^3*d^4 - 48*A^3*B^3*a^4*b^3*c^2*d^5 + 36*A^4*B^2*a^2*b^5*c^3*d^4 + 
 36*A^4*B^2*a^3*b^4*c^2*d^5 - 8*A*B^5*a^2*b^5*c^6*d - 8*A*B^5*a^6*b*c^2*d^ 
5 + 4*A^2*B^4*a*b^6*c^6*d + 4*A^2*B^4*a^6*b*c*d^6 - 16*A^5*B*a*b^6*c^3*d^4 
 - 16*A^5*B*a^3*b^4*c*d^6) + (-(A^4*b^4 + B^4*a^4 + 6*A^2*B^2*a^2*b^2 - 4* 
A*B^3*a^3*b - 4*A^3*B*a*b^3)/(256*a*b^7*c^4 + 256*a^5*b^3*d^4 - 1024*a^2*b 
^6*c^3*d - 1024*a^4*b^4*c*d^3 + 1536*a^3*b^5*c^2*d^2))^(3/4)*(512*A^3*a^3* 
b^7*c^4*d^6 - 768*A^3*a^2*b^8*c^5*d^5 - x*(-(A^4*b^4 + B^4*a^4 + 6*A^2*B^2 
*a^2*b^2 - 4*A*B^3*a^3*b - 4*A^3*B*a*b^3)/(256*a*b^7*c^4 + 256*a^5*b^3*d^4 
 - 1024*a^2*b^6*c^3*d - 1024*a^4*b^4*c*d^3 + 1536*a^3*b^5*c^2*d^2))^(1/4)* 
(4096*A^2*a^2*b^9*c^6*d^5 - 7168*A^2*a^3*b^8*c^5*d^6 + 8192*A^2*a^4*b^7*c^ 
4*d^7 - 7168*A^2*a^5*b^6*c^3*d^8 + 4096*A^2*a^6*b^5*c^2*d^9 - 2048*B^2*a^3 
*b^8*c^7*d^4 + 8192*B^2*a^4*b^7*c^6*d^5 - 12288*B^2*a^5*b^6*c^5*d^6 + 8192 
*B^2*a^6*b^5*c^4*d^7 - 2048*B^2*a^7*b^4*c^3*d^8 - 1024*A^2*a*b^10*c^7*d^4 
- 1024*A^2*a^7*b^4*c*d^10 + 2048*A*B*a^2*b^9*c^7*d^4 - 6144*A*B*a^3*b^8...
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.30 \[ \int \frac {x^2 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {\sqrt {2}\, \left (-2 \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right )+2 \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right )+\mathrm {log}\left (-d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {c}+\sqrt {d}\, x^{2}\right )-\mathrm {log}\left (d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {c}+\sqrt {d}\, x^{2}\right )\right )}{8 d^{\frac {3}{4}} c^{\frac {1}{4}}} \] Input:

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

Output:

(d**(1/4)*c**(3/4)*sqrt(2)*( - 2*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt( 
d)*x)/(d**(1/4)*c**(1/4)*sqrt(2))) + 2*atan((d**(1/4)*c**(1/4)*sqrt(2) + 2 
*sqrt(d)*x)/(d**(1/4)*c**(1/4)*sqrt(2))) + log( - d**(1/4)*c**(1/4)*sqrt(2 
)*x + sqrt(c) + sqrt(d)*x**2) - log(d**(1/4)*c**(1/4)*sqrt(2)*x + sqrt(c) 
+ sqrt(d)*x**2)))/(8*c*d)