\(\int \frac {x^2}{(c+d x) (a+b x^4)} \, dx\) [84]

Optimal result

Integrand size = 20, antiderivative size = 333 \[ \int \frac {x^2}{(c+d x) \left (a+b x^4\right )} \, dx=\frac {\sqrt {a} d^3 \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {b} \left (b c^4+a d^4\right )}-\frac {c \left (\sqrt {b} c^2-\sqrt {a} d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^4+a d^4\right )}+\frac {c \left (\sqrt {b} c^2-\sqrt {a} d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^4+a d^4\right )}-\frac {c \left (\sqrt {b} c^2+\sqrt {a} d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a}+\sqrt {b} x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^4+a d^4\right )}+\frac {c^2 d \log (c+d x)}{b c^4+a d^4}-\frac {c^2 d \log \left (a+b x^4\right )}{4 \left (b c^4+a d^4\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.11 \[ \int \frac {x^2}{(c+d x) \left (a+b x^4\right )} \, dx=\frac {-2 \left (\sqrt {2} b^{3/4} c^3-\sqrt {2} \sqrt {a} \sqrt [4]{b} c d^2+2 a^{3/4} d^3\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \left (\sqrt {2} b^{3/4} c^3-\sqrt {2} \sqrt {a} \sqrt [4]{b} c d^2-2 a^{3/4} d^3\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt [4]{b} c \left (8 \sqrt [4]{a} \sqrt [4]{b} c d \log (c+d x)+\sqrt {2} \left (\sqrt {b} c^2+\sqrt {a} d^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )-\sqrt {2} \sqrt {b} c^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )-\sqrt {2} \sqrt {a} d^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )-2 \sqrt [4]{a} \sqrt [4]{b} c d \log \left (a+b x^4\right )\right )}{8 \sqrt [4]{a} \sqrt {b} \left (b c^4+a d^4\right )} \] Input:

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

Output:

(-2*(Sqrt[2]*b^(3/4)*c^3 - Sqrt[2]*Sqrt[a]*b^(1/4)*c*d^2 + 2*a^(3/4)*d^3)* 
ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(Sqrt[2]*b^(3/4)*c^3 - Sqrt[2] 
*Sqrt[a]*b^(1/4)*c*d^2 - 2*a^(3/4)*d^3)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^( 
1/4)] + b^(1/4)*c*(8*a^(1/4)*b^(1/4)*c*d*Log[c + d*x] + Sqrt[2]*(Sqrt[b]*c 
^2 + Sqrt[a]*d^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] - 
 Sqrt[2]*Sqrt[b]*c^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2 
] - Sqrt[2]*Sqrt[a]*d^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]* 
x^2] - 2*a^(1/4)*b^(1/4)*c*d*Log[a + b*x^4]))/(8*a^(1/4)*Sqrt[b]*(b*c^4 + 
a*d^4))
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {7276, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(Sqrt[a]*d^3*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(2*Sqrt[b]*(b*c^4 + a*d^4)) - 
(c*(Sqrt[b]*c^2 - Sqrt[a]*d^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2 
*Sqrt[2]*a^(1/4)*b^(1/4)*(b*c^4 + a*d^4)) + (c*(Sqrt[b]*c^2 - Sqrt[a]*d^2) 
*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(1/4)*b^(1/4)*(b*c^ 
4 + a*d^4)) + (c^2*d*Log[c + d*x])/(b*c^4 + a*d^4) + (c*(Sqrt[b]*c^2 + Sqr 
t[a]*d^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[ 
2]*a^(1/4)*b^(1/4)*(b*c^4 + a*d^4)) - (c*(Sqrt[b]*c^2 + Sqrt[a]*d^2)*Log[S 
qrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(1/4)*b^(1 
/4)*(b*c^4 + a*d^4)) - (c^2*d*Log[a + b*x^4])/(4*(b*c^4 + a*d^4))
 

Defintions of rubi rules used

rule 2009

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

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [C] (verified)

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

Time = 0.28 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.68

method	result	size
risch	\(\frac {c^{2} d \ln \left (d x +c \right )}{a \,d^{4}+b \,c^{4}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (a^{2} b^{2} d^{4}+b^{3} a \,c^{4}\right ) \textit {\_Z}^{4}+4 a \,b^{2} c^{2} d \,\textit {\_Z}^{3}+2 a b \,d^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (5 a^{2} b^{2} d^{6}-3 a \,b^{3} c^{4} d^{2}\right ) \textit {\_R}^{4}+10 \textit {\_R}^{3} a \,b^{2} c^{2} d^{3}+\left (9 a b \,d^{4}+b^{2} c^{4}\right ) \textit {\_R}^{2}-5 \textit {\_R} b \,c^{2} d +4 d^{2}\right ) x +\left (6 a^{2} b^{2} c \,d^{5}-2 a \,b^{3} c^{5} d \right ) \textit {\_R}^{4}+6 a \,b^{2} c^{3} d^{2} \textit {\_R}^{3}+8 a b c \,d^{3} \textit {\_R}^{2}-b \,c^{3} \textit {\_R} +4 c d \right )\right )}{4}\)	\(226\)
default	\(\frac {c^{2} d \ln \left (d x +c \right )}{a \,d^{4}+b \,c^{4}}+\frac {-\frac {c \,d^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {d^{3} a \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{2 \sqrt {a b}}+\frac {c^{3} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {c^{2} d \ln \left (b \,x^{4}+a \right )}{4}}{a \,d^{4}+b \,c^{4}}\)	\(281\)

Input:

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

Output:

c^2*d*ln(d*x+c)/(a*d^4+b*c^4)+1/4*sum(_R*ln(((5*a^2*b^2*d^6-3*a*b^3*c^4*d^ 
2)*_R^4+10*_R^3*a*b^2*c^2*d^3+(9*a*b*d^4+b^2*c^4)*_R^2-5*_R*b*c^2*d+4*d^2) 
*x+(6*a^2*b^2*c*d^5-2*a*b^3*c^5*d)*_R^4+6*a*b^2*c^3*d^2*_R^3+8*a*b*c*d^3*_ 
R^2-b*c^3*_R+4*c*d),_R=RootOf(1+(a^2*b^2*d^4+a*b^3*c^4)*_Z^4+4*a*b^2*c^2*d 
*_Z^3+2*a*b*d^2*_Z^2))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 21.93 (sec) , antiderivative size = 259898, normalized size of antiderivative = 780.47 \[ \int \frac {x^2}{(c+d x) \left (a+b x^4\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.05 \[ \int \frac {x^2}{(c+d x) \left (a+b x^4\right )} \, dx=\frac {c^{2} d \log \left (d x + c\right )}{b c^{4} + a d^{4}} - \frac {\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} c^{2} d + \sqrt {a} b^{\frac {3}{2}} c^{3} + a b c d^{2}\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} c^{2} d - \sqrt {a} b^{\frac {3}{2}} c^{3} - a b c d^{2}\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} c^{3} - \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} c d^{2} - 2 \, a^{\frac {3}{2}} b d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} c^{3} - \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} c d^{2} + 2 \, a^{\frac {3}{2}} b d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}}}{8 \, {\left (b c^{4} + a d^{4}\right )}} \] Input:

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

Output:

c^2*d*log(d*x + c)/(b*c^4 + a*d^4) - 1/8*(sqrt(2)*(sqrt(2)*a^(3/4)*b^(5/4) 
*c^2*d + sqrt(a)*b^(3/2)*c^3 + a*b*c*d^2)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4 
)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(5/4)) + sqrt(2)*(sqrt(2)*a^(3/4)*b^(5/4 
)*c^2*d - sqrt(a)*b^(3/2)*c^3 - a*b*c*d^2)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/ 
4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(5/4)) - 2*(sqrt(2)*a^(3/4)*b^(7/4)*c^3 
 - sqrt(2)*a^(5/4)*b^(5/4)*c*d^2 - 2*a^(3/2)*b*d^3)*arctan(1/2*sqrt(2)*(2* 
sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(a^(3/4)*sqrt( 
sqrt(a)*sqrt(b))*b^(5/4)) - 2*(sqrt(2)*a^(3/4)*b^(7/4)*c^3 - sqrt(2)*a^(5/ 
4)*b^(5/4)*c*d^2 + 2*a^(3/2)*b*d^3)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt 
(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(b)) 
*b^(5/4)))/(b*c^4 + a*d^4)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.22 \[ \int \frac {x^2}{(c+d x) \left (a+b x^4\right )} \, dx=\frac {c^{2} d^{2} \log \left ({\left | d x + c \right |}\right )}{b c^{4} d + a d^{5}} - \frac {c^{2} d \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, {\left (b c^{4} + a d^{4}\right )}} - \frac {{\left (\sqrt {2} a b^{2} d - \left (a b^{3}\right )^{\frac {3}{4}} c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a b^{3} c^{2} + \sqrt {2} \sqrt {a b} a b^{2} d^{2} - 2 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c d\right )}} + \frac {{\left (\sqrt {2} a b^{2} d + \left (a b^{3}\right )^{\frac {3}{4}} c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a b^{3} c^{2} + \sqrt {2} \sqrt {a b} a b^{2} d^{2} + 2 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c d\right )}} - \frac {{\left (\left (a b^{3}\right )^{\frac {1}{4}} a b c d^{2} + \left (a b^{3}\right )^{\frac {3}{4}} c^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{4 \, {\left (\sqrt {2} a b^{3} c^{4} + \sqrt {2} a^{2} b^{2} d^{4}\right )}} + \frac {{\left (\left (a b^{3}\right )^{\frac {1}{4}} a b c d^{2} + \left (a b^{3}\right )^{\frac {3}{4}} c^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{4 \, {\left (\sqrt {2} a b^{3} c^{4} + \sqrt {2} a^{2} b^{2} d^{4}\right )}} \] Input:

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

Output:

c^2*d^2*log(abs(d*x + c))/(b*c^4*d + a*d^5) - 1/4*c^2*d*log(abs(b*x^4 + a) 
)/(b*c^4 + a*d^4) - 1/2*(sqrt(2)*a*b^2*d - (a*b^3)^(3/4)*c)*arctan(1/2*sqr 
t(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(sqrt(2)*a*b^3*c^2 + sqrt(2) 
*sqrt(a*b)*a*b^2*d^2 - 2*(a*b^3)^(1/4)*a*b^2*c*d) + 1/2*(sqrt(2)*a*b^2*d + 
 (a*b^3)^(3/4)*c)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/ 
4))/(sqrt(2)*a*b^3*c^2 + sqrt(2)*sqrt(a*b)*a*b^2*d^2 + 2*(a*b^3)^(1/4)*a*b 
^2*c*d) - 1/4*((a*b^3)^(1/4)*a*b*c*d^2 + (a*b^3)^(3/4)*c^3)*log(x^2 + sqrt 
(2)*x*(a/b)^(1/4) + sqrt(a/b))/(sqrt(2)*a*b^3*c^4 + sqrt(2)*a^2*b^2*d^4) + 
 1/4*((a*b^3)^(1/4)*a*b*c*d^2 + (a*b^3)^(3/4)*c^3)*log(x^2 - sqrt(2)*x*(a/ 
b)^(1/4) + sqrt(a/b))/(sqrt(2)*a*b^3*c^4 + sqrt(2)*a^2*b^2*d^4)
 

Mupad [B] (verification not implemented)

Time = 23.71 (sec) , antiderivative size = 823, normalized size of antiderivative = 2.47 \[ \int \frac {x^2}{(c+d x) \left (a+b x^4\right )} \, dx=\left (\sum _{k=1}^4\ln \left (a\,b^2\,d\,\left (c\,d+d^2\,x-\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )\,b\,c^3+{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^2\,b^2\,c^4\,x\,4+{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^2\,a\,b\,d^4\,x\,36-{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^4\,a\,b^3\,c^5\,d\,128-\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )\,b\,c^2\,d\,x\,5+{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^3\,a\,b^2\,c^3\,d^2\,96+{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^4\,a^2\,b^2\,c\,d^5\,384+{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^4\,a^2\,b^2\,d^6\,x\,320+{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^2\,a\,b\,c\,d^3\,32+{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^3\,a\,b^2\,c^2\,d^3\,x\,160-{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^4\,a\,b^3\,c^4\,d^2\,x\,192\right )\right )\,\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )\right )+\frac {c^2\,d\,\ln \left (c+d\,x\right )}{b\,c^4+a\,d^4} \] Input:

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

Output:

symsum(log(a*b^2*d*(c*d + d^2*x - root(256*a^2*b^2*d^4*z^4 + 256*a*b^3*c^4 
*z^4 + 256*a*b^2*c^2*d*z^3 + 32*a*b*d^2*z^2 + 1, z, k)*b*c^3 + 4*root(256* 
a^2*b^2*d^4*z^4 + 256*a*b^3*c^4*z^4 + 256*a*b^2*c^2*d*z^3 + 32*a*b*d^2*z^2 
 + 1, z, k)^2*b^2*c^4*x + 36*root(256*a^2*b^2*d^4*z^4 + 256*a*b^3*c^4*z^4 
+ 256*a*b^2*c^2*d*z^3 + 32*a*b*d^2*z^2 + 1, z, k)^2*a*b*d^4*x - 128*root(2 
56*a^2*b^2*d^4*z^4 + 256*a*b^3*c^4*z^4 + 256*a*b^2*c^2*d*z^3 + 32*a*b*d^2* 
z^2 + 1, z, k)^4*a*b^3*c^5*d - 5*root(256*a^2*b^2*d^4*z^4 + 256*a*b^3*c^4* 
z^4 + 256*a*b^2*c^2*d*z^3 + 32*a*b*d^2*z^2 + 1, z, k)*b*c^2*d*x + 96*root( 
256*a^2*b^2*d^4*z^4 + 256*a*b^3*c^4*z^4 + 256*a*b^2*c^2*d*z^3 + 32*a*b*d^2 
*z^2 + 1, z, k)^3*a*b^2*c^3*d^2 + 384*root(256*a^2*b^2*d^4*z^4 + 256*a*b^3 
*c^4*z^4 + 256*a*b^2*c^2*d*z^3 + 32*a*b*d^2*z^2 + 1, z, k)^4*a^2*b^2*c*d^5 
 + 320*root(256*a^2*b^2*d^4*z^4 + 256*a*b^3*c^4*z^4 + 256*a*b^2*c^2*d*z^3 
+ 32*a*b*d^2*z^2 + 1, z, k)^4*a^2*b^2*d^6*x + 32*root(256*a^2*b^2*d^4*z^4 
+ 256*a*b^3*c^4*z^4 + 256*a*b^2*c^2*d*z^3 + 32*a*b*d^2*z^2 + 1, z, k)^2*a* 
b*c*d^3 + 160*root(256*a^2*b^2*d^4*z^4 + 256*a*b^3*c^4*z^4 + 256*a*b^2*c^2 
*d*z^3 + 32*a*b*d^2*z^2 + 1, z, k)^3*a*b^2*c^2*d^3*x - 192*root(256*a^2*b^ 
2*d^4*z^4 + 256*a*b^3*c^4*z^4 + 256*a*b^2*c^2*d*z^3 + 32*a*b*d^2*z^2 + 1, 
z, k)^4*a*b^3*c^4*d^2*x))*root(256*a^2*b^2*d^4*z^4 + 256*a*b^3*c^4*z^4 + 2 
56*a*b^2*c^2*d*z^3 + 32*a*b*d^2*z^2 + 1, z, k), k, 1, 4) + (c^2*d*log(c + 
d*x))/(a*d^4 + b*c^4)
 

Reduce [F]

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

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

Output:

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